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11534:
12534:{\displaystyle {\begin{aligned}{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial q_{i}}}=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial {q}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {q}_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial Q_{k}}{\partial q_{i}}}\right)\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right)-{\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial Q_{k}}{\partial q_{i}}}=0\\\end{aligned}}}
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16153:{\displaystyle {\begin{aligned}G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} &\,{\stackrel {\text{def}}{=}}\,d(F_{*}(\mathbf {q} )){\dot {\mathbf {q} }}=\left(\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i,j=1}^{n}{\dot {\mathbf {q} }}=\left(\sum _{j=1}^{n}{\dot {q}}_{j}\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i=1,\ldots ,n}^{T}\\&=\left(\sum _{k=1}^{n}dq_{k}\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i=1,\ldots ,n}^{T}=\left(\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i,k=1}^{n}d\mathbf {q} .\end{aligned}}}
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7029:{\displaystyle {\begin{aligned}\int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t&=\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)\,\mathrm {d} t\\&=\sum _{j=1}^{n}\left_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t.\end{aligned}}}
19378:{\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {\partial L}{\partial q_{i}}}=0\\&{\frac {\partial ^{2}L}{\partial q_{j}\partial {\dot {q}}_{i}}}{\frac {dq_{j}}{dt}}+{\frac {\partial ^{2}L}{\partial {\dot {q}}_{j}\partial {\dot {q}}_{i}}}{\frac {d{\dot {q}}_{j}}{dt}}+{\frac {\partial L}{\partial t}}-{\frac {\partial L}{\partial q_{i}}}=0\\&\sum _{j}W_{ij}(q,{\dot {q}},t){\ddot {q}}_{j}={\frac {\partial L}{\partial q_{i}}}-{\frac {\partial L}{\partial t}}-\sum _{j}{\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial q_{j}}}{\dot {q}}_{j},\\\end{aligned}}}
18022:{\displaystyle {\begin{aligned}{\frac {dL}{dt}}&={\dot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}+{\frac {\partial L}{\partial t}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right){\dot {\mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}-{\dot {L}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\mathbf {\dot {q}} -L\right)={\frac {dH}{dt}}\end{aligned}}}
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4233:
16571:{\displaystyle {\begin{aligned}dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&={\frac {\partial L}{\partial \mathbf {Q} }}d\mathbf {Q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}d{\dot {\mathbf {Q} }}+{\frac {\partial L}{\partial t}}dt\\&=\left({\frac {\partial L}{\partial \mathbf {Q} }}F_{*}(\mathbf {q} )+{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}G(\mathbf {q} ,{\dot {\mathbf {q} }})\right)d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}.\end{aligned}}}
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25336:. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in a
8793:
3933:{\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right),\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},}
4123:, the curves of extremal length between two points in space (these may end up being minimal, that is the shortest paths, but not necessarily). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces
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10069:{\displaystyle {\begin{aligned}S'&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L'(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt\\&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt+\int _{t_{\text{st}}}^{t_{\text{fin}}}{\frac {\mathrm {d} f(\mathbf {q} (t),t)}{\mathrm {d} t}}\,dt\\&=S+f(P_{\text{fin}},t_{\text{fin}})-f(P_{\text{st}},t_{\text{st}}),\end{aligned}}}
8580:{\displaystyle {\frac {\partial L'}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\mathbf {r} }}_{k}}}=0\quad \Rightarrow \quad {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}
25639:
2593:{\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right),\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)}
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see for a comparison of
Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in. Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an
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1445:, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The
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8262:{\displaystyle \int _{t_{1}}^{t_{2}}\delta L'\mathrm {d} t=\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}
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9056:{\displaystyle \underbrace {{\frac {\partial T}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\mathbf {r} }}_{k}}}} _{-\mathbf {F} _{k}}+\underbrace {-{\frac {\partial V}{\partial \mathbf {r} _{k}}}} _{\mathbf {N} _{k}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}
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21408:{\displaystyle {\begin{array}{rll}&x_{\mathrm {pend} }=x+\ell \sin \theta &\quad \Rightarrow \quad {\dot {x}}_{\mathrm {pend} }={\dot {x}}+\ell {\dot {\theta }}\cos \theta \\&y_{\mathrm {pend} }=-\ell \cos \theta &\quad \Rightarrow \quad {\dot {y}}_{\mathrm {pend} }=\ell {\dot {\theta }}\sin \theta .\end{array}}}
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22056:{\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta .\end{array}}}
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6345:{\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right),\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}},}
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25403:. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and
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in terms of the original
Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized
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This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the
Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example,
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for the position and speed of every object, which allows the writing down of a general form of lagrangian (total kinetic energy minus potential energy of the system) and summing this over all possible paths of motion of the particles yielded a formula for the 'action', which he minimized to give a
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Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an
5257:
This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the
Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized
987:
of the torus, motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations. Lagrangian mechanics adopts energy rather than force as its basic ingredient,, leading to more abstract equations capable of tackling more complex problems.
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25520:. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then the
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It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the
Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle,
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Instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is one of several
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These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example,
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It is known that the
Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.
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5766:. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.
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it must be replaced as a whole by a function consistent with special relativity (scalar under
Lorentz transformations) or general relativity (4-scalar). Where a magnetic field is present, the expression for the potential energy needs restating. And for dissipative forces (e.g.,
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13765:{\displaystyle T={\frac {m}{2}}v^{2}={\frac {m}{2}}\sum _{i,j}\left({\frac {\partial {\vec {r}}}{\partial q_{i}}}{\dot {q}}_{i}\right)\cdot \left({\frac {\partial {\vec {r}}}{\partial q_{j}}}{\dot {q}}_{j}\right)={\frac {m}{2}}\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}
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If one arrives at this equation using
Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates
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individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.
4461:, i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it.
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At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time.
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for details. However, from the physical point-of-view there is an obstacle to include time derivatives higher than the first order, which is implied by
Ostrogradsky's construction of a canonical formalism for nondegenerate higher derivative Lagrangians, see
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11401:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {Q_{i}}}{\partial q_{j}}}\right)=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{k}\partial q_{j}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial t\partial q_{j}}}(\mathbf {q} ,t)}
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978:
Newton's laws and the concept of forces are the usual starting point for teaching about mechanical systems. This method works well for many problems, but for others the approach is nightmarishly complicated. For example, in calculation of the motion of a
20247:{\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial V}{\partial x}},\quad {\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}},\quad {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)=m{\ddot {x}},}
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7584:{\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}
25595:. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.
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is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy
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constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations.
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a potential can be found as shown. This may not always be possible for non-conservative forces, and
Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations.
20921:
14038:{\displaystyle \sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=\sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial T}{\partial {\dot {q}}_{k}}}={\frac {m}{2}}\left(2\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\right)=2T}
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2288:{\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}
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11154:{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial q_{j}}}=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{j}\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial q_{j}\partial t}}(\mathbf {q} ,t)}
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The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.
18776:
This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction,
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When the constraint equations are non-integrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to
5441:{\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}.}
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5248:{\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}.}
3150:{\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}},\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}.}
16746:{\displaystyle dL(\mathbf {q} ,{\dot {\mathbf {q} }},t)={\frac {\partial L}{\partial \mathbf {q} }}d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}dt.}
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enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated.
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If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either
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There is no mathematical reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives, see
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generalized set of equations. This summed quantity is minimized along the path that the particle actually takes. This choice eliminates the need for the constraint force to enter into the resultant generalized
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The constraint forces can be complicated, since they generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations.
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should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by
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8756:{\displaystyle {\frac {\partial L'}{\partial \lambda _{i}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\lambda }}_{i}}}=0\quad \Rightarrow \quad f_{i}(\mathbf {r} _{k},t)=0.}
4785:
25854:{\displaystyle {\frac {\delta }{\delta \mathbf {r} _{k}}}\equiv {\frac {\partial }{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}}
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The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate Lagrangian
22736:. We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the
3562:
Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of
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and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are
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are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like
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Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero:
4200:
24260:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}+{\frac {\partial D}{\partial {\dot {q}}_{j}}}=0.}
19741:
If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates.
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The constraint forces can either be eliminated from the equations of motion, so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.
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the particle accelerates due to forces acting on it and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D
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is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates.
14912:{\displaystyle \left\{\sum _{i=1}^{n}{\dot {Q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial Q_{i}}}\right|_{F(\mathbf {q} )}\right)\ {\biggl |}\ {\dot {Q}}_{i}\in \mathbb {R} \right\},}
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3529:
it. Where the mass is varying, the equation needs to be generalised to take the time derivative of the momentum. In three spatial dimensions, this is a system of three coupled second-order
14744:{\displaystyle \left\{\sum _{i=1}^{n}{\dot {q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial q_{i}}}\right|_{\mathbf {q} }\right)\ {\biggl |}\ {\dot {q}}_{i}\in \mathbb {R} \right\}}
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coupled second-order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for.
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22970:{\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}}
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12924:
7927:{\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t).}
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of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any external influences. For
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17323:
17144:
13219:{\displaystyle p_{z}={\frac {\partial L}{\partial {\dot {z}}}},\quad p_{s}={\frac {\partial L}{\partial {\dot {s}}}},\quad p_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}},}
6360:, but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An
26101:
Fraser, Craig. "J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics". Archive for History of Exact Sciences, vol. 28, no. 3, 1983, pp. 197–241. JSTOR,
14589:
23667:
13422:
This quantity will be equivalent to energy if the generalized coordinates are natural coordinates, i.e., they have no explicit time dependence when expressing position vector:
28730:
22623:
19655:
19579:
2953:
of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the
106:
18054:
15117:
18692:
15078:
12567:
24290:
are more complex and have additional terms in their Lagrangians. Not only can the fields form non conservative potentials, these potentials can also be velocity dependent.
17084:
15142:
28735:
27976:
18370:{\displaystyle V(\alpha \mathbf {r} _{1},\alpha \mathbf {r} _{2},\ldots ,\alpha \mathbf {r} _{N})=\alpha ^{N}V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}
1324:{\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}.}
25203:
are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity in
19608:
19532:
7367:
of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation. This will not be given here.
7296:
23799:." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the
16855:
25444:
21455:
21155:
18530:
16758:
25870:
Here the virtual displacements are assumed reversible, it is possible for some systems to have non-reversible virtual displacements that violate this principle, see
24480:{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\phi -{\frac {\partial \mathbf {A} }{\partial t}},\quad \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}
19390:
10814:{\displaystyle {\dot {Q_{i}}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial Q_{k}}{\partial t}}(\mathbf {q} ,t)}
4471:
1639:
1482:
18172:
17462:{\displaystyle {\dot {\mathbf {Q} }}{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}={\dot {\mathbf {q} }}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}
1140:
coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written
23815:
inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal
25689:
23442:
17140:
13294:
22209:
21435:
21135:
21115:
19503:
17544:
17520:
17487:
1609:
1589:
18511:
Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The length
4892:
There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in an
3172:
depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so
13265:
is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.
28354:
1350:
862:
25527:
28723:
28359:
25074:
The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.
19692:
923:) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.
25217:
is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.
23784:
and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of
12701:
8292:
are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement
25314:, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.
12907:{\displaystyle {\dot {p}}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0}
9426:{\displaystyle L'(\mathbf {q} ,{\dot {\mathbf {q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t)+{\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}},}
27167:
20326:
10329:{\displaystyle L'(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} (\mathbf {Q} ,t),{\dot {\mathbf {q} }}(\mathbf {Q} ,{\dot {\mathbf {Q} }},t),t),}
2936:{\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)={\big (}x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t{\big )}.}
1873:), another function must be introduced alongside Lagrangian often referred to as a "Rayleigh dissipation function" to account for the loss of energy.
25886:
24884:{\displaystyle L\rightarrow L+q\left({\dot {\mathbf {r} }}\cdot {\boldsymbol {\nabla }}+{\frac {\partial }{\partial t}}\right)f=L+q{\frac {df}{dt}},}
20257:
15147:
14496:
7108:
27693:
3359:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}}
27969:
1487:
7336:
the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. These ideas in turn lead to the
28083:
27913:
27868:
25658:
1003:
For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a
7949:, so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives
29031:
28718:
17242:
7181:
5930:{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}},}
28801:
23843:(i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom.
21604:{\displaystyle T={\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left({\dot {x}}_{\mathrm {pend} }^{2}+{\dot {y}}_{\mathrm {pend} }^{2}\right)}
20052:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\partial L}{\partial x}},}
17054:{\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}
5753:{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}}
4164:
2639:
772:
15037:{\displaystyle \textstyle F_{*}(\mathbf {q} )=\left(\left.{\frac {\partial F_{i}}{\partial q_{j}}}\right|_{\mathbf {q} }\right)_{i,j=1}^{n}}
4117:
it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are
28536:
13226:
are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this case
10448:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {Q}}_{i}}}={\frac {\partial L'}{\partial Q_{i}}}.}
9617:{\textstyle {\frac {\partial f(\mathbf {q} ,t)}{\partial t}}+\sum _{i}{\frac {\partial f(\mathbf {q} ,t)}{\partial q_{i}}}{\dot {q}}_{i}.}
6060:{\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0.}
29120:
27962:
5453:
361:
27840:
9436:
4457:, are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system
28581:
27834:
480:
20827:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})-mr^{2}\sin \theta \cos \theta \,{\dot {\varphi }}^{2}=0,}
11521:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial Q_{i}}{\partial q_{k}}}\right)={\frac {\partial {\dot {Q}}_{i}}{\partial q_{k}}}}
22670:
7308:
Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the
1000:. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.
26646:
25419:
In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In
15329:{\displaystyle d{\dot {\mathbf {Q} }}=G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} +F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }},}
13425:
28088:
27613:
25683:
4081:
3949:
453:
7090:
are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of
28497:
20927:
coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum
20698:{\displaystyle m{\ddot {r}}-mr({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\varphi }}^{2})+{\frac {\partial V}{\partial r}}=0,}
19943:{\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-V(x,y,z).}
4066:{\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}}
3484:
1815:
If there is some external field or external driving force changing with time, the potential changes with time, so most generally
23109:
12914:
and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case of
3175:
2607:
with respect to the indicated variables (not a derivative with respect to the entire vector). Each overdot is a shorthand for a
28999:
27918:
4146:, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense.
38:
25226:
28098:
27781:
27747:
27726:
27623:
27453:
27361:
27225:
27070:
27018:
26824:
26593:
26366:
23811:), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as
21614:
20561:{\displaystyle L={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2})-V(r).}
7269:], [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is
24009:
are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If
23531:
reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity
2708:
In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of
28664:
27903:
27795:
18712:
14291:{\displaystyle a_{ij}={\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}}
25589:
18696:
If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the system
14197:{\displaystyle H=\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-L=2T-(T-V)=T+V=E}
10647:{\displaystyle d{Q_{i}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}d{q_{k}}+{\frac {\partial Q_{k}}{\partial t}}dt,}
28118:
28113:
27861:
23401:{\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell .}
435:
14424:
4430:{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}
24997:{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} .}
18502:{\displaystyle {\frac {\alpha ^{2}}{\beta ^{2}}}=\alpha ^{N}\quad \Rightarrow \quad \beta =\alpha ^{1-{\frac {N}{2}}}.}
4883:{\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}.}
24749:{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +{\boldsymbol {\nabla }}f,\quad \phi \rightarrow \phi -{\dot {f}},}
21026:{\displaystyle p_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=mr^{2}\sin ^{2}\theta {\dot {\varphi }},}
12668:{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q}}_{i}}}-{\frac {{\partial }L}{\partial Q_{i}}}=0.}
28794:
27898:
27703:
27682:
27663:
27644:
27424:
27405:
27382:
27340:
27319:
27298:
27275:
27254:
27204:
27181:
27131:
27112:
27091:
27044:
26234:
25323:
23823:
forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently."
1865:
765:
18648:
812:(also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer
28531:
28215:
22184:{\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=(M+m){\dot {x}}+m\ell {\dot {\theta }}\cos \theta ,}
20372:
13415:{\displaystyle H={\biggl (}\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}{\biggr )}-L.}
10920:{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial {\dot {q}}_{k}}}={\frac {\partial Q_{i}}{\partial q_{k}}}}
3465:
For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of
727:
28278:
28038:
25208:
23860:
10467:
7272:
3441:. Both equations are linear in the Lagrangian, but generally are nonlinear coupled equations in the coordinates.
101:
23152:
Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates
9165:
thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers.
9158:{\displaystyle \mathbf {C} _{k}=\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}},}
28947:
28205:
27854:
27153:
25365:
16755:
It was mentioned earlier that Lagrangians do not depend on the choice of configuration space coordinates, i.e.
5021:{\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}},}
4778:
by converting to virtual displacements in the generalized coordinates. These are related in the same form as a
3530:
3426:
1426:
356:
96:
24582:{\displaystyle L={\tfrac {1}{2}}m{\dot {\mathbf {r} }}^{2}+q\,{\dot {\mathbf {r} }}\cdot \mathbf {A} -q\phi ,}
22494:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left+m\ell ({\dot {x}}{\dot {\theta }}+g)\sin \theta =0;}
22321:{\displaystyle (M+m){\ddot {x}}+m\ell {\ddot {\theta }}\cos \theta -m\ell {\dot {\theta }}^{2}\sin \theta =0.}
17232:{\displaystyle \textstyle \sum \limits _{i=1}^{n}{\frac {\partial L}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}}
29151:
29141:
28210:
28103:
25714:
23864:
21137:
be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle
18197:
of the coordinates and independent of time, and all position vectors are scaled by the same nonzero constant
14311:
5783:
2950:
838:
26057:{\displaystyle C_{k\,x}\delta x_{k}\neq 0,\quad C_{k\,y}\delta y_{k}\neq 0,\quad C_{k\,z}\delta z_{k}\neq 0}
23661:
23292:{\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu \left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r),}
22636:
17088:
1452:
28787:
28764:
28108:
27058:
18145:
17328:
17297:
4286:
943:
in this case is gravity. Notice the initial position of the bead on the wire can lead to different motions.
758:
745:
506:
429:
351:
194:
22580:{\displaystyle {\ddot {\theta }}+{\frac {\ddot {x}}{\ell }}\cos \theta +{\frac {g}{\ell }}\sin \theta =0.}
9272:, then two Lagrangians describing the same system can differ by the "total time derivative" of a function
7329:
6410:, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,
2093:. In either case, the Lagrangian always has implicit time dependence through the generalized coordinates.
1988:
depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics
951:
Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation
29084:
28150:
25871:
25289:
24598:
24105:{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}}
23808:
19953:
14556:
3239:
2011:
or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian
983:
rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like the
809:
425:
226:
3533:
to solve, since there are three components in this vector equation. The solution is the position vector
204:
29146:
28915:
28888:
28768:
28232:
23632:{\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}.}
14592:
4643:{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}
1997:
904:
645:
534:
460:
320:
253:
27565:
Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of post-Newtonian radiation and reaction".
22593:
19613:
19537:
4920:, and can without loss of generality be converted into the generalized analogues by the definition of
2698:
or total derivatives need to be used to relate the velocity component to the corresponding coordinate
2642:, and derivatives of the Lagrangian are taken with respect to these separately according to the usual
28628:
28561:
28556:
28519:
28050:
28023:
27993:
25388:
25384:
25369:
24333:
15083:
7321:
4899:
The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces
4102:, each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates.
1773:
399:
27518:
Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action".
15047:
12546:
29021:
28485:
28480:
28389:
28349:
28065:
25704:
25294:
20916:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}\sin ^{2}\theta \,{\dot {\varphi }})=0.}
17064:
15122:
14421:
Besides this result, the proof below shows that, under such change of coordinates, the derivatives
10139:
7360:
7247:
3438:
1985:
675:
519:
27828:
27634:
26426:
20407:
is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), and
16948:{\displaystyle dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=dL(\mathbf {q} ,{\dot {\mathbf {q} }},t),}
28674:
25668:
25192:
24597:
that the Lagrangian is the kinetic energy minus the potential energy is incorrect. Combined with
23993:{\displaystyle D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k},}
19584:
19508:
17523:
7345:
7313:
7101:
3564:
2730:
2611:. This procedure does increase the number of equations to solve compared to Newton's laws, from 3
1129:
625:
389:
28310:
26346:
16845:{\displaystyle L(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t).}
14414:{\displaystyle E(\mathbf {q} ,{\dot {\mathbf {q} }},t)=E(\mathbf {Q} ,{\dot {\mathbf {Q} }},t).}
13033:{\displaystyle L(r,\theta ,{\dot {s}},{\dot {z}},{\dot {r}},{\dot {\theta }},{\dot {\phi }},t),}
818:
816:
in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus,
665:
29105:
28967:
28679:
28669:
28643:
28465:
28460:
28455:
28450:
28408:
28371:
28317:
28043:
28033:
28011:
27908:
26282:
25735:
25420:
24671:{\displaystyle m{\ddot {\mathbf {r} }}=q\mathbf {E} +q{\dot {\mathbf {r} }}\times \mathbf {B} }
19478:{\displaystyle W_{ij}={\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial {\dot {q}}_{j}}}}
7353:
7309:
6078:
3478:
3370:
2766:
992:
670:
487:
27372:
27351:
27330:
27309:
27286:
27102:
27081:
25199:
to the original variables. For example, given a set of generalized coordinates, the variables
25184:{\displaystyle H=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L}
4244:
4232:
2630:
coupled second-order differential equations in the position coordinates and multipliers, plus
29110:
29079:
28942:
28863:
28596:
28576:
28514:
28443:
28222:
28200:
28175:
28075:
27877:
27393:
27265:
27242:
27192:
27054:
26806:
25673:
25663:
25653:
25404:
25200:
25196:
25083:
24307:
21440:
21140:
19665:
The following examples apply Lagrange's equations of the second kind to mechanical problems.
18188:
13297:
7337:
4766:. Since the generalized coordinates are independent, we can avoid the complications with the
3588:
2643:
1877:
1063:
813:
789:
680:
655:
341:
159:
7606:, but the constraints are still assumed to be holonomic. As always the end points are fixed
4465:
is the work done along a virtual displacement for any force (constraint or non-constraint).
3400:
of the system. The number of equations has decreased compared to Newtonian mechanics, from 3
1614:
1449:
Lagrangian for a system of particles in the absence of an electromagnetic field is given by
29069:
28893:
28853:
28754:
28648:
28633:
28548:
28475:
28433:
28423:
28183:
28137:
27584:
27537:
27482:
26658:
25709:
25604:
25337:
25015:
24681:
23752:{\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}.}
18194:
18051:
does not vary with time evolution of particle, indeed, an integral of motion, meaning that
12915:
11531:
Starting from Euler Lagrange equations in initial set of generalized coordinates, we have:
10340:
for partial differentiation, Lagrange's equations are invariant under this transformation;
6361:
4722:
4441:
4282:
1772:
For those non-conservative forces which can be derived from an appropriate potential (e.g.
700:
660:
568:
564:
556:
546:
329:
85:
27658:. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing.
26191:
26082:
The Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000.
23639:
which is the equation of motion for a one-dimensional problem in which a particle of mass
12569:
is invertible, it follows that the form of the Euler-Lagrange equation is invariant i.e.,
695:
8:
29044:
28903:
28898:
28883:
28858:
28835:
28811:
28623:
28492:
28403:
28290:
28261:
27985:
27923:
27893:
27888:
26584:
26167:
24771:
is any scalar function of space and time, the aforementioned Lagrangian transforms like:
10337:
7679:
3959:
3397:
2307:
1332:
997:
916:
912:
805:
475:
416:
394:
139:
134:
129:
29:
27588:
27541:
27486:
26811:. World Scientific Lecture Notes in Physics. Vol. 81. WORLD SCIENTIFIC. p. 7.
26662:
25423:, the physical system is not a set of discrete particles, but rather a continuous field
18127:{\displaystyle H(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)={\text{constant of time}}.}
17122:
13276:
13072:
are all absent in the Lagrangian even though their velocities are not. Then the momenta
29100:
28977:
28972:
28925:
28760:
28413:
28399:
28381:
28364:
28344:
28327:
28237:
28060:
27938:
27933:
27600:
27574:
27553:
27527:
27506:
27472:
25686:, the general topic of finding a Lagrangian for a system given the equations of motion.
25678:
25624:
25620:
25612:
25608:
25439:
25333:
25329:
25311:
25305:
25214:
25019:
24895:
23042:
22737:
22194:
21420:
21120:
21100:
19678:
19488:
17547:
17529:
17505:
17472:
7364:
6353:
4779:
4294:
4220:
4143:
3470:
3422:
2601:
1740:
1594:
1574:
605:
346:
221:
189:
149:
27636:
Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics
25027:
is not gauge invariant, and therefore not a measurable physical quantity; However, if
21669:
2054:. If neither the potential nor the kinetic energy depend on time, then the Lagrangian
29115:
29064:
29009:
28989:
28875:
28586:
28322:
28300:
28227:
28188:
28165:
27954:
27811:
27790:
27777:
27743:
27722:
27699:
27678:
27659:
27640:
27619:
27604:
27557:
27498:
27449:
27420:
27401:
27378:
27357:
27336:
27315:
27294:
27271:
27250:
27221:
27200:
27177:
27171:
27163:
27149:
27141:
27127:
27108:
27087:
27066:
27040:
27014:
26820:
26674:
26589:
26362:
26302:
26230:
25699:
25392:
25349:
25204:
25057:
given here is the conserved momentum, while the measurable physical kinetic momentum
25011:
24298:
23800:
22728:
7302:
7099:
must also be zero. Then we obtain the equations of motion. This can be summarized by
4921:
4290:
3542:
1744:
1163:
of each particle is how fast the particle moves along its path of motion, and is the
1133:
615:
572:
529:
524:
465:
241:
231:
124:
27510:
23527:
This equation is identical to the radial equation obtained using Newton's laws in a
22069:
is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is
21165:
18134:
Hence, if the chosen coordinates were natural coordinates, the energy is conserved.
8765:
For the case of a conservative force given by the gradient of some potential energy
5786:(only some are shown). The path taken by the system (red) has a stationary action (δ
29054:
28994:
28920:
28504:
28295:
28283:
28266:
28244:
28193:
28145:
28006:
27715:
27592:
27545:
27494:
27490:
27441:
26812:
26666:
26354:
26294:
26163:
25694:
25644:
25353:
25007:
24905:
24590:
24311:
24294:
22682:
21157:
from the vertical. The coordinates and velocity components of the pendulum bob are
19686:
12684:
7341:
7325:
7266:
7175:
6069:
However, the Euler–Lagrange equations can only account for non-conservative forces
5637:
are independent and nonzero, the coefficients can be equated to zero, resulting in
4135:
3430:
2954:
1736:
1336:
984:
897:
847:
710:
690:
635:
630:
576:
551:
406:
264:
209:
184:
23795:. Their presence or absence depends, not upon the particular problem at hand, but
4153:
acting on the particle, which in turn requires the resultant non-constraint force
3454:
935:
Bead constrained to move on a frictionless wire. The wire exerts a reaction force
29074:
28984:
28930:
28830:
28610:
28566:
28393:
28337:
28332:
28271:
28093:
28018:
27737:
27215:
27199:(Reprint of University of Toronto 1970 4th ed.). Courier Dover. p. 43.
26999:
26988:
26319:
Bersani AM, Caressa P. Lagrangian descriptions of dissipative systems: a review.
26298:
25521:
25380:
25361:
24279:
22626:
4139:
2608:
1164:
1036:
705:
650:
600:
595:
514:
26358:
25505:{\displaystyle {\mathcal {L}}(\phi ,\nabla \phi ,{\dot {\phi }},\mathbf {r} ,t)}
19657:
but also, the Hamiltonian equations of motions will not take the standard form.
18601:{\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-{\frac {N}{2}}}.}
8600:
Euler-Lagrange equations for the new Lagrangian return the constraint equations
7077:, the terms not integrated are zero. If in addition the entire time integral of
4665:
might be connected by a constraint equation, which prevents us from setting the
29004:
28959:
28937:
28825:
28254:
27943:
27596:
27032:
25592:
25512:
defined in terms of the field and its space and time derivatives at a location
24377:
24355:
23149:
which states the center of mass moves in a straight line at constant velocity.
22761:
22630:
15209:{\displaystyle {\dot {\mathbf {Q} }}=F_{*}(\mathbf {q} ){\dot {\mathbf {q} }}.}
7349:
7317:
4538:{\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0,}
1564:
1004:
893:
732:
640:
541:
258:
27549:
27445:
24489:
The Lagrangian of a massive charged test particle in an electromagnetic field
24310:, is the prototypical example of a velocity-dependent potential. The electric
7724:, and adding the results to the original Lagrangian, gives the new Lagrangian
1683:
is the energy of the system's motion and is a function only of the velocities
29135:
29014:
28571:
28526:
28509:
28155:
28028:
26678:
26306:
25861:
is used. Throughout this article only partial and total derivatives are used.
25373:
25357:
24602:
24594:
22686:
21081:
Sketch of the situation with definition of the coordinates (click to enlarge)
21077:
19581:. If the matrix is non-invertible, it would not be possible to represent all
17578:
satisfies the Euler–Lagrange equations). Taking the total time-derivative of
17499:
7682:
can be used to include the constraints. Multiplying each constraint equation
4298:
4086:
3469:
particles, all of these equations apply to each particle in the system). The
2765:, by expressing each position vector, and hence the position coordinates, as
620:
447:
28779:
2638:
In the Lagrangian, the position coordinates and velocity components are all
29039:
28591:
28470:
28249:
28160:
28055:
27502:
27028:
26155:
25340:
way, it may be possible if a particular frame of reference is singled out.
23791:"Since such quantities are not true physical forces, they are often called
23761:
Of course, if one remains entirely within the one-dimensional formulation,
23518:{\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}.}
23305:
is a cyclic coordinate with the corresponding conserved (angular) momentum
23032:
21061:
The Lagrangian in two-dimensional polar coordinates is recovered by fixing
7333:
4462:
3458:
1990:
can only be applied to systems whose constraints, if any, are all holonomic
1864:
is applicable to many important classes of system, but not everywhere. For
1344:
920:
685:
610:
299:
179:
27463:
Galley, Chad R. (2013). "Classical Mechanics of Nonconservative Systems".
27314:(Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover. p. 156.
5450:
Now D'Alembert's principle is in the generalized coordinates as required,
931:
28702:
28618:
27691:
23846:
In a more general formulation, the forces could be both conservative and
23840:
5774:
1671:
27846:
25603:
The action principle, and the Lagrangian formalism, are tied closely to
17287:{\displaystyle \textstyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}}
4305:
particles the virtual work, i.e. the work along a virtual displacement,
29059:
29049:
28638:
26102:
10134:, if we change these variables to a new set of generalized coordinates
9634:′ produce the same equations of motion since the corresponding actions
2695:
1410:{\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}},}
470:
144:
26670:
25577:{\displaystyle L(t)=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {r} }
20399:
as commonly used in physics (ISO 80000-2:2019 convention), where
3373:, which can also be used in mechanics. Substituting in the Lagrangian
1747:), it is a function of the position vectors of the particles only, so
947:
28910:
28686:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
28001:
27435:
26283:"Synchronous Lagrangian variational principles in General Relativity"
25396:
23847:
19732:{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}V(\mathbf {r} ).}
18700:
into the sum of non-interacting Lagrangians, plus another Lagrangian
1568:
492:
17294:
are contravariant 1-tensors. Applying both sides of the equation to
12760:{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}
8271:
The introduced multipliers can be found so that the coefficients of
5806:
that depends on positions and velocities. If the generalized forces
5769:
5258:
coordinates and velocities can be found to give the desired result:
785:
24287:
24283:
19952:
The equations of motion for the particle are found by applying the
19682:
9484:{\textstyle {\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}}}
7174:
The time integral of the Lagrangian is another quantity called the
4119:
1870:
1441:
in the system. The central quantity of Lagrangian mechanics is the
1160:
411:
294:
269:
27579:
27532:
27477:
25438:
defined over a region of 3D space. Associated with the field is a
25077:
20411:
is the azimuthal angle, the Lagrangian for a central potential is
4721:
are linked together by a constraint equation, so are those of the
4673:
26816:
26808:
Classical and Quantum Dynamics of Constrained Hamiltonian Systems
26226:
25400:
20364:{\displaystyle m{\ddot {\mathbf {r} }}=-{\boldsymbol {\nabla }}V}
797:
384:
237:
154:
27774:
Variational methods with applications in science and engineering
25282:
23830:
19505:
is non-singular, the above equations can be solved to represent
12784:, it follows immediately from the Euler–Lagrange equations that
7265:
and returns a scalar value. Its dimensions are the same as [
5937:
equating to Lagrange's equations and defining the Lagrangian as
1876:
One or more of the particles may each be subject to one or more
28741:
European Community on Computational Methods in Applied Sciences
28305:
28123:
26427:
https://scholar.harvard.edu/files/david-morin/files/cmchap6.pdf
25929:{\displaystyle \mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0}
25408:
20568:
So, in spherical coordinates, the Euler–Lagrange equations are
20308:{\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}},}
18418:). The entire Lagrangian has been scaled by the same factor if
14546:{\displaystyle d\mathbf {Q} =F_{*}(\mathbf {q} )d\mathbf {q} ,}
7165:{\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0.}
4301:
in 1743 to solve dynamical problems. The principle asserts for
1952:
each of which could apply to any of the particles. If particle
1438:
443:
289:
199:
26645:
Hanc, Jozef; Taylor, Edwin F.; Tuleja, Slavomir (2005-07-01).
14301:
13480:{\displaystyle \mathbf {r} =\mathbf {r} (q_{1},\cdots ,q_{n})}
7654:. What cannot be done is to simply equate the coefficients of
7363:
if the constraint equations can be put into a certain form, a
5790:= 0) under small changes in the configuration of the system (δ
23041:
the potential of the radial force, which depends only on the
20323:
coordinates. Collecting the equations in vector form we find
9178:
The Lagrangian of a given system is not unique. A Lagrangian
1556:{\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}
1340:
980:
939:
on the bead to keep it on the wire. The non-constraint force
825:
Lagrangian mechanics describes a mechanical system as a pair
279:
274:
216:
28736:
International Council for Industrial and Applied Mathematics
27437:
Introduction to the classical theory of particles and fields
7593:
Hamilton's principle is still valid even if the coordinates
7251:; it takes in the Lagrangian function for all times between
1880:; such a constraint is described by an equation of the form
1670:
is the magnitude squared of its velocity, equivalent to the
991:
Particularly, Lagrange's approach was to set up independent
27329:
Zak, Michail; Zbilut, Joseph P.; Meyers, Ronald E. (1997).
24275:
14958:
14812:
14656:
12678:
5798:
For a non-conservative force which depends on velocity, it
4271:
confined to a curve. The resultant non-constraint force is
4216:
2726:. We can transform each position vector to a common set of
1012:
284:
247:
27293:. Vol. 1 (2nd ed.). Courier Dover. p. 160.
18786:
tends to zero reducing to the non-interacting case above.
14298:
is a symmetric matrix that is defined for the derivation.
13300:
of the corresponding mechanical system is, by definition,
11528:. The two derived relations can be employed in the proof.
3228:{\displaystyle T=T(\mathbf {q} ,{\dot {\mathbf {q} }},t).}
1136:
to uniquely define the location of a point, so there are 3
970:
is the tension in the rod. Again the non-constraint force
26479:
23850:. If an appropriate transformation can be found from the
10125:
7936:
The Lagrange multipliers are arbitrary functions of time
6364:
with respect to time can transfer the time derivative of
4149:
However, we still need to know the total resultant force
3444:
77:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
27400:(3rd ed.). Cambridge University Press. p. 48.
26281:
Cremaschini, Claudio; Tessarotto, Massimo (2015-06-30).
25082:
A closely related formulation of classical mechanics is
17582:
along this extremal and using the EL equations leads to
7238:{\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,}
27615:
Engineering Dynamics: From the Lagrangian to Simulation
26690:
26688:
26421:
Morin, D. (2007). Chapter 6: The Lagrangian Method. In
26345:
Pfeiffer, Friedrich (2008), Pfeiffer, Friedrich (ed.),
25690:
Lagrangian and Eulerian specification of the flow field
25227:
Position and momentum space § Lagrangian mechanics
23656:
and a second outward force, called in this context the
18793:
18519:
in the original trajectory corresponds to a new length
12918:. Such coordinates are called "cyclic" or "ignorable".
7370:
4195:{\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} .}
27984:
27692:
M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006).
26966:
26964:
25033:
is cyclic (i.e. Lagrangian is independent of position
24503:
17246:
17148:
14926:
9497:
9439:
4138:, the above form of Newton's law also carries over to
4090:
of the curvilinear coordinate system. All the indices
850:
28355:
Numerical methods for ordinary differential equations
27193:"II §5 Auxiliary conditions: the Lagrangian λ-method"
27126:(5th ed.). Imperial College Press. p. 236.
26728:
26726:
26280:
26264:
26262:
26260:
26258:
25946:
25889:
25744:
25530:
25447:
25092:
24920:
24777:
24690:
24611:
24495:
24404:
24118:
24019:
23877:
23670:
23548:
23445:
23311:
23183:
23112:
22816:
22639:
22596:
22507:
22340:
22217:
22197:
22191:
and the Lagrange equation for the support coordinate
22075:
21667:
21617:
21463:
21443:
21423:
21163:
21143:
21123:
21103:
20933:
20839:
20710:
20574:
20417:
20329:
20260:
20065:
19966:
19755:
19695:
19616:
19587:
19540:
19511:
19491:
19393:
18804:
18715:
18651:
18533:
18424:
18230:
18148:
18057:
17588:
17532:
17508:
17475:
17362:
17331:
17300:
17245:
17147:
17125:
17091:
17067:
16961:
16858:
16761:
16586:
16168:
15342:
15222:
15150:
15125:
15086:
15050:
14925:
14757:
14601:
14559:
14499:
14427:
14330:
14210:
14050:
13777:
13493:
13428:
13306:
13279:
13078:
12927:
12790:
12704:
12575:
12549:
11537:
11414:
11167:
10936:
10827:
10660:
10527:
10470:
10346:
10189:
9648:
9293:
9069:
8796:
8606:
8298:
7955:
7730:
7398:
7275:
7184:
7111:
6416:
6091:
5965:
5825:
5651:
5456:
5264:
5034:
4929:
4788:
4551:
4474:
4323:
4167:
3968:
3600:
3487:
3262:
3178:
2963:
2775:
2331:
2116:
1617:
1597:
1577:
1490:
1455:
1353:
1173:
41:
28731:
Société de Mathématiques Appliquées et Industrielles
28724:
Japan Society for Industrial and Applied Mathematics
28360:
Numerical methods for partial differential equations
28074:
27039:(3rd ed.). Butterworth Heinemann. p. 134.
26685:
25634:
21049:
can all vary with time, but only in such a way that
18798:
From the Euler-Lagrange equations, it follows that:
18627:
of the overall system is the sum of the Lagrangians
4653:
arbitrary coordinate system since the displacements
28547:
27176:(3rd ed.). San Francisco, CA: Addison Wesley.
27162:
27148:(2nd ed.). San Francisco, CA: Addison Wesley.
26961:
26744:
26323:. 2021;26(6):785-803. Doi: 10.1177/1081286520971834
22727:are in orbit about each other due to an attractive
18527:′ in the new trajectory, given by the ratios
5621:{\displaystyle \sum _{j=1}^{n}\left\delta q_{j}=0,}
3425:, the results of the partial derivatives are still
27714:
27695:General Relativity: An Introduction for Physicists
26723:
26255:
26056:
25928:
25853:
25576:
25504:
25195:on the Lagrangian, which introduces new variables
25183:
25069:
24996:
24883:
24748:
24670:
24581:
24479:
24259:
24104:
23992:
23751:
23631:
23517:
23400:
23291:
23141:
22969:
22660:
22617:
22579:
22493:
22320:
22203:
22183:
22055:
21653:
21603:
21449:
21429:
21407:
21149:
21129:
21109:
21025:
20915:
20826:
20697:
20560:
20363:
20307:
20246:
20051:
19942:
19731:
19649:
19602:
19573:
19526:
19497:
19477:
19377:
18766:
18686:
18600:
18501:
18369:
18166:
18126:
18021:
17538:
17514:
17481:
17461:
17348:
17317:
17286:
17231:
17134:
17111:
17078:
17053:
16947:
16844:
16745:
16570:
16152:
15328:
15216:After differentiation involving the product rule,
15208:
15136:
15111:
15072:
15036:
14911:
14743:
14583:
14545:
14464:{\displaystyle \partial L/\partial {\dot {q}}_{i}}
14463:
14413:
14290:
14196:
14037:
13764:
13479:
13414:
13288:
13218:
13032:
12906:
12759:
12667:
12561:
12533:
11520:
11400:
11153:
10919:
10813:
10646:
10513:
10447:
10328:
10068:
9616:
9483:
9425:
9157:
9055:
8755:
8579:
8261:
7926:
7583:
7290:
7237:
7164:
7028:
6344:
6077:The Euler–Lagrange equations also follow from the
6059:
5929:
5752:
5620:
5440:
5247:
5020:
4882:
4642:
4537:
4429:
4194:
4065:
3932:
3509:
3429:in the position coordinates of the particles. The
3358:
3227:
3149:
2935:
2654:velocity component of particle 2, defined by
2592:
2287:
1992:. Three examples of nonholonomic constraints are:
1633:
1603:
1583:
1555:
1476:
1409:
1323:
903:The stationary action principle requires that the
856:
76:
27766:Classical mechanics of particles and rigid bodies
27672:
27564:
27517:
27398:Theoretical Astrophysics: Astrophysical processes
27240:
26956:
26945:
26647:"Variational mechanics in one and two dimensions"
26461:
26353:, Berlin, Heidelberg: Springer, pp. 85–186,
26185:
26183:
25051:fields are uniform, then this canonical momentum
24891:which still produces the same Lorentz force law.
22676:
20378:
18137:
17239:is a well-defined linear form whose coefficients
16075:
15922:
15734:
15539:
14866:
14701:
13398:
13315:
13248:is also a translational momentum along the curve
5770:Euler–Lagrange equations and Hamilton's principle
1437:Instead of forces, Lagrangian mechanics uses the
29133:
27356:. Cambridge University Press. pp. 118–119.
26644:
26638:
23427:can vary with time, but only in such a way that
21072:
20375:for a particle subject to a conservative force.
12683:An important property of the Lagrangian is that
5802:be possible to find a potential energy function
5762:These equations are equivalent to Newton's laws
3952:of the resultant force acting on the particle, Γ
1429:in the positions of the particles to solve for.
27812:"Cambridge Lecture Notes on Classical Dynamics"
27328:
27217:Theoretical Mechanics of Particles and Continua
27121:
26887:
26620:
26533:
26223:50 ideas you really need to know : science
25659:Fundamental lemma of the calculus of variations
25078:Alternative formulations of classical mechanics
21417:The generalized coordinates can be taken to be
9168:
4674:Equations of motion from D'Alembert's principle
28719:Society for Industrial and Applied Mathematics
27841:Constrained motion and generalized coordinates
27698:. Cambridge University Press. pp. 79–80.
27053:
27027:
26792:
26780:
26756:
26732:
26180:
19749:The Lagrangian of the particle can be written
18709:containing information about the interaction,
13261:is an angular momentum in the plane the angle
8776:coordinates only, substituting the Lagrangian
1895:If the number of constraints in the system is
28809:
28795:
27970:
27862:
27284:
27213:
26717:
26545:
26521:
25283:Higher derivatives of generalized coordinates
25220:
23831:Extensions to include non-conservative forces
4682:, then since the coordinates of the position
2925:
2825:
1421:-particle system in 3 dimensions, there are 3
766:
28537:Supersymmetric theory of stochastic dynamics
27914:Lagrange's identity (boundary value problem)
27611:
27287:"§1.4 Lagrange equations of the second kind"
27243:"§ 3.2 Lagrange equations of the first kind"
27086:(2nd ed.). Cambridge University Press.
26220:
25018:. From this expression, we can see that the
24593:. This is a good example of when the common
18142:Under all these circumstances, the constant
12921:For example, a system may have a Lagrangian
9208:. If one restricts as above to trajectories
7678:are not independent. Instead, the method of
27835:Joseph Louis de Lagrange - Œuvres complètes
27712:
27639:. Cambridge University Press. p. 267.
27391:
27122:Kibble, T. W. B.; Berkshire, F. H. (2004).
27104:Classical Dynamics: A Contemporary Approach
27100:
26851:
26449:
26064:because of the constraint equations on the
25317:
25255:obtains the generalized momenta Lagrangian
21457:. The kinetic energy of the system is then
21093:, which is attached to a support with mass
14302:Invariance under coordinate transformations
10514:{\displaystyle Q_{i}=Q_{i}(\mathbf {q} ,t)}
10457:This may simplify the equations of motion.
4073:is the kinetic energy of the particle, and
3510:{\displaystyle \mathbf {F} =m\mathbf {a} ,}
28802:
28788:
27977:
27963:
27869:
27855:
27307:
26875:
26804:
26581:
26443:
26149:
26147:
25328:Lagrangian mechanics can be formulated in
23142:{\displaystyle M{\ddot {\mathbf {R} }}=0,}
22625:should give the equations of motion for a
12691:"canonically conjugate to" the coordinate
7597:is expressed in are not independent, here
773:
759:
27876:
27831:Excellent interactive explanation/webpage
27776:. Cambridge: Cambridge University Press.
27721:. first Dover Publications 1978 edition.
27578:
27531:
27520:International Journal of Modern Physics A
27476:
27377:. University Science Books. p. 297.
27241:Dvorak, R.; Freistetter, Florian (2005).
27140:
27079:
26863:
26768:
26706:
26694:
26632:
26608:
26569:
26557:
26509:
26497:
26485:
26473:
26398:
26386:
26332:
26138:
26126:
26114:
26029:
25992:
25955:
25556:
25414:
24914:field (known as the potential momentum):
24544:
23645:is subjected to the inward central force
20891:
20795:
20640:
20517:
18035:does not explicitly depend on time, then
15407:
15390:
14897:
14732:
14471:change as coefficients of a linear form.
11311:
11064:
10744:
9958:
9869:
9768:
8244:
7566:
7223:
7147:
7010:
6724:
6456:
4211:
3421:Although the equations of motion include
2957:of its position with respect to time, is
2769:of the generalized coordinates and time:
27433:
27247:Chaos and stability in planetary systems
26997:
26986:
26923:
26431:
26344:
25684:Inverse problem for Lagrangian mechanics
25524:of the Lagrangian density over 3D space
23662:centrifugal force#Other uses of the term
22671:stepping through the results iteratively
21076:
19744:
18610:
18182:
17522:does not explicitly depend on time. The
12679:Cyclic coordinates and conserved momenta
9182:can be multiplied by a nonzero constant
5773:
4215:
3453:
946:
930:
784:
27654:Torby, Bruce (1984). "Energy Methods".
27414:
27349:
27197:The variational principles of mechanics
27190:
27008:
26899:
26805:Rothe, Heinz J; Rothe, Klaus D (2010).
26575:
26410:
26144:
25940:subject to a constraint force, however
25310:Lagrangian mechanics can be applied to
23433:is constant. The Lagrange equation for
21654:{\displaystyle V=mgy_{\mathrm {pend} }}
17498:In Lagrangian mechanics, the system is
13052:is an arc length along some curve, and
10183:′ is a function of the new coordinates
10130:Given a set of generalized coordinates
7940:, but not functions of the coordinates
7359:Hamilton's principle can be applied to
5953:Lagrange's equations of the second kind
3245:Lagrange's equations of the second kind
1899:, then each constraint has an equation
29134:
29000:Atomic, molecular, and optical physics
27771:
27632:
27462:
27370:
27270:(3rd ed.). Springer. p. 61.
27214:Fetter, A. L.; Walecka, J. D. (1980).
26934:
26911:
26839:
22661:{\displaystyle {\ddot {\theta }}\to 0}
18615:For a given system, if two subsystems
17112:{\displaystyle {\dot {\mathbf {q} }},}
10126:Invariance under point transformations
8589:Lagrange's equations of the first kind
5628:and since these virtual displacements
3445:From Newtonian to Lagrangian mechanics
2098:Lagrange's equations of the first kind
1132:, each position vector requires three
28783:
27958:
27850:
27829:Principle of least action interactive
27673:Foster, J; Nightingale, J.D. (1995).
27653:
27263:
26970:
26268:
26249:
23835:
21097:, which can move along a line in the
19668:
18767:{\displaystyle L=L_{A}+L_{B}+L_{AB}.}
17349:{\displaystyle {\dot {\mathbf {Q} }}}
17318:{\displaystyle {\dot {\mathbf {q} }}}
9186:and shifted by an arbitrary constant
4678:If there are constraints on particle
911:must remain at a stationary point (a
455:Newton's law of universal gravitation
27675:A Short Course in General Relativity
27308:Hildebrand, Francis Begnaud (1992).
27101:Saletan, E. J.; José, J. V. (1998).
26216:
26214:
26168:10.1103/PhysRevPhysEducRes.15.020126
26153:
26103:http://www.jstor.org/stable/41133689
25598:
25343:
25191:and can be obtained by performing a
24274:A test particle is a particle whose
22330:The Lagrange equation for the angle
18794:Consequences of singular Lagrangians
18623:are non-interacting, the Lagrangian
15044:is the Jacobian. In the coordinates
12687:can easily be read off from it. The
7371:Lagrange multipliers and constraints
7312:to mechanical problems, such as the
4157:plus the resultant constraint force
2600:are each shorthands for a vector of
1776:), the velocities will appear also,
900:energy of the system, respectively.
27919:Lagrange's trigonometric identities
27739:Schaum's Outline of Tensor Calculus
27735:
27394:"§2.3.2 Motion in a rotating frame"
26437:
26321:Mathematics and Mechanics of Solids
26221:Parsons, Paul; Dixon, Gail (2016).
24269:
17150:
14584:{\displaystyle F_{*}(\mathbf {q} )}
14324:, i.e. (using natural coordinates)
13235:is a translational momentum in the
3481:of 1687, in modern vector notation
436:Mechanics of planar particle motion
44:
13:
27986:Industrial and applied mathematics
27758:
27080:Hand, L. N.; Finch, J. D. (1998).
26287:The European Physical Journal Plus
25824:
25820:
25808:
25802:
25778:
25774:
25559:
25551:
25464:
25450:
25150:
25142:
25006:This relation is also used in the
24940:
24932:
24829:
24825:
24440:
24430:
24226:
24218:
24196:
24188:
24153:
24145:
24128:
24122:
24077:
24069:
24047:
24039:
23680:
23677:
23584:
23574:
23343:
23328:
22350:
22344:
22100:
22092:
21645:
21642:
21639:
21636:
21585:
21582:
21579:
21576:
21547:
21544:
21541:
21538:
21365:
21362:
21359:
21356:
21310:
21307:
21304:
21301:
21242:
21239:
21236:
21233:
21184:
21181:
21178:
21175:
20958:
20950:
20849:
20843:
20720:
20714:
20677:
20669:
20403:is the radial distance to origin,
20293:
20285:
20201:
20193:
20176:
20170:
20127:
20119:
20103:
20095:
20077:
20069:
20037:
20029:
20001:
19993:
19976:
19970:
19450:
19428:
19414:
19333:
19311:
19297:
19271:
19263:
19241:
19233:
19126:
19118:
19103:
19095:
19028:
19006:
18992:
18933:
18920:
18906:
18875:
18867:
18836:
18828:
17951:
17943:
17904:
17896:
17853:
17845:
17789:
17781:
17743:
17735:
17716:
17708:
17684:
17676:
17645:
17637:
17436:
17428:
17388:
17380:
17258:
17250:
17181:
17173:
17028:
17020:
16973:
16965:
16725:
16717:
16674:
16666:
16641:
16633:
16552:
16544:
16480:
16472:
16402:
16394:
16356:
16348:
16315:
16307:
16264:
16256:
16231:
16223:
16056:
16043:
16022:
15903:
15890:
15869:
15715:
15702:
15681:
15520:
15507:
15486:
14978:
14963:
14820:
14816:
14664:
14660:
14439:
14428:
14272:
14262:
14240:
14230:
14113:
14105:
13908:
13900:
13829:
13821:
13642:
13625:
13574:
13557:
13371:
13363:
13195:
13187:
13149:
13141:
13103:
13095:
13048:are lengths along straight lines,
12882:
12874:
12843:
12835:
12822:
12816:
12729:
12721:
12643:
12634:
12602:
12594:
12505:
12490:
12457:
12449:
12414:
12406:
12334:
12319:
12272:
12263:
12231:
12207:
12188:
12179:
12143:
12119:
12072:
12064:
12033:
12009:
11977:
11969:
11899:
11875:
11847:
11838:
11813:
11798:
11779:
11770:
11733:
11709:
11681:
11673:
11609:
11600:
11568:
11560:
11502:
11478:
11452:
11437:
11365:
11359:
11338:
11278:
11265:
11244:
11207:
11190:
11125:
11112:
11091:
11031:
11018:
10997:
10964:
10940:
10901:
10886:
10855:
10831:
10785:
10770:
10711:
10696:
10626:
10611:
10574:
10559:
10426:
10413:
10382:
10369:
10356:
10350:
9948:
9912:
9576:
9551:
9526:
9501:
9471:
9444:
9410:
9383:
9134:
9119:
9026:
9011:
8935:
8927:
8863:
8855:
8842:
8836:
8812:
8804:
8677:
8664:
8651:
8645:
8623:
8610:
8550:
8535:
8471:
8463:
8450:
8444:
8420:
8412:
8371:
8358:
8345:
8339:
8315:
8302:
8246:
8203:
8188:
8124:
8116:
8103:
8097:
8073:
8065:
7997:
7568:
7516:
7508:
7495:
7489:
7465:
7457:
7225:
7149:
7012:
6967:
6959:
6946:
6940:
6918:
6910:
6781:
6773:
6726:
6681:
6673:
6660:
6654:
6605:
6597:
6579:
6573:
6538:
6530:
6458:
6329:
6303:
6286:
6269:
6181:
6173:
6138:
6130:
6026:
6018:
6005:
5999:
5977:
5969:
5908:
5900:
5869:
5861:
5848:
5842:
5734:
5726:
5695:
5687:
5674:
5668:
5570:
5562:
5531:
5523:
5510:
5504:
5419:
5411:
5380:
5372:
5359:
5353:
5331:
5314:
5166:
5149:
4999:
4982:
4848:
4831:
4053:
4036:
4022:
4005:
3917:
3900:
3849:
3841:
3810:
3802:
3789:
3783:
3744:
3727:
3713:
3696:
3671:
3650:
3627:
3369:are mathematical results from the
3340:
3332:
3297:
3289:
3272:
3266:
3135:
3118:
3077:
3060:
3007:
2990:
2560:
2556:
2526:
2522:
2492:
2488:
2451:
2447:
2420:
2416:
2395:
2391:
2370:
2366:
2338:
2334:
2258:
2243:
2179:
2171:
2158:
2152:
2128:
2120:
16:Formulation of classical mechanics
14:
29163:
28216:Stochastic differential equations
27904:Lagrange's theorem (group theory)
27809:
27803:
27419:. Addison-Wesley Publishers Ltd.
27353:Computational continuum mechanics
27267:Information and self-organization
26745:Goldstein, Poole & Safko 2002
26211:
26189:
25324:Relativistic Lagrangian mechanics
23797:upon the coordinate system chosen
17061:This demonstrates that, for each
9204:will describe the same motion as
9173:
2306:labels the particles, there is a
1866:relativistic Lagrangian mechanics
1591:particles. Each particle labeled
1571:Σ of the kinetic energies of the
1417:applies to each particle. For an
28532:Supersymmetric quantum mechanics
27713:Synge, J.L.; Schild, A. (1949).
27332:From instability to intelligence
25910:
25892:
25832:
25783:
25756:
25637:
25570:
25489:
25086:. The Hamiltonian is defined by
24987:
24970:
24947:
24922:
24816:
24802:
24708:
24700:
24692:
24664:
24650:
24636:
24619:
24563:
24549:
24522:
24470:
24462:
24454:
24434:
24417:
24406:
23120:
23100:The Euler–Lagrange equation for
22936:
22905:
22845:
22618:{\displaystyle {\ddot {x}}\to 0}
20383:Using the spherical coordinates
20354:
20337:
19719:
19708:
19697:
19650:{\displaystyle ({\dot {q}},q,t)}
19574:{\displaystyle ({\dot {q}},q,t)}
18354:
18333:
18318:
18284:
18260:
18242:
18085:
18065:
17975:
17972:
17960:
17957:
17862:
17859:
17832:
17815:
17798:
17795:
17693:
17690:
17663:
17649:
17624:
17443:
17415:
17395:
17367:
17336:
17325:and using the above formula for
17305:
17096:
17069:
17035:
17007:
16980:
16923:
16912:
16880:
16869:
16852:One implication of this is that
16820:
16809:
16780:
16769:
16701:
16681:
16656:
16645:
16608:
16597:
16528:
16514:
16487:
16462:
16440:
16429:
16409:
16381:
16360:
16291:
16271:
16246:
16235:
16194:
16183:
16139:
16082:
15929:
15741:
15597:
15546:
15442:
15428:
15382:
15365:
15354:
15313:
15299:
15278:
15261:
15250:
15230:
15193:
15182:
15155:
15130:
15000:
14941:
14848:
14686:
14574:
14536:
14525:
14504:
14478:For a coordinate transformation
14389:
14378:
14349:
14338:
14310:, the energy is invariant under
14266:
14234:
13438:
13430:
11385:
11298:
11138:
11051:
10798:
10731:
10498:
10464:For a coordinate transformation
10295:
10284:
10270:
10250:
10242:
10213:
10202:
9982:
9923:
9841:
9821:
9740:
9720:
9665:
9561:
9511:
9455:
9394:
9357:
9346:
9317:
9306:
9139:
9072:
9031:
8962:
8940:
8902:
8871:
8817:
8728:
8555:
8479:
8425:
8379:
8320:
8234:
8208:
8132:
8078:
7902:
7813:
7789:
7765:
7750:
7556:
7524:
7470:
6352:which has a form similar to the
5815:can be derived from a potential
5319:
5298:
5154:
5112:
5076:
5058:
4987:
4966:
4908:along the virtual displacements
4836:
4794:
4624:
4603:
4578:
4516:
4498:
4411:
4390:
4365:
4350:
4243:
4231:
4185:
4177:
4169:
3500:
3489:
3473:for a particle of constant mass
3449:
3203:
3192:
3123:
3065:
3023:
2904:
2874:
2844:
2807:
2793:
2778:
2646:(e.g. the partial derivative of
2459:
2343:
2263:
2187:
2133:
1382:
1358:
1297:
1276:
1244:
1223:
1197:
1176:
740:
739:
726:
59:
29121:Timeline of physics discoveries
28414:Stochastic variational calculus
28206:Ordinary differential equations
27656:Advanced Dynamics for Engineers
27291:Fundamental formulas of physics
26957:Birnholtz, Hadar & Kol 2013
26950:
26946:Birnholtz, Hadar & Kol 2014
26939:
26928:
26917:
26905:
26893:
26881:
26869:
26857:
26845:
26833:
26798:
26786:
26774:
26762:
26750:
26738:
26711:
26700:
26626:
26614:
26602:
26563:
26551:
26539:
26527:
26515:
26503:
26491:
26467:
26455:
26415:
26404:
26392:
26380:
26338:
26326:
26313:
26274:
26156:"Teaching Energy Before Forces"
26076:
26020:
25983:
25877:
25864:
25728:
25209:Hamiltonian (quantum mechanics)
25070:Other contexts and formulations
24718:
24452:
23071:. The Lagrangian splits into a
22740:; the separation of the bodies
21340:
21336:
21217:
21213:
20166:
20115:
19677:moves under the influence of a
19387:where the matrix is defined as
18466:
18462:
18377:and time is scaled by a factor
17493:
15112:{\displaystyle {\dot {Q}}_{i},}
13170:
13124:
8712:
8708:
8408:
8404:
7379:can be varied in the Cartesian
6388:), in the process exchanging d(
6236:
5643:generalized equations of motion
3873:
3531:ordinary differential equations
3427:ordinary differential equations
3020:
2444:
1860:As already noted, this form of
1427:ordinary differential equations
926:
28211:Partial differential equations
28084:Arbitrary-precision arithmetic
27899:Lagrange's four-square theorem
27495:10.1103/PhysRevLett.110.174301
27311:Methods of applied mathematics
27107:. Cambridge University Press.
27063:The Classical Theory of Fields
26588:. Vintage books. p. 474.
26243:
26132:
26120:
26108:
26095:
25734:Sometimes in this context the
25540:
25534:
25499:
25455:
25368:can be understood in terms of
25366:principle of stationary action
24781:
24722:
24696:
23658:(Lagrangian) centrifugal force
23283:
23277:
22945:
22941:
22931:
22927:
22677:Two-body central force problem
22652:
22609:
22470:
22437:
22420:
22368:
22230:
22218:
22133:
22121:
21337:
21214:
20904:
20859:
20758:
20730:
20660:
20599:
20552:
20546:
20537:
20434:
20379:Polar coordinates in 2D and 3D
19934:
19916:
19907:
19841:
19822:
19759:
19723:
19715:
19644:
19617:
19568:
19541:
19205:
19178:
18687:{\displaystyle L=L_{A}+L_{B}.}
18463:
18364:
18313:
18294:
18234:
18138:Kinetic and potential energies
18110:
18101:
18095:
18075:
18069:
18061:
17502:if and only if its Lagrangian
17011:
17003:
16939:
16908:
16896:
16865:
16836:
16805:
16796:
16765:
16624:
16593:
16518:
16510:
16450:
16425:
16385:
16377:
16210:
16179:
15435:
15432:
15424:
15411:
15375:
15350:
15303:
15295:
15271:
15246:
15186:
15178:
15073:{\displaystyle {\dot {q}}_{i}}
14945:
14937:
14852:
14844:
14578:
14570:
14529:
14521:
14405:
14374:
14365:
14334:
14173:
14161:
13634:
13566:
13474:
13442:
13024:
12931:
12562:{\displaystyle q\rightarrow Q}
12553:
12543:Since the transformation from
11395:
11381:
11308:
11294:
11148:
11134:
11061:
11047:
10808:
10794:
10741:
10727:
10508:
10494:
10320:
10311:
10280:
10260:
10246:
10238:
10229:
10198:
10056:
10030:
10021:
9995:
9986:
9978:
9942:
9933:
9927:
9919:
9866:
9857:
9851:
9831:
9825:
9817:
9765:
9756:
9750:
9730:
9724:
9716:
9669:
9661:
9571:
9557:
9521:
9507:
9465:
9451:
9404:
9390:
9373:
9342:
9333:
9302:
8744:
8723:
8709:
8405:
7918:
7897:
7884:
7878:
7841:
7745:
6323:
6307:
4613:
4573:
4400:
4345:
3219:
3188:
2914:
2900:
2884:
2870:
2854:
2840:
2817:
2803:
2091:explicitly independent of time
1674:of the velocity with itself.
1:
28099:Interactive geometry software
27934:Lagrange's mean value theorem
27440:. Berlin, Germany: Springer.
27235:The Principle of Least Action
26979:
26888:Zak, Zbilut & Meyers 1997
26462:Foster & Nightingale 1995
25715:Restricted three-body problem
24908:plus a contribution from the
21073:Pendulum on a movable support
20373:Newton's second law of motion
18193:If the potential energy is a
17469:The invariance of the energy
17079:{\displaystyle \mathbf {q} ,}
15137:{\displaystyle d\mathbf {Q} }
10076:with the last two components
5764:for the non-constraint forces
2733:, conveniently written as an
2316:for each constraint equation
1432:
362:Koopman–von Neumann mechanics
27742:. McGraw Hill Professional.
27371:Taylor, John Robert (2005).
25721:
21611:and the potential energy is
21085:Consider a pendulum of mass
9169:Properties of the Lagrangian
3237:With these definitions, the
1567:of the system, equaling the
1331:In Newtonian mechanics, the
430:Non-inertial reference frame
7:
29085:Quantum information science
28151:Computational number theory
28114:Numerical-analysis software
27392:Padmanabhan, Thanu (2000).
27191:Lanczos, Cornelius (1986).
26651:American Journal of Physics
26621:Kibble & Berkshire 2004
26534:Kibble & Berkshire 2004
26359:10.1007/978-3-540-79436-3_3
26154:Ball, Philip (2019-09-13).
25630:
23809:inertial frame of reference
23542:from this radial equation,
19660:
19603:{\displaystyle {\ddot {q}}}
19527:{\displaystyle {\ddot {q}}}
9212:over a given time interval
7291:{\displaystyle \delta S=0.}
3525:the resultant force acting
907:of the system derived from
864:within that space called a
810:stationary-action principle
357:Appell's equation of motion
227:Inertial frame of reference
10:
29168:
28916:Classical electromagnetism
27677:(2nd ed.). Springer.
27597:10.1103/PhysRevD.88.104037
27350:Shabana, Ahmed A. (2008).
27249:. Birkhäuser. p. 24.
26793:Landau & Lifshitz 1976
26781:Landau & Lifshitz 1976
26757:Landau & Lifshitz 1976
26733:Landau & Lifshitz 1976
26351:Mechanical System Dynamics
26299:10.1140/epjp/i2015-15123-4
25607:, which connects physical
25321:
25303:
25224:
25221:Momentum space formulation
24013:is defined this way, then
22680:
18523:′ traversed in time
18398:are scaled by a factor of
18186:
12777:depend on some coordinate
8283:are zero, even though the
7332:around the same time, and
7081:is zero, then because the
3170:in generalized coordinates
29093:
29030:
28958:
28874:
28846:
28818:
28749:
28711:
28695:
28657:
28609:
28557:Algebra of physical space
28422:
28380:
28174:
28136:
28024:Automated theorem proving
27992:
27884:
27799:. 3rd ed., Pearson, 2002.
27736:Kay, David (April 1988).
27550:10.1142/S0217751X14501322
27446:10.1007/978-3-540-40934-2
27415:Doughty, Noel A. (1990).
27335:. Springer. p. 202.
27220:. Dover. pp. 53–57.
27170:; Safko, John L. (2002).
26718:Fetter & Walecka 1980
26546:Fetter & Walecka 1980
26522:Fetter & Walecka 1980
25389:principle of least action
25385:path integral formulation
25370:constructive interference
25299:
24334:magnetic vector potential
23871:, of the following form:
22810:. The Lagrangian is then
17546:of a closed system is an
15119:the previous formula for
13268:
9190:, and the new Lagrangian
7707:by a Lagrange multiplier
4260:and virtual displacement
2712:coordinates is therefore
2052:explicitly time-dependent
1956:is subject to constraint
1774:electromagnetic potential
1066:are often sufficient, so
29022:Condensed matter physics
28350:Numerical linear algebra
27434:Kosyakov, B. P. (2007).
26998:Lagrange, J. L. (1815).
26987:Lagrange, J. L. (1811).
26089:
25705:Non-autonomous mechanics
25318:Relativistic formulation
25295:Ostrogradsky instability
25039:), which happens if the
22760:and the location of the
22629:that is at rest in some
7361:nonholonomic constraints
5957:Euler–Lagrange equations
4289:, introduced in 1708 by
4281:A fundamental result in
3537:of the particle at time
3521:is its acceleration and
3439:implicit differentiation
3240:Euler–Lagrange equations
2096:With these definitions,
1986:Nonholonomic constraints
974:in this case is gravity.
520:Rotating reference frame
352:Hamilton–Jacobi equation
28089:Finite element analysis
28039:Constraint satisfaction
27526:(24): 1450132–1450190.
27465:Physical Review Letters
27009:Penrose, Roger (2007).
26450:Synge & Schild 1949
26105:. Accessed 3 Nov. 2023.
25872:Udwadia–Kalaba equation
25738:denoted and defined as
25669:Generalized coordinates
25356:and quantum-mechanical
25290:Euler–Lagrange equation
25193:Legendre transformation
24898:(conjugate to position
24599:Euler–Lagrange equation
23786:using polar coordinates
21450:{\displaystyle \theta }
21150:{\displaystyle \theta }
19954:Euler–Lagrange equation
18406:and the kinetic energy
17526:states that the energy
17524:energy conservation law
10160:which is invertible as
9230:} and fixed end points
7314:Brachistochrone problem
5778:As the system evolves,
4267:for a particle of mass
4250:Two degrees of freedom.
3950:contravariant component
3565:curvilinear coordinates
2731:generalized coordinates
1130:three-dimensional space
993:generalized coordinates
966:, the constraint force
461:Newton's laws of motion
321:Newton's laws of motion
29106:Nobel Prize in Physics
28968:Relativistic mechanics
28644:Mathematical economics
28318:Multivariable calculus
28201:Differential equations
28044:Constraint programming
28034:Computational geometry
27772:Cassel, Kevin (2013).
27764:Gupta, Kiran Chandra,
27633:Gannon, Terry (2006).
27618:. New York: Springer.
27417:Lagrangian Interaction
27289:. In DH Menzel (ed.).
27285:Henry Zatzkis (1960).
26192:"Lagrangian mechanics"
26058:
25930:
25855:
25736:variational derivative
25615:of a physical system.
25578:
25506:
25421:classical field theory
25415:Classical field theory
25185:
25119:
24998:
24885:
24750:
24672:
24583:
24481:
24306:, interacting with an
24261:
24106:
23994:
23935:
23914:
23788:. As Hildebrand says:
23753:
23633:
23519:
23410:The radial coordinate
23402:
23293:
23143:
22971:
22709:with position vectors
22662:
22619:
22581:
22495:
22322:
22205:
22185:
22057:
21661:giving the Lagrangian
21655:
21605:
21451:
21431:
21409:
21151:
21131:
21111:
21082:
21065:to the constant value
21027:
20917:
20828:
20699:
20562:
20365:
20315:and similarly for the
20309:
20248:
20053:
19944:
19733:
19651:
19604:
19575:
19528:
19499:
19479:
19379:
18768:
18688:
18602:
18503:
18389:, then the velocities
18371:
18168:
18128:
18023:
17540:
17516:
17483:
17463:
17350:
17319:
17288:
17233:
17169:
17136:
17113:
17080:
17055:
16949:
16846:
16747:
16572:
16154:
16017:
15864:
15830:
15676:
15636:
15481:
15330:
15210:
15138:
15113:
15074:
15038:
14913:
14783:
14745:
14627:
14585:
14547:
14465:
14415:
14306:At every time instant
14292:
14198:
14082:
14039:
13877:
13798:
13766:
13481:
13416:
13340:
13290:
13220:
13034:
12908:
12761:
12669:
12563:
12535:
11522:
11402:
11155:
10930:It also follows that:
10921:
10815:
10648:
10515:
10449:
10330:
10070:
9618:
9485:
9427:
9159:
9105:
9057:
8997:
8757:
8581:
8521:
8263:
8174:
8056:
7928:
7867:
7585:
7448:
7338:variational principles
7310:calculus of variations
7292:
7239:
7166:
7038:Now, if the condition
7030:
6901:
6763:
6521:
6346:
6121:
6079:calculus of variations
6061:
5931:
5795:
5782:traces a path through
5754:
5622:
5477:
5442:
5285:
5249:
5218:
5145:
5109:
5055:
5022:
4963:
4884:
4827:
4644:
4572:
4539:
4495:
4431:
4344:
4287:D'Alembert's principle
4238:One degree of freedom.
4224:
4212:D'Alembert's principle
4196:
4067:
3934:
3511:
3462:
3371:calculus of variations
3360:
3229:
3151:
3056:
2937:
2594:
2289:
2229:
2000:or use other methods.
1635:
1634:{\displaystyle m_{k},}
1605:
1585:
1557:
1527:
1478:
1477:{\displaystyle L=T-V,}
1411:
1339:. The second law "net
1325:
1167:of its position, thus
1035:, each particle has a
975:
944:
858:
844:and a smooth function
793:
488:Simple harmonic motion
401:Euler's laws of motion
195:D'Alembert's principle
78:
29111:Philosophy of physics
28597:Supersymmetry algebra
28582:Representation theory
28577:Renormalization group
28223:Differential geometry
28104:Optimization software
28076:Mathematical software
27878:Joseph-Louis Lagrange
27612:Roger F Gans (2013).
27168:Poole, Charles P. Jr.
26864:Hand & Finch 1998
26695:Hand & Finch 1998
26633:Hand & Finch 1998
26570:Hand & Finch 1998
26558:Hand & Finch 1998
26510:Hand & Finch 1998
26498:Hand & Finch 1998
26474:Hand & Finch 1998
26423:The Lagrangian Method
26399:Hand & Finch 1998
26387:Hand & Finch 1998
26333:Hand & Finch 1998
26139:Hand & Finch 1998
26127:Hand & Finch 1998
26115:Hand & Finch 1998
26059:
25931:
25856:
25674:Hamiltonian mechanics
25664:Functional derivative
25654:Canonical coordinates
25579:
25507:
25201:canonically conjugate
25197:canonically conjugate
25186:
25099:
25084:Hamiltonian mechanics
24999:
24886:
24751:
24673:
24584:
24482:
24354:are defined from the
24308:electromagnetic field
24293:The Lagrangian for a
24262:
24107:
23995:
23915:
23894:
23803:for a curved motion.
23754:
23634:
23520:
23416:and angular velocity
23403:
23294:
23144:
22972:
22691:Two bodies of masses
22663:
22620:
22582:
22496:
22323:
22206:
22186:
22058:
21656:
21606:
21452:
21432:
21410:
21152:
21132:
21112:
21080:
21028:
20918:
20829:
20700:
20563:
20366:
20310:
20249:
20054:
19945:
19745:Cartesian coordinates
19734:
19652:
19605:
19576:
19529:
19500:
19480:
19380:
18769:
18689:
18611:Interacting particles
18603:
18504:
18372:
18189:Mechanical similarity
18183:Mechanical similarity
18169:
18167:{\displaystyle E=T+V}
18129:
18024:
17541:
17517:
17484:
17464:
17351:
17320:
17289:
17234:
17149:
17137:
17114:
17081:
17056:
16950:
16847:
16748:
16573:
16155:
15997:
15844:
15810:
15656:
15616:
15461:
15331:
15211:
15139:
15114:
15075:
15039:
14914:
14763:
14746:
14607:
14586:
14548:
14466:
14416:
14293:
14199:
14062:
14040:
13857:
13778:
13767:
13482:
13417:
13320:
13291:
13221:
13035:
12909:
12762:
12670:
12564:
12536:
11523:
11403:
11156:
10922:
10816:
10649:
10516:
10450:
10331:
10179:, the new Lagrangian
10071:
9619:
9486:
9428:
9160:
9085:
9058:
8977:
8758:
8582:
8501:
8264:
8154:
8036:
7929:
7847:
7586:
7428:
7293:
7240:
7167:
7031:
6881:
6743:
6501:
6347:
6101:
6085:of the Lagrangian is
6062:
5932:
5777:
5755:
5623:
5457:
5443:
5265:
5250:
5198:
5125:
5089:
5035:
5023:
4943:
4885:
4807:
4723:virtual displacements
4645:
4552:
4540:
4475:
4459:at an instant of time
4442:virtual displacements
4432:
4324:
4219:
4197:
4068:
3935:
3589:tensor index notation
3512:
3457:
3431:total time derivative
3361:
3230:
3168:, the kinetic energy
3152:
3036:
2938:
2644:differentiation rules
2640:independent variables
2626:, because there are 3
2595:
2290:
2209:
1878:holonomic constraints
1636:
1606:
1586:
1558:
1507:
1479:
1412:
1326:
1064:Cartesian coordinates
1011:point particles with
950:
934:
859:
814:Joseph-Louis Lagrange
790:Joseph-Louis Lagrange
788:
342:Hamiltonian mechanics
160:Statistical mechanics
79:
29152:Mathematical physics
29142:Lagrangian mechanics
29070:Mathematical physics
28649:Mathematical finance
28634:Social choice theory
28549:Algebraic structures
28498:in quantum mechanics
28434:Analytical mechanics
28400:Stochastic processes
28372:Variational calculus
28184:Approximation theory
28109:Statistical software
27929:Lagrangian mechanics
27083:Analytical Mechanics
27001:Mécanique analytique
26990:Mécanique analytique
26347:"Constraint Systems"
25944:
25887:
25742:
25609:conserved quantities
25528:
25445:
25360:are related via the
25338:manifestly covariant
25090:
25016:quantum field theory
24918:
24775:
24688:
24682:gauge transformation
24609:
24493:
24402:
24116:
24017:
23875:
23865:dissipation function
23668:
23546:
23443:
23309:
23181:
23110:
22814:
22637:
22594:
22505:
22338:
22215:
22195:
22073:
21665:
21615:
21461:
21441:
21421:
21161:
21141:
21121:
21101:
20931:
20837:
20708:
20572:
20415:
20327:
20258:
20063:
19964:
19753:
19693:
19614:
19610:'s as a function of
19585:
19538:
19509:
19489:
19391:
18802:
18713:
18649:
18645:for the subsystems:
18531:
18422:
18228:
18195:homogeneous function
18146:
18055:
17586:
17553:More precisely, let
17530:
17506:
17473:
17360:
17329:
17298:
17243:
17145:
17123:
17089:
17065:
16959:
16856:
16759:
16584:
16166:
16162:In vector notation,
15340:
15220:
15148:
15123:
15084:
15048:
14923:
14755:
14751:to the vector space
14599:
14595:of the vector space
14557:
14497:
14425:
14328:
14208:
14048:
13775:
13491:
13426:
13304:
13277:
13076:
12925:
12788:
12702:
12689:generalized momentum
12685:conserved quantities
12573:
12547:
11535:
11412:
11165:
10934:
10825:
10658:
10525:
10468:
10344:
10187:
10140:point transformation
9646:
9495:
9437:
9291:
9067:
8794:
8769:, a function of the
8604:
8296:
7953:
7728:
7680:Lagrange multipliers
7666:to zero because the
7396:
7320:in 1696, as well as
7273:
7182:
7109:
7102:Hamilton's principle
6414:
6362:integration by parts
6089:
5963:
5823:
5649:
5639:Lagrange's equations
5454:
5262:
5032:
4927:
4786:
4549:
4472:
4321:
4283:analytical mechanics
4165:
4082:covariant components
3966:
3962:of the second kind,
3598:
3485:
3260:
3250:Lagrange's equations
3176:
2961:
2773:
2650:with respect to the
2329:
2114:
2104:Lagrange's equations
1692:, not the positions
1615:
1595:
1575:
1488:
1453:
1351:
1171:
870:. For many systems,
848:
819:Mécanique analytique
804:is a formulation of
802:Lagrangian mechanics
565:Angular acceleration
557:Rotational frequency
337:Lagrangian mechanics
330:Analytical mechanics
86:Second law of motion
39:
29045:Atmospheric physics
28884:Classical mechanics
28812:branches of physics
28624:Operations research
28493:Perturbation theory
28291:Multilinear algebra
28262:Functional analysis
28119:Numerical libraries
28051:Computational logic
27924:Lagrange multiplier
27909:Lagrange's identity
27894:Lagrange polynomial
27889:Lagrange multiplier
27796:Classical Mechanics
27589:2013PhRvD..88j4037B
27542:2014IJMPA..2950132B
27487:2013PhRvL.110q4301G
27374:Classical mechanics
27173:Classical Mechanics
27146:Classical Mechanics
27124:Classical Mechanics
27035:(15 January 1976).
27011:The Road to Reality
26663:2005AmJPh..73..603H
26585:The Road to Reality
26582:R. Penrose (2007).
22997:is the total mass,
21595:
21557:
19673:A particle of mass
17572:. (In other words,
16580:On the other hand,
16134:
15987:
15793:
15592:
15032:
14314:coordinate changes
14312:configuration space
13273:Given a Lagrangian
13060:are angles. Notice
10821:which implies that
10654:which implies that
9907:
9813:
9707:
8035:
7984:
7427:
7219:
7140:
6880:
6848:
6500:
6449:
5784:configuration space
4297:, and developed by
3960:Christoffel symbols
3479:Newton's second law
3423:partial derivatives
3398:equations of motion
2951:configuration space
2602:partial derivatives
2308:Lagrange multiplier
1998:Newtonian mechanics
1741:conservative forces
1552:
1333:equations of motion
998:system of equations
839:configuration space
806:classical mechanics
417:Harmonic oscillator
395:Equations of motion
30:Classical mechanics
24:Part of a series on
29101:History of physics
28761:Mathematics portal
28658:Other applications
28382:Probability theory
28365:Validated numerics
28345:Numerical analysis
28238:Geometric analysis
28228:Differential forms
28061:Information theory
27939:Lagrange stability
27793:, Herbert, et al.
27164:Goldstein, Herbert
27142:Goldstein, Herbert
26902:, pp. 118–119
26866:, pp. 140–141
26054:
25926:
25851:
25679:Hamiltonian optics
25625:general relativity
25621:special relativity
25574:
25502:
25440:Lagrangian density
25405:Fermat's principle
25334:general relativity
25330:special relativity
25312:geometrical optics
25306:Hamiltonian optics
25215:Routhian mechanics
25181:
25020:canonical momentum
24994:
24896:canonical momentum
24881:
24746:
24668:
24601:, it produces the
24579:
24512:
24477:
24257:
24102:
23990:
23836:Dissipative forces
23749:
23629:
23515:
23398:
23289:
23139:
23045:of the separation
22967:
22966:
22952:
22879:
22865:
22738:Jacobi coordinates
22658:
22615:
22577:
22491:
22318:
22201:
22181:
22053:
22051:
21651:
21601:
21561:
21523:
21447:
21427:
21405:
21403:
21147:
21127:
21107:
21083:
21023:
20913:
20824:
20695:
20558:
20361:
20305:
20244:
20049:
19940:
19729:
19679:conservative force
19669:Conservative force
19647:
19600:
19571:
19524:
19495:
19475:
19375:
19373:
19292:
19164:
18764:
18684:
18598:
18515:traversed in time
18499:
18367:
18164:
18124:
18031:If the Lagrangian
18019:
18017:
17548:integral of motion
17536:
17512:
17479:
17459:
17346:
17315:
17284:
17283:
17229:
17228:
17135:{\displaystyle t,}
17132:
17109:
17076:
17051:
16945:
16842:
16743:
16568:
16566:
16150:
16148:
15991:
15804:
15610:
15455:
15326:
15206:
15134:
15109:
15070:
15034:
15033:
14951:
14909:
14741:
14581:
14543:
14476:
14461:
14411:
14288:
14194:
14035:
13969:
13762:
13710:
13548:
13477:
13412:
13289:{\displaystyle L,}
13286:
13216:
13030:
12904:
12769:If the Lagrangian
12757:
12665:
12559:
12531:
12529:
12378:
11941:
11644:
11518:
11398:
11239:
11151:
10992:
10917:
10811:
10692:
10644:
10555:
10511:
10462:
10445:
10326:
10066:
10064:
9879:
9785:
9679:
9642:′ are related via
9614:
9547:
9481:
9423:
9155:
9053:
8973:
8957:
8913:
8894:
8753:
8577:
8259:
8007:
7956:
7924:
7581:
7399:
7365:linear combination
7288:
7235:
7191:
7162:
7112:
7026:
7024:
6852:
6764:
6472:
6421:
6354:total differential
6342:
6057:
5927:
5796:
5750:
5618:
5438:
5245:
5018:
4922:generalized forces
4880:
4780:total differential
4640:
4535:
4427:
4295:static equilibrium
4225:
4192:
4144:general relativity
4063:
3930:
3543:initial conditions
3507:
3471:equation of motion
3463:
3356:
3225:
3147:
2949:is a point in the
2933:
2590:
2285:
1631:
1601:
1581:
1553:
1538:
1474:
1407:
1343:equals mass times
1321:
1007:. For a system of
976:
945:
854:
794:
733:Physics portal
347:Routhian mechanics
222:Frame of reference
74:
29147:Dynamical systems
29129:
29128:
29116:Physics education
29065:Materials science
29032:Interdisciplinary
28990:Quantum mechanics
28777:
28776:
28611:Decision sciences
28605:
28604:
28587:Spacetime algebra
28279:Harmonic analysis
28245:Dynamical systems
28189:Clifford analysis
28166:Discrete geometry
28132:
28131:
27952:
27951:
27783:978-1-107-02258-4
27749:978-0-07-033484-7
27728:978-0-486-63612-2
27625:978-1-4614-3929-5
27567:Physical Review D
27455:978-3-540-40933-5
27363:978-0-521-88569-0
27264:Haken, H (2006).
27227:978-0-486-43261-8
27072:978-0-7506-2768-9
27020:978-0-679-77631-4
27013:. Vintage books.
26826:978-981-4299-64-0
26671:10.1119/1.1848516
26623:, p. 234–235
26595:978-0-679-77631-4
26452:, p. 150–152
26368:978-3-540-79436-3
26199:www.astro.uvic.ca
26141:, pp. 46, 51
25849:
25839:
25816:
25794:
25767:
25710:Plateau's problem
25700:Lagrangian system
25605:Noether's theorem
25599:Noether's theorem
25482:
25393:quantum mechanics
25350:quantum mechanics
25344:Quantum mechanics
25205:quantum mechanics
25173:
25163:
25130:
25012:quantum mechanics
24977:
24958:
24954:
24876:
24836:
24809:
24740:
24657:
24626:
24556:
24529:
24511:
24447:
24299:electrical charge
24249:
24239:
24210:
24176:
24166:
24136:
24100:
24090:
24061:
23978:
23959:
23892:
23863:suggests using a
23801:centripetal force
23744:
23705:
23624:
23592:
23561:
23509:
23492:
23462:
23386:
23359:
23355:
23338:
23257:
23225:
23205:
23191:
23127:
22962:
22912:
22895:
22885:
22883:
22875:
22852:
22835:
22825:
22823:
22729:central potential
22649:
22606:
22560:
22538:
22533:
22517:
22461:
22449:
22417:
22380:
22358:
22294:
22263:
22242:
22204:{\displaystyle x}
22166:
22145:
22116:
22112:
22016:
21991:
21968:
21953:
21929:
21898:
21831:
21787:
21769:
21744:
21724:
21709:
21571:
21533:
21513:
21493:
21478:
21430:{\displaystyle x}
21386:
21351:
21278:
21260:
21228:
21130:{\displaystyle x}
21110:{\displaystyle x}
21017:
20974:
20970:
20901:
20857:
20806:
20755:
20728:
20684:
20651:
20612:
20587:
20528:
20479:
20447:
20432:
20344:
20300:
20273:
20238:
20217:
20213:
20184:
20160:
20143:
20139:
20110:
20084:
20059:with derivatives
20044:
20017:
20013:
19984:
19898:
19876:
19854:
19836:
19819:
19804:
19789:
19681:derived from the
19629:
19597:
19553:
19534:as a function of
19521:
19498:{\displaystyle W}
19473:
19463:
19441:
19359:
19347:
19324:
19283:
19278:
19255:
19218:
19196:
19155:
19140:
19110:
19087:
19069:
19051:
19041:
19019:
18983:
18956:
18946:
18889:
18859:
18849:
18823:
18591:
18570:
18547:
18492:
18447:
18119:
18092:
18013:
17978:
17967:
17963:
17933:
17911:
17883:
17869:
17865:
17839:
17822:
17805:
17801:
17772:
17750:
17723:
17700:
17696:
17670:
17654:
17631:
17611:
17539:{\displaystyle E}
17515:{\displaystyle L}
17482:{\displaystyle E}
17454:
17450:
17422:
17406:
17402:
17374:
17343:
17312:
17281:
17271:
17219:
17204:
17194:
17103:
17046:
17042:
16991:
16987:
16930:
16887:
16827:
16787:
16732:
16708:
16692:
16688:
16650:
16615:
16559:
16535:
16498:
16494:
16447:
16420:
16416:
16365:
16322:
16298:
16282:
16278:
16240:
16201:
16098:
16070:
15945:
15917:
15729:
15647:
15604:
15534:
15449:
15404:
15402:
15372:
15320:
15268:
15237:
15200:
15162:
15097:
15061:
14992:
14884:
14873:
14863:
14834:
14794:
14719:
14708:
14698:
14678:
14638:
14474:
14452:
14396:
14356:
14286:
14254:
14136:
14126:
14093:
14012:
13993:
13954:
13944:
13931:
13921:
13888:
13852:
13842:
13809:
13753:
13734:
13695:
13693:
13668:
13656:
13637:
13600:
13588:
13569:
13533:
13531:
13508:
13394:
13384:
13351:
13252:is measured, and
13211:
13207:
13165:
13161:
13119:
13115:
13015:
13000:
12985:
12970:
12955:
12916:Noether's theorem
12896:
12866:
12856:
12830:
12801:
12752:
12742:
12657:
12625:
12615:
12589:
12519:
12480:
12470:
12437:
12427:
12397:
12369:
12348:
12310:
12295:
12285:
12254:
12244:
12220:
12202:
12166:
12156:
12132:
12110:
12095:
12085:
12056:
12046:
12022:
12000:
11990:
11960:
11932:
11915:
11888:
11870:
11860:
11829:
11793:
11756:
11746:
11722:
11704:
11694:
11663:
11635:
11623:
11591:
11581:
11555:
11516:
11491:
11466:
11428:
11408:which imply that
11379:
11322:
11292:
11230:
11221:
11181:
11132:
11075:
11045:
10983:
10978:
10959:
10915:
10878:
10868:
10850:
10792:
10755:
10725:
10683:
10677:
10633:
10588:
10546:
10460:
10440:
10405:
10395:
10364:
10302:
10277:
10220:
10053:
10040:
10018:
10005:
9956:
9903:
9891:
9848:
9809:
9797:
9747:
9703:
9691:
9626:Both Lagrangians
9602:
9590:
9538:
9533:
9479:
9418:
9364:
9324:
9150:
9042:
8951:
8919:
8917:
8888:
8878:
8850:
8828:
8799:
8797:
8700:
8690:
8659:
8637:
8566:
8496:
8486:
8458:
8436:
8396:
8386:
8353:
8331:
8219:
8149:
8139:
8111:
8089:
7820:
7796:
7541:
7531:
7503:
7481:
7388:coordinates, for
7340:of mechanics, of
7303:action principles
6990:
6980:
6954:
6932:
6804:
6794:
6704:
6694:
6668:
6628:
6618:
6587:
6552:
6337:
6294:
6250:
6219:
6204:
6194:
6152:
6049:
6039:
6013:
5991:
5922:
5892:
5882:
5856:
5748:
5718:
5708:
5682:
5584:
5554:
5544:
5518:
5433:
5403:
5393:
5367:
5345:
5180:
5013:
4862:
4291:Jacques Bernoulli
4256:Constraint force
4061:
4030:
3983:
3925:
3884:
3863:
3833:
3823:
3797:
3752:
3721:
3665:
3593:"Lagrangian form"
3541:, subject to the
3354:
3320:
3310:
3280:
3210:
3142:
3103:
3091:
3015:
2974:
2583:
2573:
2549:
2539:
2515:
2505:
2476:
2466:
2434:
2409:
2384:
2354:
2274:
2204:
2194:
2166:
2144:
1745:Newtonian gravity
1604:{\displaystyle k}
1584:{\displaystyle N}
1505:
1402:
1316:
1263:
1216:
905:action functional
783:
782:
530:Centrifugal force
525:Centripetal force
481:Euler's equations
466:Relative velocity
242:Moment of inertia
72:
46:
29159:
29055:Chemical physics
28995:Particle physics
28921:Classical optics
28804:
28797:
28790:
28781:
28780:
28562:Feynman integral
28545:
28544:
28505:Potential theory
28394:random variables
28284:Fourier analysis
28267:Operator algebra
28194:Clifford algebra
28146:Computer algebra
28072:
28071:
27979:
27972:
27965:
27956:
27955:
27871:
27864:
27857:
27848:
27847:
27825:
27823:
27822:
27787:
27753:
27732:
27720:
27709:
27688:
27669:
27650:
27629:
27608:
27582:
27561:
27535:
27514:
27480:
27459:
27430:
27411:
27388:
27367:
27346:
27325:
27304:
27281:
27260:
27231:
27210:
27187:
27159:
27137:
27118:
27097:
27076:
27065:. Elsevier Ltd.
27059:Lifshitz, Evgeny
27050:
27024:
27005:
26994:
26974:
26968:
26959:
26954:
26948:
26943:
26937:
26932:
26926:
26921:
26915:
26909:
26903:
26897:
26891:
26885:
26879:
26873:
26867:
26861:
26855:
26852:Padmanabhan 2000
26849:
26843:
26837:
26831:
26830:
26802:
26796:
26790:
26784:
26778:
26772:
26766:
26760:
26754:
26748:
26742:
26736:
26730:
26721:
26720:, pp. 68–70
26715:
26709:
26704:
26698:
26692:
26683:
26682:
26642:
26636:
26630:
26624:
26618:
26612:
26606:
26600:
26599:
26579:
26573:
26567:
26561:
26555:
26549:
26543:
26537:
26531:
26525:
26519:
26513:
26507:
26501:
26495:
26489:
26488:, pp. 16–18
26483:
26477:
26471:
26465:
26459:
26453:
26447:
26441:
26435:
26429:
26419:
26413:
26408:
26402:
26396:
26390:
26384:
26378:
26377:
26376:
26375:
26342:
26336:
26330:
26324:
26317:
26311:
26310:
26278:
26272:
26266:
26253:
26247:
26241:
26240:
26229:. pp. 4–7.
26218:
26209:
26208:
26206:
26205:
26196:
26187:
26178:
26177:
26175:
26174:
26151:
26142:
26136:
26130:
26129:, pp. 18–20
26124:
26118:
26112:
26106:
26099:
26083:
26080:
26074:
26063:
26061:
26060:
26055:
26047:
26046:
26034:
26033:
26010:
26009:
25997:
25996:
25973:
25972:
25960:
25959:
25935:
25933:
25932:
25927:
25919:
25918:
25913:
25901:
25900:
25895:
25881:
25875:
25868:
25862:
25860:
25858:
25857:
25852:
25850:
25848:
25847:
25846:
25841:
25840:
25835:
25830:
25819:
25817:
25815:
25811:
25805:
25800:
25795:
25793:
25792:
25791:
25786:
25773:
25768:
25766:
25765:
25764:
25759:
25746:
25732:
25695:Lagrangian point
25647:
25645:Astronomy portal
25642:
25641:
25640:
25583:
25581:
25580:
25575:
25573:
25568:
25567:
25562:
25555:
25554:
25511:
25509:
25508:
25503:
25492:
25484:
25483:
25475:
25454:
25453:
25437:
25277:
25254:
25190:
25188:
25187:
25182:
25174:
25172:
25171:
25170:
25165:
25164:
25156:
25148:
25140:
25138:
25137:
25132:
25131:
25123:
25118:
25113:
25065:
25056:
25050:
25044:
25038:
25032:
25026:
25010:prescription in
25008:minimal coupling
25003:
25001:
25000:
24995:
24990:
24979:
24978:
24973:
24968:
24959:
24957:
24956:
24955:
24950:
24945:
24938:
24930:
24925:
24913:
24906:kinetic momentum
24903:
24890:
24888:
24887:
24882:
24877:
24875:
24867:
24859:
24842:
24838:
24837:
24835:
24824:
24819:
24811:
24810:
24805:
24800:
24770:
24755:
24753:
24752:
24747:
24742:
24741:
24733:
24711:
24703:
24695:
24677:
24675:
24674:
24669:
24667:
24659:
24658:
24653:
24648:
24639:
24628:
24627:
24622:
24617:
24591:minimal coupling
24588:
24586:
24585:
24580:
24566:
24558:
24557:
24552:
24547:
24537:
24536:
24531:
24530:
24525:
24520:
24513:
24504:
24486:
24484:
24483:
24478:
24473:
24465:
24457:
24448:
24446:
24438:
24437:
24428:
24420:
24409:
24397:
24375:
24353:
24331:
24312:scalar potential
24305:
24295:charged particle
24270:Electromagnetism
24266:
24264:
24263:
24258:
24250:
24248:
24247:
24246:
24241:
24240:
24232:
24224:
24216:
24211:
24209:
24208:
24207:
24194:
24186:
24181:
24177:
24175:
24174:
24173:
24168:
24167:
24159:
24151:
24143:
24137:
24135:
24131:
24125:
24120:
24111:
24109:
24108:
24103:
24101:
24099:
24098:
24097:
24092:
24091:
24083:
24075:
24067:
24062:
24060:
24059:
24058:
24045:
24037:
24029:
24028:
23999:
23997:
23996:
23991:
23986:
23985:
23980:
23979:
23971:
23967:
23966:
23961:
23960:
23952:
23948:
23947:
23934:
23929:
23913:
23908:
23893:
23885:
23783:
23766:
23758:
23756:
23755:
23750:
23745:
23743:
23742:
23741:
23728:
23727:
23718:
23713:
23712:
23707:
23706:
23698:
23685:
23684:
23683:
23655:
23644:
23638:
23636:
23635:
23630:
23625:
23623:
23622:
23621:
23608:
23607:
23598:
23593:
23591:
23587:
23581:
23577:
23571:
23563:
23562:
23554:
23541:
23524:
23522:
23521:
23516:
23511:
23510:
23502:
23493:
23491:
23483:
23475:
23470:
23469:
23464:
23463:
23455:
23438:
23432:
23426:
23415:
23407:
23405:
23404:
23399:
23388:
23387:
23379:
23376:
23375:
23360:
23358:
23357:
23356:
23348:
23341:
23340:
23339:
23336:
23326:
23321:
23320:
23304:
23298:
23296:
23295:
23290:
23270:
23266:
23265:
23264:
23259:
23258:
23250:
23246:
23245:
23233:
23232:
23227:
23226:
23218:
23206:
23198:
23193:
23192:
23189:
23176:
23174:
23163:
23148:
23146:
23145:
23140:
23129:
23128:
23123:
23118:
23105:
23096:
23083:
23070:
23068:
23052:
23040:
23030:
22996:
22976:
22974:
22973:
22968:
22965:
22964:
22963:
22960:
22953:
22948:
22944:
22939:
22934:
22920:
22919:
22914:
22913:
22908:
22903:
22896:
22888:
22878:
22877:
22876:
22873:
22866:
22861:
22860:
22859:
22854:
22853:
22848:
22843:
22836:
22828:
22809:
22759:
22735:
22726:
22717:
22708:
22699:
22683:Two-body problem
22667:
22665:
22664:
22659:
22651:
22650:
22642:
22624:
22622:
22621:
22616:
22608:
22607:
22599:
22586:
22584:
22583:
22578:
22561:
22553:
22539:
22534:
22526:
22524:
22519:
22518:
22510:
22501:and simplifying
22500:
22498:
22497:
22492:
22463:
22462:
22454:
22451:
22450:
22442:
22427:
22423:
22419:
22418:
22410:
22407:
22406:
22382:
22381:
22373:
22359:
22357:
22353:
22347:
22342:
22327:
22325:
22324:
22319:
22302:
22301:
22296:
22295:
22287:
22265:
22264:
22256:
22244:
22243:
22235:
22210:
22208:
22207:
22202:
22190:
22188:
22187:
22182:
22168:
22167:
22159:
22147:
22146:
22138:
22117:
22115:
22114:
22113:
22105:
22098:
22090:
22085:
22084:
22062:
22060:
22059:
22054:
22052:
22024:
22023:
22018:
22017:
22009:
22005:
22004:
21992:
21984:
21970:
21969:
21961:
21955:
21954:
21946:
21937:
21936:
21931:
21930:
21922:
21918:
21914:
21899:
21891:
21883:
21858:
21854:
21853:
21852:
21847:
21843:
21833:
21832:
21824:
21809:
21808:
21803:
21799:
21789:
21788:
21780:
21771:
21770:
21762:
21745:
21737:
21732:
21731:
21726:
21725:
21717:
21710:
21702:
21694:
21660:
21658:
21657:
21652:
21650:
21649:
21648:
21610:
21608:
21607:
21602:
21600:
21596:
21594:
21589:
21588:
21573:
21572:
21564:
21556:
21551:
21550:
21535:
21534:
21526:
21514:
21506:
21501:
21500:
21495:
21494:
21486:
21479:
21471:
21456:
21454:
21453:
21448:
21436:
21434:
21433:
21428:
21414:
21412:
21411:
21406:
21404:
21388:
21387:
21379:
21370:
21369:
21368:
21353:
21352:
21344:
21315:
21314:
21313:
21293:
21280:
21279:
21271:
21262:
21261:
21253:
21247:
21246:
21245:
21230:
21229:
21221:
21189:
21188:
21187:
21167:
21156:
21154:
21153:
21148:
21136:
21134:
21133:
21128:
21117:-direction. Let
21116:
21114:
21113:
21108:
21032:
21030:
21029:
21024:
21019:
21018:
21010:
21001:
21000:
20991:
20990:
20975:
20973:
20972:
20971:
20963:
20956:
20948:
20943:
20942:
20922:
20920:
20919:
20914:
20903:
20902:
20894:
20884:
20883:
20874:
20873:
20858:
20856:
20852:
20846:
20841:
20833:
20831:
20830:
20825:
20814:
20813:
20808:
20807:
20799:
20776:
20775:
20757:
20756:
20748:
20745:
20744:
20729:
20727:
20723:
20717:
20712:
20704:
20702:
20701:
20696:
20685:
20683:
20675:
20667:
20659:
20658:
20653:
20652:
20644:
20633:
20632:
20620:
20619:
20614:
20613:
20605:
20589:
20588:
20580:
20567:
20565:
20564:
20559:
20536:
20535:
20530:
20529:
20521:
20510:
20509:
20500:
20499:
20487:
20486:
20481:
20480:
20472:
20468:
20467:
20455:
20454:
20449:
20448:
20440:
20433:
20425:
20398:
20370:
20368:
20367:
20362:
20357:
20346:
20345:
20340:
20335:
20314:
20312:
20311:
20306:
20301:
20299:
20291:
20283:
20275:
20274:
20266:
20253:
20251:
20250:
20245:
20240:
20239:
20231:
20222:
20218:
20216:
20215:
20214:
20206:
20199:
20191:
20185:
20183:
20179:
20173:
20168:
20162:
20161:
20153:
20144:
20142:
20141:
20140:
20132:
20125:
20117:
20111:
20109:
20101:
20093:
20085:
20083:
20075:
20067:
20058:
20056:
20055:
20050:
20045:
20043:
20035:
20027:
20022:
20018:
20016:
20015:
20014:
20006:
19999:
19991:
19985:
19983:
19979:
19973:
19968:
19949:
19947:
19946:
19941:
19906:
19905:
19900:
19899:
19891:
19884:
19883:
19878:
19877:
19869:
19862:
19861:
19856:
19855:
19847:
19837:
19829:
19821:
19820:
19812:
19806:
19805:
19797:
19791:
19790:
19782:
19738:
19736:
19735:
19730:
19722:
19711:
19700:
19687:scalar potential
19656:
19654:
19653:
19648:
19631:
19630:
19622:
19609:
19607:
19606:
19601:
19599:
19598:
19590:
19580:
19578:
19577:
19572:
19555:
19554:
19546:
19533:
19531:
19530:
19525:
19523:
19522:
19514:
19504:
19502:
19501:
19496:
19485:. If the matrix
19484:
19482:
19481:
19476:
19474:
19472:
19471:
19470:
19465:
19464:
19456:
19449:
19448:
19443:
19442:
19434:
19426:
19422:
19421:
19411:
19406:
19405:
19384:
19382:
19381:
19376:
19374:
19367:
19366:
19361:
19360:
19352:
19348:
19346:
19345:
19344:
19332:
19331:
19326:
19325:
19317:
19309:
19305:
19304:
19294:
19291:
19279:
19277:
19269:
19261:
19256:
19254:
19253:
19252:
19239:
19231:
19226:
19225:
19220:
19219:
19211:
19198:
19197:
19189:
19177:
19176:
19163:
19151:
19141:
19139:
19138:
19137:
19124:
19116:
19111:
19109:
19101:
19093:
19088:
19086:
19078:
19077:
19076:
19071:
19070:
19062:
19054:
19052:
19050:
19049:
19048:
19043:
19042:
19034:
19027:
19026:
19021:
19020:
19012:
19004:
19000:
18999:
18989:
18984:
18982:
18974:
18973:
18972:
18959:
18957:
18955:
18954:
18953:
18948:
18947:
18939:
18932:
18931:
18918:
18914:
18913:
18903:
18900:
18890:
18888:
18887:
18886:
18873:
18865:
18860:
18858:
18857:
18856:
18851:
18850:
18842:
18834:
18826:
18824:
18822:
18811:
18808:
18773:
18771:
18770:
18765:
18760:
18759:
18744:
18743:
18731:
18730:
18693:
18691:
18690:
18685:
18680:
18679:
18667:
18666:
18607:
18605:
18604:
18599:
18594:
18593:
18592:
18584:
18575:
18571:
18566:
18558:
18548:
18543:
18535:
18508:
18506:
18505:
18500:
18495:
18494:
18493:
18485:
18461:
18460:
18448:
18446:
18445:
18436:
18435:
18426:
18376:
18374:
18373:
18368:
18363:
18362:
18357:
18342:
18341:
18336:
18327:
18326:
18321:
18309:
18308:
18293:
18292:
18287:
18269:
18268:
18263:
18251:
18250:
18245:
18223:
18173:
18171:
18170:
18165:
18133:
18131:
18130:
18125:
18120:
18118:constant of time
18117:
18094:
18093:
18088:
18083:
18068:
18046:
18028:
18026:
18025:
18020:
18018:
18014:
18012:
18004:
17996:
17991:
17987:
17980:
17979:
17971:
17968:
17966:
17965:
17964:
17956:
17949:
17941:
17934:
17932:
17921:
17912:
17910:
17902:
17894:
17885:
17884:
17876:
17870:
17868:
17867:
17866:
17858:
17851:
17843:
17841:
17840:
17835:
17830:
17824:
17823:
17818:
17813:
17810:
17806:
17804:
17803:
17802:
17794:
17787:
17779:
17773:
17771:
17760:
17751:
17749:
17741:
17733:
17724:
17722:
17714:
17706:
17701:
17699:
17698:
17697:
17689:
17682:
17674:
17672:
17671:
17666:
17661:
17655:
17653:
17652:
17643:
17635:
17633:
17632:
17627:
17622:
17612:
17610:
17602:
17594:
17577:
17567:
17545:
17543:
17542:
17537:
17521:
17519:
17518:
17513:
17488:
17486:
17485:
17480:
17468:
17466:
17465:
17460:
17455:
17453:
17452:
17451:
17446:
17441:
17434:
17426:
17424:
17423:
17418:
17413:
17407:
17405:
17404:
17403:
17398:
17393:
17386:
17378:
17376:
17375:
17370:
17365:
17355:
17353:
17352:
17347:
17345:
17344:
17339:
17334:
17324:
17322:
17321:
17316:
17314:
17313:
17308:
17303:
17293:
17291:
17290:
17285:
17282:
17280:
17279:
17278:
17273:
17272:
17264:
17256:
17248:
17238:
17236:
17235:
17230:
17227:
17226:
17221:
17220:
17212:
17205:
17203:
17202:
17201:
17196:
17195:
17187:
17179:
17171:
17168:
17163:
17141:
17139:
17138:
17133:
17118:
17116:
17115:
17110:
17105:
17104:
17099:
17094:
17085:
17083:
17082:
17077:
17072:
17060:
17058:
17057:
17052:
17047:
17045:
17044:
17043:
17038:
17033:
17026:
17018:
17010:
17002:
17001:
16992:
16990:
16989:
16988:
16983:
16978:
16971:
16963:
16954:
16952:
16951:
16946:
16932:
16931:
16926:
16921:
16915:
16889:
16888:
16883:
16878:
16872:
16851:
16849:
16848:
16843:
16829:
16828:
16823:
16818:
16812:
16789:
16788:
16783:
16778:
16772:
16752:
16750:
16749:
16744:
16733:
16731:
16723:
16715:
16710:
16709:
16704:
16699:
16693:
16691:
16690:
16689:
16684:
16679:
16672:
16664:
16659:
16651:
16649:
16648:
16639:
16631:
16617:
16616:
16611:
16606:
16600:
16577:
16575:
16574:
16569:
16567:
16560:
16558:
16550:
16542:
16537:
16536:
16531:
16526:
16517:
16509:
16508:
16499:
16497:
16496:
16495:
16490:
16485:
16478:
16470:
16465:
16457:
16453:
16449:
16448:
16443:
16438:
16432:
16421:
16419:
16418:
16417:
16412:
16407:
16400:
16392:
16384:
16376:
16375:
16366:
16364:
16363:
16354:
16346:
16333:
16323:
16321:
16313:
16305:
16300:
16299:
16294:
16289:
16283:
16281:
16280:
16279:
16274:
16269:
16262:
16254:
16249:
16241:
16239:
16238:
16229:
16221:
16203:
16202:
16197:
16192:
16186:
16159:
16157:
16156:
16151:
16149:
16142:
16133:
16128:
16111:
16107:
16106:
16105:
16100:
16099:
16091:
16087:
16086:
16085:
16079:
16078:
16071:
16069:
16068:
16067:
16055:
16054:
16041:
16040:
16039:
16030:
16029:
16019:
16016:
16011:
15986:
15981:
15958:
15954:
15953:
15952:
15947:
15946:
15938:
15934:
15933:
15932:
15926:
15925:
15918:
15916:
15915:
15914:
15902:
15901:
15888:
15887:
15886:
15877:
15876:
15866:
15863:
15858:
15843:
15842:
15829:
15824:
15797:
15792:
15787:
15764:
15760:
15759:
15758:
15746:
15745:
15744:
15738:
15737:
15730:
15728:
15727:
15726:
15714:
15713:
15700:
15699:
15698:
15689:
15688:
15678:
15675:
15670:
15655:
15654:
15649:
15648:
15640:
15635:
15630:
15606:
15605:
15600:
15595:
15591:
15586:
15569:
15565:
15564:
15563:
15551:
15550:
15549:
15543:
15542:
15535:
15533:
15532:
15531:
15519:
15518:
15505:
15504:
15503:
15494:
15493:
15483:
15480:
15475:
15451:
15450:
15445:
15440:
15431:
15423:
15422:
15406:
15405:
15403:
15400:
15398:
15393:
15385:
15374:
15373:
15368:
15363:
15357:
15335:
15333:
15332:
15327:
15322:
15321:
15316:
15311:
15302:
15294:
15293:
15281:
15270:
15269:
15264:
15259:
15253:
15239:
15238:
15233:
15228:
15215:
15213:
15212:
15207:
15202:
15201:
15196:
15191:
15185:
15177:
15176:
15164:
15163:
15158:
15153:
15143:
15141:
15140:
15135:
15133:
15118:
15116:
15115:
15110:
15105:
15104:
15099:
15098:
15090:
15079:
15077:
15076:
15071:
15069:
15068:
15063:
15062:
15054:
15043:
15041:
15040:
15035:
15031:
15026:
15009:
15005:
15004:
15003:
14997:
14993:
14991:
14990:
14989:
14976:
14975:
14974:
14961:
14944:
14936:
14935:
14918:
14916:
14915:
14910:
14905:
14901:
14900:
14892:
14891:
14886:
14885:
14877:
14871:
14870:
14869:
14861:
14860:
14856:
14855:
14851:
14839:
14835:
14833:
14832:
14831:
14815:
14802:
14801:
14796:
14795:
14787:
14782:
14777:
14750:
14748:
14747:
14742:
14740:
14736:
14735:
14727:
14726:
14721:
14720:
14712:
14706:
14705:
14704:
14696:
14695:
14691:
14690:
14689:
14683:
14679:
14677:
14676:
14675:
14659:
14646:
14645:
14640:
14639:
14631:
14626:
14621:
14590:
14588:
14587:
14582:
14577:
14569:
14568:
14552:
14550:
14549:
14544:
14539:
14528:
14520:
14519:
14507:
14492:
14470:
14468:
14467:
14462:
14460:
14459:
14454:
14453:
14445:
14438:
14420:
14418:
14417:
14412:
14398:
14397:
14392:
14387:
14381:
14358:
14357:
14352:
14347:
14341:
14323:
14297:
14295:
14294:
14289:
14287:
14285:
14284:
14283:
14270:
14269:
14260:
14255:
14253:
14252:
14251:
14238:
14237:
14228:
14223:
14222:
14203:
14201:
14200:
14195:
14142:
14138:
14137:
14135:
14134:
14133:
14128:
14127:
14119:
14111:
14103:
14101:
14100:
14095:
14094:
14086:
14081:
14076:
14044:
14042:
14041:
14036:
14025:
14021:
14020:
14019:
14014:
14013:
14005:
14001:
14000:
13995:
13994:
13986:
13982:
13981:
13968:
13945:
13937:
13932:
13930:
13929:
13928:
13923:
13922:
13914:
13906:
13898:
13896:
13895:
13890:
13889:
13881:
13876:
13871:
13853:
13851:
13850:
13849:
13844:
13843:
13835:
13827:
13819:
13817:
13816:
13811:
13810:
13802:
13797:
13792:
13771:
13769:
13768:
13763:
13761:
13760:
13755:
13754:
13746:
13742:
13741:
13736:
13735:
13727:
13723:
13722:
13709:
13694:
13686:
13681:
13677:
13676:
13675:
13670:
13669:
13661:
13657:
13655:
13654:
13653:
13640:
13639:
13638:
13630:
13623:
13613:
13609:
13608:
13607:
13602:
13601:
13593:
13589:
13587:
13586:
13585:
13572:
13571:
13570:
13562:
13555:
13547:
13532:
13524:
13519:
13518:
13509:
13501:
13486:
13484:
13483:
13478:
13473:
13472:
13454:
13453:
13441:
13433:
13421:
13419:
13418:
13413:
13402:
13401:
13395:
13393:
13392:
13391:
13386:
13385:
13377:
13369:
13361:
13359:
13358:
13353:
13352:
13344:
13339:
13334:
13319:
13318:
13295:
13293:
13292:
13287:
13225:
13223:
13222:
13217:
13212:
13210:
13209:
13208:
13200:
13193:
13185:
13180:
13179:
13166:
13164:
13163:
13162:
13154:
13147:
13139:
13134:
13133:
13120:
13118:
13117:
13116:
13108:
13101:
13093:
13088:
13087:
13039:
13037:
13036:
13031:
13017:
13016:
13008:
13002:
13001:
12993:
12987:
12986:
12978:
12972:
12971:
12963:
12957:
12956:
12948:
12913:
12911:
12910:
12905:
12897:
12895:
12894:
12893:
12880:
12872:
12867:
12865:
12864:
12863:
12858:
12857:
12849:
12841:
12833:
12831:
12829:
12825:
12819:
12814:
12809:
12808:
12803:
12802:
12794:
12766:
12764:
12763:
12758:
12753:
12751:
12750:
12749:
12744:
12743:
12735:
12727:
12719:
12714:
12713:
12674:
12672:
12671:
12666:
12658:
12656:
12655:
12654:
12641:
12637:
12631:
12626:
12624:
12623:
12622:
12617:
12616:
12608:
12600:
12592:
12590:
12588:
12577:
12568:
12566:
12565:
12560:
12540:
12538:
12537:
12532:
12530:
12520:
12518:
12517:
12516:
12503:
12502:
12501:
12488:
12486:
12482:
12481:
12479:
12478:
12477:
12472:
12471:
12463:
12455:
12447:
12442:
12438:
12436:
12435:
12434:
12429:
12428:
12420:
12412:
12404:
12398:
12396:
12385:
12377:
12358:
12354:
12353:
12349:
12347:
12346:
12345:
12332:
12331:
12330:
12317:
12311:
12309:
12298:
12296:
12294:
12293:
12292:
12287:
12286:
12278:
12270:
12266:
12260:
12255:
12253:
12252:
12251:
12246:
12245:
12237:
12229:
12228:
12227:
12222:
12221:
12213:
12205:
12203:
12201:
12200:
12199:
12186:
12182:
12176:
12171:
12167:
12165:
12164:
12163:
12158:
12157:
12149:
12141:
12140:
12139:
12134:
12133:
12125:
12117:
12111:
12109:
12098:
12096:
12094:
12093:
12092:
12087:
12086:
12078:
12070:
12062:
12057:
12055:
12054:
12053:
12048:
12047:
12039:
12031:
12030:
12029:
12024:
12023:
12015:
12007:
12005:
12001:
11999:
11998:
11997:
11992:
11991:
11983:
11975:
11967:
11961:
11959:
11948:
11940:
11921:
11917:
11916:
11914:
11913:
11912:
11907:
11897:
11896:
11895:
11890:
11889:
11881:
11873:
11871:
11869:
11868:
11867:
11862:
11861:
11853:
11845:
11841:
11835:
11830:
11828:
11827:
11826:
11821:
11811:
11810:
11809:
11796:
11794:
11792:
11791:
11790:
11777:
11773:
11767:
11762:
11758:
11757:
11755:
11754:
11753:
11748:
11747:
11739:
11731:
11730:
11729:
11724:
11723:
11715:
11707:
11705:
11703:
11702:
11701:
11696:
11695:
11687:
11679:
11671:
11664:
11662:
11651:
11643:
11624:
11622:
11621:
11620:
11607:
11603:
11597:
11592:
11590:
11589:
11588:
11583:
11582:
11574:
11566:
11558:
11556:
11554:
11543:
11527:
11525:
11524:
11519:
11517:
11515:
11514:
11513:
11500:
11499:
11498:
11493:
11492:
11484:
11476:
11471:
11467:
11465:
11464:
11463:
11450:
11449:
11448:
11435:
11429:
11427:
11416:
11407:
11405:
11404:
11399:
11388:
11380:
11378:
11377:
11376:
11357:
11356:
11355:
11346:
11345:
11335:
11330:
11329:
11324:
11323:
11315:
11301:
11293:
11291:
11290:
11289:
11277:
11276:
11263:
11262:
11261:
11252:
11251:
11241:
11238:
11226:
11222:
11220:
11219:
11218:
11205:
11204:
11203:
11202:
11188:
11182:
11180:
11169:
11160:
11158:
11157:
11152:
11141:
11133:
11131:
11124:
11123:
11110:
11109:
11108:
11099:
11098:
11088:
11083:
11082:
11077:
11076:
11068:
11054:
11046:
11044:
11043:
11042:
11030:
11029:
11016:
11015:
11014:
11005:
11004:
10994:
10991:
10979:
10977:
10976:
10975:
10962:
10961:
10960:
10955:
10954:
10945:
10938:
10926:
10924:
10923:
10918:
10916:
10914:
10913:
10912:
10899:
10898:
10897:
10884:
10879:
10877:
10876:
10875:
10870:
10869:
10861:
10853:
10852:
10851:
10846:
10845:
10836:
10829:
10820:
10818:
10817:
10812:
10801:
10793:
10791:
10783:
10782:
10781:
10768:
10763:
10762:
10757:
10756:
10748:
10734:
10726:
10724:
10723:
10722:
10709:
10708:
10707:
10694:
10691:
10679:
10678:
10673:
10672:
10663:
10653:
10651:
10650:
10645:
10634:
10632:
10624:
10623:
10622:
10609:
10604:
10603:
10602:
10589:
10587:
10586:
10585:
10572:
10571:
10570:
10557:
10554:
10542:
10541:
10540:
10520:
10518:
10517:
10512:
10501:
10493:
10492:
10480:
10479:
10454:
10452:
10451:
10446:
10441:
10439:
10438:
10437:
10424:
10423:
10411:
10406:
10404:
10403:
10402:
10397:
10396:
10388:
10380:
10379:
10367:
10365:
10363:
10359:
10353:
10348:
10335:
10333:
10332:
10327:
10304:
10303:
10298:
10293:
10287:
10279:
10278:
10273:
10268:
10253:
10245:
10222:
10221:
10216:
10211:
10205:
10197:
10178:
10159:
10117:
10096:
10075:
10073:
10072:
10067:
10065:
10055:
10054:
10051:
10042:
10041:
10038:
10020:
10019:
10016:
10007:
10006:
10003:
9985:
9968:
9957:
9955:
9951:
9945:
9926:
9915:
9909:
9906:
9905:
9904:
9901:
9894:
9893:
9892:
9889:
9850:
9849:
9844:
9839:
9824:
9812:
9811:
9810:
9807:
9800:
9799:
9798:
9795:
9778:
9749:
9748:
9743:
9738:
9723:
9715:
9706:
9705:
9704:
9701:
9694:
9693:
9692:
9689:
9668:
9660:
9623:
9621:
9620:
9615:
9610:
9609:
9604:
9603:
9595:
9591:
9589:
9588:
9587:
9574:
9564:
9549:
9546:
9534:
9532:
9524:
9514:
9499:
9490:
9488:
9487:
9482:
9480:
9478:
9474:
9468:
9458:
9447:
9441:
9432:
9430:
9429:
9424:
9419:
9417:
9413:
9407:
9397:
9386:
9380:
9366:
9365:
9360:
9355:
9349:
9326:
9325:
9320:
9315:
9309:
9301:
9286:
9271:
9250:
9229:
9203:
9164:
9162:
9161:
9156:
9151:
9149:
9148:
9147:
9142:
9132:
9131:
9130:
9117:
9115:
9114:
9104:
9099:
9081:
9080:
9075:
9062:
9060:
9059:
9054:
9043:
9041:
9040:
9039:
9034:
9024:
9023:
9022:
9009:
9007:
9006:
8996:
8991:
8972:
8971:
8970:
8965:
8958:
8953:
8952:
8950:
8949:
8948:
8943:
8933:
8925:
8912:
8911:
8910:
8905:
8895:
8890:
8889:
8887:
8886:
8885:
8880:
8879:
8874:
8869:
8861:
8853:
8851:
8849:
8845:
8839:
8834:
8829:
8827:
8826:
8825:
8820:
8810:
8802:
8789:
8762:
8760:
8759:
8754:
8737:
8736:
8731:
8722:
8721:
8701:
8699:
8698:
8697:
8692:
8691:
8683:
8675:
8674:
8662:
8660:
8658:
8654:
8648:
8643:
8638:
8636:
8635:
8634:
8621:
8620:
8608:
8586:
8584:
8583:
8578:
8567:
8565:
8564:
8563:
8558:
8548:
8547:
8546:
8533:
8531:
8530:
8520:
8515:
8497:
8495:
8494:
8493:
8488:
8487:
8482:
8477:
8469:
8461:
8459:
8457:
8453:
8447:
8442:
8437:
8435:
8434:
8433:
8428:
8418:
8410:
8397:
8395:
8394:
8393:
8388:
8387:
8382:
8377:
8369:
8368:
8356:
8354:
8352:
8348:
8342:
8337:
8332:
8330:
8329:
8328:
8323:
8313:
8312:
8300:
8268:
8266:
8265:
8260:
8249:
8243:
8242:
8237:
8225:
8221:
8220:
8218:
8217:
8216:
8211:
8201:
8200:
8199:
8186:
8184:
8183:
8173:
8168:
8150:
8148:
8147:
8146:
8141:
8140:
8135:
8130:
8122:
8114:
8112:
8110:
8106:
8100:
8095:
8090:
8088:
8087:
8086:
8081:
8071:
8063:
8055:
8050:
8034:
8033:
8032:
8022:
8021:
8020:
8000:
7995:
7983:
7982:
7981:
7971:
7970:
7969:
7933:
7931:
7930:
7925:
7911:
7910:
7905:
7896:
7895:
7877:
7876:
7866:
7861:
7828:
7827:
7822:
7821:
7816:
7811:
7804:
7803:
7798:
7797:
7792:
7787:
7774:
7773:
7768:
7759:
7758:
7753:
7738:
7706:
7649:
7590:
7588:
7587:
7582:
7571:
7565:
7564:
7559:
7547:
7543:
7542:
7540:
7539:
7538:
7533:
7532:
7527:
7522:
7514:
7506:
7504:
7502:
7498:
7492:
7487:
7482:
7480:
7479:
7478:
7473:
7463:
7455:
7447:
7442:
7426:
7425:
7424:
7414:
7413:
7412:
7326:Daniel Bernoulli
7297:
7295:
7294:
7289:
7267:angular momentum
7244:
7242:
7241:
7236:
7228:
7218:
7217:
7216:
7206:
7205:
7204:
7171:
7169:
7168:
7163:
7152:
7139:
7138:
7137:
7127:
7126:
7125:
7072:
7035:
7033:
7032:
7027:
7025:
7015:
7009:
7008:
6996:
6992:
6991:
6989:
6988:
6987:
6982:
6981:
6973:
6965:
6957:
6955:
6953:
6949:
6943:
6938:
6933:
6931:
6930:
6929:
6916:
6908:
6900:
6895:
6879:
6878:
6877:
6867:
6866:
6865:
6847:
6846:
6845:
6835:
6834:
6833:
6823:
6819:
6818:
6817:
6805:
6803:
6802:
6801:
6796:
6795:
6787:
6779:
6771:
6762:
6757:
6736:
6729:
6723:
6719:
6718:
6717:
6705:
6703:
6702:
6701:
6696:
6695:
6687:
6679:
6671:
6669:
6667:
6663:
6657:
6652:
6647:
6643:
6642:
6641:
6629:
6627:
6626:
6625:
6620:
6619:
6611:
6603:
6595:
6588:
6586:
6582:
6576:
6571:
6566:
6565:
6553:
6551:
6550:
6549:
6536:
6528:
6520:
6515:
6499:
6498:
6497:
6487:
6486:
6485:
6461:
6448:
6447:
6446:
6436:
6435:
6434:
6351:
6349:
6348:
6343:
6338:
6336:
6332:
6326:
6322:
6321:
6306:
6300:
6295:
6293:
6289:
6283:
6282:
6281:
6272:
6266:
6258:
6257:
6252:
6251:
6243:
6232:
6228:
6227:
6226:
6221:
6220:
6212:
6205:
6203:
6202:
6201:
6196:
6195:
6187:
6179:
6171:
6166:
6165:
6153:
6151:
6150:
6149:
6136:
6128:
6120:
6115:
6066:
6064:
6063:
6058:
6050:
6048:
6047:
6046:
6041:
6040:
6032:
6024:
6016:
6014:
6012:
6008:
6002:
5997:
5992:
5990:
5989:
5988:
5975:
5967:
5950:
5936:
5934:
5933:
5928:
5923:
5921:
5920:
5919:
5906:
5898:
5893:
5891:
5890:
5889:
5884:
5883:
5875:
5867:
5859:
5857:
5855:
5851:
5845:
5840:
5835:
5834:
5759:
5757:
5756:
5751:
5749:
5747:
5746:
5745:
5732:
5724:
5719:
5717:
5716:
5715:
5710:
5709:
5701:
5693:
5685:
5683:
5681:
5677:
5671:
5666:
5661:
5660:
5627:
5625:
5624:
5619:
5608:
5607:
5595:
5591:
5590:
5586:
5585:
5583:
5582:
5581:
5568:
5560:
5555:
5553:
5552:
5551:
5546:
5545:
5537:
5529:
5521:
5519:
5517:
5513:
5507:
5502:
5492:
5491:
5476:
5471:
5447:
5445:
5444:
5439:
5434:
5432:
5431:
5430:
5417:
5409:
5404:
5402:
5401:
5400:
5395:
5394:
5386:
5378:
5370:
5368:
5366:
5362:
5356:
5351:
5346:
5344:
5343:
5342:
5329:
5328:
5327:
5322:
5312:
5307:
5306:
5301:
5295:
5294:
5284:
5279:
5254:
5252:
5251:
5246:
5241:
5240:
5228:
5227:
5217:
5212:
5194:
5193:
5181:
5179:
5178:
5177:
5164:
5163:
5162:
5157:
5147:
5144:
5139:
5121:
5120:
5115:
5108:
5103:
5085:
5084:
5079:
5067:
5066:
5061:
5054:
5049:
5027:
5025:
5024:
5019:
5014:
5012:
5011:
5010:
4997:
4996:
4995:
4990:
4980:
4975:
4974:
4969:
4962:
4957:
4939:
4938:
4889:
4887:
4886:
4881:
4876:
4875:
4863:
4861:
4860:
4859:
4846:
4845:
4844:
4839:
4829:
4826:
4821:
4803:
4802:
4797:
4765:
4720:
4649:
4647:
4646:
4641:
4633:
4632:
4627:
4612:
4611:
4606:
4600:
4599:
4587:
4586:
4581:
4571:
4566:
4544:
4542:
4541:
4536:
4525:
4524:
4519:
4507:
4506:
4501:
4494:
4489:
4436:
4434:
4433:
4428:
4420:
4419:
4414:
4399:
4398:
4393:
4387:
4386:
4374:
4373:
4368:
4359:
4358:
4353:
4343:
4338:
4247:
4235:
4201:
4199:
4198:
4193:
4188:
4180:
4172:
4136:curved spacetime
4133:
4116:
4072:
4070:
4069:
4064:
4062:
4060:
4056:
4050:
4049:
4048:
4039:
4033:
4031:
4029:
4025:
4019:
4018:
4017:
4008:
4002:
4000:
3999:
3984:
3976:
3939:
3937:
3936:
3931:
3926:
3924:
3920:
3914:
3913:
3912:
3903:
3897:
3892:
3891:
3886:
3885:
3877:
3869:
3865:
3864:
3862:
3861:
3860:
3847:
3839:
3834:
3832:
3831:
3830:
3825:
3824:
3816:
3808:
3800:
3798:
3796:
3792:
3786:
3781:
3774:
3773:
3758:
3754:
3753:
3751:
3747:
3741:
3740:
3739:
3730:
3724:
3722:
3720:
3716:
3710:
3709:
3708:
3699:
3693:
3691:
3690:
3682:
3679:
3678:
3666:
3664:
3663:
3662:
3653:
3647:
3646:
3645:
3636:
3635:
3630:
3623:
3610:
3609:
3586:
3559:
3516:
3514:
3513:
3508:
3503:
3492:
3417:
3395:
3365:
3363:
3362:
3357:
3355:
3353:
3352:
3351:
3338:
3330:
3325:
3321:
3319:
3318:
3317:
3312:
3311:
3303:
3295:
3287:
3281:
3279:
3275:
3269:
3264:
3234:
3232:
3231:
3226:
3212:
3211:
3206:
3201:
3195:
3156:
3154:
3153:
3148:
3143:
3141:
3133:
3132:
3131:
3126:
3116:
3111:
3110:
3105:
3104:
3096:
3092:
3090:
3089:
3088:
3075:
3074:
3073:
3068:
3058:
3055:
3050:
3032:
3031:
3026:
3016:
3014:
3010:
3004:
3003:
3002:
2993:
2987:
2982:
2981:
2976:
2975:
2967:
2955:total derivative
2942:
2940:
2939:
2934:
2929:
2928:
2907:
2899:
2898:
2877:
2869:
2868:
2847:
2839:
2838:
2829:
2828:
2810:
2802:
2801:
2796:
2787:
2786:
2781:
2764:
2725:
2693:
2676:
2625:
2606:
2599:
2597:
2596:
2591:
2589:
2585:
2584:
2582:
2581:
2580:
2575:
2574:
2566:
2555:
2550:
2548:
2547:
2546:
2541:
2540:
2532:
2521:
2516:
2514:
2513:
2512:
2507:
2506:
2498:
2487:
2477:
2475:
2474:
2473:
2468:
2467:
2462:
2457:
2446:
2440:
2436:
2435:
2433:
2432:
2431:
2415:
2410:
2408:
2407:
2406:
2390:
2385:
2383:
2382:
2381:
2365:
2355:
2353:
2352:
2351:
2346:
2333:
2294:
2292:
2291:
2286:
2275:
2273:
2272:
2271:
2266:
2256:
2255:
2254:
2241:
2239:
2238:
2228:
2223:
2205:
2203:
2202:
2201:
2196:
2195:
2190:
2185:
2177:
2169:
2167:
2165:
2161:
2155:
2150:
2145:
2143:
2142:
2141:
2136:
2126:
2118:
2088:
2049:
1982:
1951:
1933:
1916:
1894:
1857:
1814:
1771:
1737:potential energy
1729:
1682:
1669:
1640:
1638:
1637:
1632:
1627:
1626:
1610:
1608:
1607:
1602:
1590:
1588:
1587:
1582:
1562:
1560:
1559:
1554:
1551:
1546:
1537:
1536:
1526:
1521:
1506:
1498:
1483:
1481:
1480:
1475:
1447:non-relativistic
1416:
1414:
1413:
1408:
1403:
1401:
1400:
1399:
1386:
1385:
1380:
1379:
1369:
1361:
1330:
1328:
1327:
1322:
1317:
1315:
1307:
1306:
1305:
1300:
1290:
1285:
1284:
1279:
1264:
1262:
1254:
1253:
1252:
1247:
1237:
1232:
1231:
1226:
1217:
1215:
1207:
1206:
1205:
1200:
1190:
1185:
1184:
1179:
1158:
1127:
1096:
985:angular velocity
965:
883:
863:
861:
860:
855:
837:consisting of a
836:
775:
768:
761:
748:
743:
742:
735:
731:
730:
636:Johann Bernoulli
631:Daniel Bernoulli
552:Tangential speed
456:
432:
407:Fictitious force
402:
254:Mechanical power
244:
185:Angular momentum
83:
81:
80:
75:
73:
71:
63:
62:
53:
48:
47:
21:
20:
29167:
29166:
29162:
29161:
29160:
29158:
29157:
29156:
29132:
29131:
29130:
29125:
29089:
29075:Medical physics
29026:
28985:Nuclear physics
28954:
28948:Non-equilibrium
28870:
28842:
28814:
28808:
28778:
28773:
28745:
28707:
28691:
28653:
28601:
28567:Poisson algebra
28543:
28425:
28418:
28376:
28272:Operator theory
28170:
28128:
28094:Tensor software
28070:
28019:Automata theory
27988:
27983:
27953:
27948:
27880:
27875:
27820:
27818:
27806:
27784:
27761:
27759:Further reading
27756:
27750:
27729:
27717:Tensor Calculus
27706:
27685:
27666:
27647:
27626:
27456:
27427:
27408:
27385:
27364:
27343:
27322:
27301:
27278:
27257:
27228:
27207:
27184:
27156:
27134:
27115:
27094:
27073:
27047:
27033:Lifshitz, E. M.
27021:
26982:
26977:
26969:
26962:
26955:
26951:
26944:
26940:
26933:
26929:
26922:
26918:
26910:
26906:
26898:
26894:
26886:
26882:
26876:Hildebrand 1992
26874:
26870:
26862:
26858:
26850:
26846:
26838:
26834:
26827:
26803:
26799:
26791:
26787:
26779:
26775:
26767:
26763:
26755:
26751:
26743:
26739:
26731:
26724:
26716:
26712:
26705:
26701:
26697:, p. 44–45
26693:
26686:
26643:
26639:
26631:
26627:
26619:
26615:
26607:
26603:
26596:
26580:
26576:
26572:, p. 15–17
26568:
26564:
26556:
26552:
26544:
26540:
26532:
26528:
26520:
26516:
26508:
26504:
26496:
26492:
26484:
26480:
26472:
26468:
26460:
26456:
26448:
26444:
26436:
26432:
26420:
26416:
26409:
26405:
26397:
26393:
26389:, p. 60–61
26385:
26381:
26373:
26371:
26369:
26343:
26339:
26335:, p. 36–40
26331:
26327:
26318:
26314:
26279:
26275:
26267:
26256:
26248:
26244:
26237:
26219:
26212:
26203:
26201:
26194:
26188:
26181:
26172:
26170:
26162:. p. 100.
26152:
26145:
26137:
26133:
26125:
26121:
26113:
26109:
26100:
26096:
26092:
26087:
26086:
26081:
26077:
26072:
26042:
26038:
26025:
26021:
26005:
26001:
25988:
25984:
25968:
25964:
25951:
25947:
25945:
25942:
25941:
25914:
25909:
25908:
25896:
25891:
25890:
25888:
25885:
25884:
25883:In other words
25882:
25878:
25869:
25865:
25842:
25831:
25829:
25828:
25827:
25823:
25818:
25807:
25806:
25801:
25799:
25787:
25782:
25781:
25777:
25772:
25760:
25755:
25754:
25750:
25745:
25743:
25740:
25739:
25733:
25729:
25724:
25719:
25643:
25638:
25636:
25633:
25601:
25569:
25563:
25558:
25557:
25550:
25549:
25529:
25526:
25525:
25522:volume integral
25488:
25474:
25473:
25449:
25448:
25446:
25443:
25442:
25424:
25417:
25383:discovered the
25362:Planck constant
25346:
25326:
25320:
25308:
25302:
25285:
25256:
25233:
25229:
25223:
25166:
25155:
25154:
25153:
25149:
25141:
25139:
25133:
25122:
25121:
25120:
25114:
25103:
25091:
25088:
25087:
25080:
25072:
25058:
25052:
25046:
25040:
25034:
25028:
25022:
24986:
24969:
24967:
24966:
24946:
24944:
24943:
24939:
24931:
24929:
24921:
24919:
24916:
24915:
24909:
24899:
24868:
24860:
24858:
24828:
24823:
24815:
24801:
24799:
24798:
24797:
24793:
24776:
24773:
24772:
24757:
24732:
24731:
24707:
24699:
24691:
24689:
24686:
24685:
24663:
24649:
24647:
24646:
24635:
24618:
24616:
24615:
24610:
24607:
24606:
24562:
24548:
24546:
24545:
24532:
24521:
24519:
24518:
24517:
24502:
24494:
24491:
24490:
24469:
24461:
24453:
24439:
24433:
24429:
24427:
24416:
24405:
24403:
24400:
24399:
24380:
24358:
24336:
24314:
24301:
24272:
24242:
24231:
24230:
24229:
24225:
24217:
24215:
24203:
24199:
24195:
24187:
24185:
24169:
24158:
24157:
24156:
24152:
24144:
24142:
24138:
24127:
24126:
24121:
24119:
24117:
24114:
24113:
24093:
24082:
24081:
24080:
24076:
24068:
24066:
24054:
24050:
24046:
24038:
24036:
24024:
24020:
24018:
24015:
24014:
24008:
23981:
23970:
23969:
23968:
23962:
23951:
23950:
23949:
23940:
23936:
23930:
23919:
23909:
23898:
23884:
23876:
23873:
23872:
23858:
23838:
23833:
23773:
23762:
23737:
23733:
23729:
23723:
23719:
23717:
23708:
23697:
23696:
23695:
23676:
23675:
23671:
23669:
23666:
23665:
23646:
23640:
23617:
23613:
23609:
23603:
23599:
23597:
23583:
23582:
23573:
23572:
23570:
23553:
23552:
23547:
23544:
23543:
23532:
23501:
23500:
23484:
23476:
23474:
23465:
23454:
23453:
23452:
23444:
23441:
23440:
23434:
23428:
23417:
23411:
23378:
23377:
23371:
23367:
23347:
23346:
23342:
23335:
23331:
23327:
23325:
23316:
23312:
23310:
23307:
23306:
23300:
23260:
23249:
23248:
23247:
23241:
23237:
23228:
23217:
23216:
23215:
23214:
23210:
23197:
23188:
23184:
23182:
23179:
23178:
23170:
23165:
23153:
23119:
23117:
23116:
23111:
23108:
23107:
23101:
23095:
23089:
23086:relative motion
23082:
23076:
23067:
23060:
23054:
23053:| = |
23048:
23046:
23036:
23028:
23021:
23014:
23008:
22998:
22995:
22988:
22978:
22959:
22955:
22954:
22940:
22935:
22930:
22915:
22904:
22902:
22901:
22900:
22887:
22886:
22884:
22872:
22868:
22867:
22855:
22844:
22842:
22841:
22840:
22827:
22826:
22824:
22815:
22812:
22811:
22807:
22800:
22793:
22787:
22780:
22774:
22764:
22758:
22751:
22741:
22731:
22725:
22719:
22716:
22710:
22707:
22701:
22698:
22692:
22689:
22681:Main articles:
22679:
22641:
22640:
22638:
22635:
22634:
22627:simple pendulum
22598:
22597:
22595:
22592:
22591:
22552:
22525:
22523:
22509:
22508:
22506:
22503:
22502:
22453:
22452:
22441:
22440:
22409:
22408:
22402:
22398:
22372:
22371:
22364:
22360:
22349:
22348:
22343:
22341:
22339:
22336:
22335:
22297:
22286:
22285:
22284:
22255:
22254:
22234:
22233:
22216:
22213:
22212:
22196:
22193:
22192:
22158:
22157:
22137:
22136:
22104:
22103:
22099:
22091:
22089:
22080:
22076:
22074:
22071:
22070:
22050:
22049:
22019:
22008:
22007:
22006:
22000:
21996:
21983:
21960:
21959:
21945:
21944:
21932:
21921:
21920:
21919:
21904:
21900:
21890:
21888:
21881:
21880:
21848:
21823:
21822:
21818:
21814:
21813:
21804:
21779:
21778:
21761:
21760:
21759:
21755:
21754:
21753:
21749:
21736:
21727:
21716:
21715:
21714:
21701:
21699:
21692:
21691:
21680:
21675:
21668:
21666:
21663:
21662:
21635:
21634:
21630:
21616:
21613:
21612:
21590:
21575:
21574:
21563:
21562:
21552:
21537:
21536:
21525:
21524:
21522:
21518:
21505:
21496:
21485:
21484:
21483:
21470:
21462:
21459:
21458:
21442:
21439:
21438:
21422:
21419:
21418:
21402:
21401:
21378:
21377:
21355:
21354:
21343:
21342:
21341:
21334:
21300:
21299:
21295:
21291:
21290:
21270:
21269:
21252:
21251:
21232:
21231:
21220:
21219:
21218:
21211:
21174:
21173:
21169:
21164:
21162:
21159:
21158:
21142:
21139:
21138:
21122:
21119:
21118:
21102:
21099:
21098:
21075:
21057:
21009:
21008:
20996:
20992:
20986:
20982:
20962:
20961:
20957:
20949:
20947:
20938:
20934:
20932:
20929:
20928:
20893:
20892:
20879:
20875:
20869:
20865:
20848:
20847:
20842:
20840:
20838:
20835:
20834:
20809:
20798:
20797:
20796:
20771:
20767:
20747:
20746:
20740:
20736:
20719:
20718:
20713:
20711:
20709:
20706:
20705:
20676:
20668:
20666:
20654:
20643:
20642:
20641:
20628:
20624:
20615:
20604:
20603:
20602:
20579:
20578:
20573:
20570:
20569:
20531:
20520:
20519:
20518:
20505:
20501:
20495:
20491:
20482:
20471:
20470:
20469:
20463:
20459:
20450:
20439:
20438:
20437:
20424:
20416:
20413:
20412:
20384:
20381:
20353:
20336:
20334:
20333:
20328:
20325:
20324:
20292:
20284:
20282:
20265:
20264:
20259:
20256:
20255:
20230:
20229:
20205:
20204:
20200:
20192:
20190:
20186:
20175:
20174:
20169:
20167:
20152:
20151:
20131:
20130:
20126:
20118:
20116:
20102:
20094:
20092:
20076:
20068:
20066:
20064:
20061:
20060:
20036:
20028:
20026:
20005:
20004:
20000:
19992:
19990:
19986:
19975:
19974:
19969:
19967:
19965:
19962:
19961:
19901:
19890:
19889:
19888:
19879:
19868:
19867:
19866:
19857:
19846:
19845:
19844:
19828:
19811:
19810:
19796:
19795:
19781:
19780:
19754:
19751:
19750:
19747:
19718:
19707:
19696:
19694:
19691:
19690:
19671:
19663:
19621:
19620:
19615:
19612:
19611:
19589:
19588:
19586:
19583:
19582:
19545:
19544:
19539:
19536:
19535:
19513:
19512:
19510:
19507:
19506:
19490:
19487:
19486:
19466:
19455:
19454:
19453:
19444:
19433:
19432:
19431:
19427:
19417:
19413:
19412:
19410:
19398:
19394:
19392:
19389:
19388:
19372:
19371:
19362:
19351:
19350:
19349:
19340:
19336:
19327:
19316:
19315:
19314:
19310:
19300:
19296:
19295:
19293:
19287:
19270:
19262:
19260:
19248:
19244:
19240:
19232:
19230:
19221:
19210:
19209:
19208:
19188:
19187:
19169:
19165:
19159:
19149:
19148:
19133:
19129:
19125:
19117:
19115:
19102:
19094:
19092:
19079:
19072:
19061:
19060:
19059:
19055:
19053:
19044:
19033:
19032:
19031:
19022:
19011:
19010:
19009:
19005:
18995:
18991:
18990:
18988:
18975:
18968:
18964:
18960:
18958:
18949:
18938:
18937:
18936:
18927:
18923:
18919:
18909:
18905:
18904:
18902:
18898:
18897:
18882:
18878:
18874:
18866:
18864:
18852:
18841:
18840:
18839:
18835:
18827:
18825:
18815:
18810:
18805:
18803:
18800:
18799:
18796:
18785:
18752:
18748:
18739:
18735:
18726:
18722:
18714:
18711:
18710:
18708:
18675:
18671:
18662:
18658:
18650:
18647:
18646:
18644:
18635:
18613:
18583:
18576:
18559:
18557:
18553:
18552:
18536:
18534:
18532:
18529:
18528:
18484:
18477:
18473:
18456:
18452:
18441:
18437:
18431:
18427:
18425:
18423:
18420:
18419:
18397:
18358:
18353:
18352:
18337:
18332:
18331:
18322:
18317:
18316:
18304:
18300:
18288:
18283:
18282:
18264:
18259:
18258:
18246:
18241:
18240:
18229:
18226:
18225:
18222:
18210:
18202:
18191:
18185:
18147:
18144:
18143:
18140:
18116:
18084:
18082:
18081:
18064:
18056:
18053:
18052:
18036:
18016:
18015:
18005:
17997:
17995:
17970:
17969:
17955:
17954:
17950:
17942:
17940:
17939:
17935:
17925:
17920:
17913:
17903:
17895:
17893:
17887:
17886:
17875:
17874:
17857:
17856:
17852:
17844:
17842:
17831:
17829:
17828:
17814:
17812:
17811:
17793:
17792:
17788:
17780:
17778:
17774:
17764:
17759:
17752:
17742:
17734:
17732:
17726:
17725:
17715:
17707:
17705:
17688:
17687:
17683:
17675:
17673:
17662:
17660:
17659:
17648:
17644:
17636:
17634:
17623:
17621:
17620:
17613:
17603:
17595:
17593:
17589:
17587:
17584:
17583:
17573:
17554:
17531:
17528:
17527:
17507:
17504:
17503:
17496:
17491:
17474:
17471:
17470:
17442:
17440:
17439:
17435:
17427:
17425:
17414:
17412:
17411:
17394:
17392:
17391:
17387:
17379:
17377:
17366:
17364:
17363:
17361:
17358:
17357:
17335:
17333:
17332:
17330:
17327:
17326:
17304:
17302:
17301:
17299:
17296:
17295:
17274:
17263:
17262:
17261:
17257:
17249:
17247:
17244:
17241:
17240:
17222:
17211:
17210:
17209:
17197:
17186:
17185:
17184:
17180:
17172:
17170:
17164:
17153:
17146:
17143:
17142:
17124:
17121:
17120:
17095:
17093:
17092:
17090:
17087:
17086:
17068:
17066:
17063:
17062:
17034:
17032:
17031:
17027:
17019:
17017:
17006:
16997:
16993:
16979:
16977:
16976:
16972:
16964:
16962:
16960:
16957:
16956:
16922:
16920:
16919:
16911:
16879:
16877:
16876:
16868:
16857:
16854:
16853:
16819:
16817:
16816:
16808:
16779:
16777:
16776:
16768:
16760:
16757:
16756:
16724:
16716:
16714:
16700:
16698:
16697:
16680:
16678:
16677:
16673:
16665:
16663:
16655:
16644:
16640:
16632:
16630:
16607:
16605:
16604:
16596:
16585:
16582:
16581:
16565:
16564:
16551:
16543:
16541:
16527:
16525:
16524:
16513:
16504:
16500:
16486:
16484:
16483:
16479:
16471:
16469:
16461:
16439:
16437:
16436:
16428:
16408:
16406:
16405:
16401:
16393:
16391:
16380:
16371:
16367:
16359:
16355:
16347:
16345:
16344:
16340:
16331:
16330:
16314:
16306:
16304:
16290:
16288:
16287:
16270:
16268:
16267:
16263:
16255:
16253:
16245:
16234:
16230:
16222:
16220:
16213:
16193:
16191:
16190:
16182:
16169:
16167:
16164:
16163:
16147:
16146:
16138:
16129:
16112:
16101:
16090:
16089:
16088:
16081:
16080:
16074:
16073:
16072:
16063:
16059:
16050:
16046:
16042:
16035:
16031:
16025:
16021:
16020:
16018:
16012:
16001:
15996:
15992:
15982:
15959:
15948:
15937:
15936:
15935:
15928:
15927:
15921:
15920:
15919:
15910:
15906:
15897:
15893:
15889:
15882:
15878:
15872:
15868:
15867:
15865:
15859:
15848:
15838:
15834:
15825:
15814:
15809:
15805:
15795:
15794:
15788:
15765:
15754:
15750:
15740:
15739:
15733:
15732:
15731:
15722:
15718:
15709:
15705:
15701:
15694:
15690:
15684:
15680:
15679:
15677:
15671:
15660:
15650:
15639:
15638:
15637:
15631:
15620:
15615:
15611:
15596:
15594:
15593:
15587:
15570:
15559:
15555:
15545:
15544:
15538:
15537:
15536:
15527:
15523:
15514:
15510:
15506:
15499:
15495:
15489:
15485:
15484:
15482:
15476:
15465:
15460:
15456:
15441:
15439:
15438:
15427:
15418:
15414:
15399:
15394:
15392:
15391:
15386:
15381:
15364:
15362:
15361:
15353:
15343:
15341:
15338:
15337:
15312:
15310:
15309:
15298:
15289:
15285:
15277:
15260:
15258:
15257:
15249:
15229:
15227:
15226:
15221:
15218:
15217:
15192:
15190:
15189:
15181:
15172:
15168:
15154:
15152:
15151:
15149:
15146:
15145:
15129:
15124:
15121:
15120:
15100:
15089:
15088:
15087:
15085:
15082:
15081:
15064:
15053:
15052:
15051:
15049:
15046:
15045:
15027:
15010:
14999:
14998:
14985:
14981:
14977:
14970:
14966:
14962:
14960:
14957:
14956:
14952:
14940:
14931:
14927:
14924:
14921:
14920:
14896:
14887:
14876:
14875:
14874:
14865:
14864:
14847:
14840:
14827:
14823:
14819:
14814:
14811:
14810:
14806:
14797:
14786:
14785:
14784:
14778:
14767:
14762:
14758:
14756:
14753:
14752:
14731:
14722:
14711:
14710:
14709:
14700:
14699:
14685:
14684:
14671:
14667:
14663:
14658:
14655:
14654:
14650:
14641:
14630:
14629:
14628:
14622:
14611:
14606:
14602:
14600:
14597:
14596:
14573:
14564:
14560:
14558:
14555:
14554:
14535:
14524:
14515:
14511:
14503:
14498:
14495:
14494:
14479:
14455:
14444:
14443:
14442:
14434:
14426:
14423:
14422:
14388:
14386:
14385:
14377:
14348:
14346:
14345:
14337:
14329:
14326:
14325:
14315:
14304:
14279:
14275:
14271:
14265:
14261:
14259:
14247:
14243:
14239:
14233:
14229:
14227:
14215:
14211:
14209:
14206:
14205:
14129:
14118:
14117:
14116:
14112:
14104:
14102:
14096:
14085:
14084:
14083:
14077:
14066:
14061:
14057:
14049:
14046:
14045:
14015:
14004:
14003:
14002:
13996:
13985:
13984:
13983:
13974:
13970:
13958:
13950:
13946:
13936:
13924:
13913:
13912:
13911:
13907:
13899:
13897:
13891:
13880:
13879:
13878:
13872:
13861:
13845:
13834:
13833:
13832:
13828:
13820:
13818:
13812:
13801:
13800:
13799:
13793:
13782:
13776:
13773:
13772:
13756:
13745:
13744:
13743:
13737:
13726:
13725:
13724:
13715:
13711:
13699:
13685:
13671:
13660:
13659:
13658:
13649:
13645:
13641:
13629:
13628:
13624:
13622:
13621:
13617:
13603:
13592:
13591:
13590:
13581:
13577:
13573:
13561:
13560:
13556:
13554:
13553:
13549:
13537:
13523:
13514:
13510:
13500:
13492:
13489:
13488:
13468:
13464:
13449:
13445:
13437:
13429:
13427:
13424:
13423:
13397:
13396:
13387:
13376:
13375:
13374:
13370:
13362:
13360:
13354:
13343:
13342:
13341:
13335:
13324:
13314:
13313:
13305:
13302:
13301:
13278:
13275:
13274:
13271:
13260:
13247:
13234:
13199:
13198:
13194:
13186:
13184:
13175:
13171:
13153:
13152:
13148:
13140:
13138:
13129:
13125:
13107:
13106:
13102:
13094:
13092:
13083:
13079:
13077:
13074:
13073:
13007:
13006:
12992:
12991:
12977:
12976:
12962:
12961:
12947:
12946:
12926:
12923:
12922:
12889:
12885:
12881:
12873:
12871:
12859:
12848:
12847:
12846:
12842:
12834:
12832:
12821:
12820:
12815:
12813:
12804:
12793:
12792:
12791:
12789:
12786:
12785:
12782:
12745:
12734:
12733:
12732:
12728:
12720:
12718:
12709:
12705:
12703:
12700:
12699:
12696:
12681:
12676:
12650:
12646:
12642:
12633:
12632:
12630:
12618:
12607:
12606:
12605:
12601:
12593:
12591:
12581:
12576:
12574:
12571:
12570:
12548:
12545:
12544:
12528:
12527:
12512:
12508:
12504:
12497:
12493:
12489:
12487:
12473:
12462:
12461:
12460:
12456:
12448:
12446:
12430:
12419:
12418:
12417:
12413:
12405:
12403:
12399:
12389:
12384:
12383:
12379:
12373:
12366:
12365:
12341:
12337:
12333:
12326:
12322:
12318:
12316:
12312:
12302:
12297:
12288:
12277:
12276:
12275:
12271:
12262:
12261:
12259:
12247:
12236:
12235:
12234:
12230:
12223:
12212:
12211:
12210:
12206:
12204:
12195:
12191:
12187:
12178:
12177:
12175:
12159:
12148:
12147:
12146:
12142:
12135:
12124:
12123:
12122:
12118:
12116:
12112:
12102:
12097:
12088:
12077:
12076:
12075:
12071:
12063:
12061:
12049:
12038:
12037:
12036:
12032:
12025:
12014:
12013:
12012:
12008:
12006:
11993:
11982:
11981:
11980:
11976:
11968:
11966:
11962:
11952:
11947:
11946:
11942:
11936:
11929:
11928:
11908:
11903:
11902:
11898:
11891:
11880:
11879:
11878:
11874:
11872:
11863:
11852:
11851:
11850:
11846:
11837:
11836:
11834:
11822:
11817:
11816:
11812:
11805:
11801:
11797:
11795:
11786:
11782:
11778:
11769:
11768:
11766:
11749:
11738:
11737:
11736:
11732:
11725:
11714:
11713:
11712:
11708:
11706:
11697:
11686:
11685:
11684:
11680:
11672:
11670:
11669:
11665:
11655:
11650:
11649:
11645:
11639:
11632:
11631:
11616:
11612:
11608:
11599:
11598:
11596:
11584:
11573:
11572:
11571:
11567:
11559:
11557:
11547:
11542:
11538:
11536:
11533:
11532:
11509:
11505:
11501:
11494:
11483:
11482:
11481:
11477:
11475:
11459:
11455:
11451:
11444:
11440:
11436:
11434:
11430:
11420:
11415:
11413:
11410:
11409:
11384:
11372:
11368:
11358:
11351:
11347:
11341:
11337:
11336:
11334:
11325:
11314:
11313:
11312:
11297:
11285:
11281:
11272:
11268:
11264:
11257:
11253:
11247:
11243:
11242:
11240:
11234:
11214:
11210:
11206:
11198:
11194:
11193:
11189:
11187:
11183:
11173:
11168:
11166:
11163:
11162:
11161:and similarly:
11137:
11119:
11115:
11111:
11104:
11100:
11094:
11090:
11089:
11087:
11078:
11067:
11066:
11065:
11050:
11038:
11034:
11025:
11021:
11017:
11010:
11006:
11000:
10996:
10995:
10993:
10987:
10971:
10967:
10963:
10950:
10946:
10944:
10943:
10939:
10937:
10935:
10932:
10931:
10908:
10904:
10900:
10893:
10889:
10885:
10883:
10871:
10860:
10859:
10858:
10854:
10841:
10837:
10835:
10834:
10830:
10828:
10826:
10823:
10822:
10797:
10784:
10777:
10773:
10769:
10767:
10758:
10747:
10746:
10745:
10730:
10718:
10714:
10710:
10703:
10699:
10695:
10693:
10687:
10668:
10664:
10662:
10661:
10659:
10656:
10655:
10625:
10618:
10614:
10610:
10608:
10598:
10594:
10593:
10581:
10577:
10573:
10566:
10562:
10558:
10556:
10550:
10536:
10532:
10531:
10526:
10523:
10522:
10497:
10488:
10484:
10475:
10471:
10469:
10466:
10465:
10433:
10429:
10425:
10416:
10412:
10410:
10398:
10387:
10386:
10385:
10381:
10372:
10368:
10366:
10355:
10354:
10349:
10347:
10345:
10342:
10341:
10294:
10292:
10291:
10283:
10269:
10267:
10266:
10249:
10241:
10212:
10210:
10209:
10201:
10190:
10188:
10185:
10184:
10161:
10142:
10138:according to a
10128:
10118:independent of
10115:
10108:
10098:
10094:
10087:
10077:
10063:
10062:
10050:
10046:
10037:
10033:
10015:
10011:
10002:
9998:
9981:
9966:
9965:
9947:
9946:
9922:
9911:
9910:
9908:
9900:
9896:
9895:
9888:
9884:
9883:
9840:
9838:
9837:
9820:
9806:
9802:
9801:
9794:
9790:
9789:
9776:
9775:
9739:
9737:
9736:
9719:
9708:
9700:
9696:
9695:
9688:
9684:
9683:
9672:
9664:
9653:
9649:
9647:
9644:
9643:
9605:
9594:
9593:
9592:
9583:
9579:
9575:
9560:
9550:
9548:
9542:
9525:
9510:
9500:
9498:
9496:
9493:
9492:
9470:
9469:
9454:
9443:
9442:
9440:
9438:
9435:
9434:
9409:
9408:
9393:
9382:
9381:
9379:
9356:
9354:
9353:
9345:
9316:
9314:
9313:
9305:
9294:
9292:
9289:
9288:
9273:
9269:
9258:
9252:
9248:
9237:
9231:
9227:
9220:
9213:
9191:
9176:
9171:
9143:
9138:
9137:
9133:
9126:
9122:
9118:
9116:
9110:
9106:
9100:
9089:
9076:
9071:
9070:
9068:
9065:
9064:
9035:
9030:
9029:
9025:
9018:
9014:
9010:
9008:
9002:
8998:
8992:
8981:
8966:
8961:
8960:
8959:
8944:
8939:
8938:
8934:
8926:
8924:
8920:
8918:
8906:
8901:
8900:
8896:
8881:
8870:
8868:
8867:
8866:
8862:
8854:
8852:
8841:
8840:
8835:
8833:
8821:
8816:
8815:
8811:
8803:
8801:
8800:
8798:
8795:
8792:
8791:
8777:
8775:
8732:
8727:
8726:
8717:
8713:
8693:
8682:
8681:
8680:
8676:
8667:
8663:
8661:
8650:
8649:
8644:
8642:
8630:
8626:
8622:
8613:
8609:
8607:
8605:
8602:
8601:
8599:
8559:
8554:
8553:
8549:
8542:
8538:
8534:
8532:
8526:
8522:
8516:
8505:
8489:
8478:
8476:
8475:
8474:
8470:
8462:
8460:
8449:
8448:
8443:
8441:
8429:
8424:
8423:
8419:
8411:
8409:
8389:
8378:
8376:
8375:
8374:
8370:
8361:
8357:
8355:
8344:
8343:
8338:
8336:
8324:
8319:
8318:
8314:
8305:
8301:
8299:
8297:
8294:
8293:
8291:
8282:
8245:
8238:
8233:
8232:
8212:
8207:
8206:
8202:
8195:
8191:
8187:
8185:
8179:
8175:
8169:
8158:
8142:
8131:
8129:
8128:
8127:
8123:
8115:
8113:
8102:
8101:
8096:
8094:
8082:
8077:
8076:
8072:
8064:
8062:
8061:
8057:
8051:
8040:
8028:
8024:
8023:
8016:
8012:
8011:
7996:
7988:
7977:
7973:
7972:
7965:
7961:
7960:
7954:
7951:
7950:
7948:
7906:
7901:
7900:
7891:
7887:
7872:
7868:
7862:
7851:
7823:
7812:
7810:
7809:
7808:
7799:
7788:
7786:
7785:
7784:
7769:
7764:
7763:
7754:
7749:
7748:
7731:
7729:
7726:
7725:
7715:
7700:
7691:
7683:
7677:
7665:
7644:
7637:
7625:
7618:
7607:
7605:
7567:
7560:
7555:
7554:
7534:
7523:
7521:
7520:
7519:
7515:
7507:
7505:
7494:
7493:
7488:
7486:
7474:
7469:
7468:
7464:
7456:
7454:
7453:
7449:
7443:
7432:
7420:
7416:
7415:
7408:
7404:
7403:
7397:
7394:
7393:
7387:
7375:The Lagrangian
7373:
7274:
7271:
7270:
7264:
7257:
7224:
7212:
7208:
7207:
7200:
7196:
7195:
7183:
7180:
7179:
7148:
7133:
7129:
7128:
7121:
7117:
7116:
7110:
7107:
7106:
7098:
7089:
7070:
7063:
7054:
7047:
7039:
7023:
7022:
7011:
7004:
7000:
6983:
6972:
6971:
6970:
6966:
6958:
6956:
6945:
6944:
6939:
6937:
6925:
6921:
6917:
6909:
6907:
6906:
6902:
6896:
6885:
6873:
6869:
6868:
6861:
6857:
6856:
6841:
6837:
6836:
6829:
6825:
6824:
6813:
6809:
6797:
6786:
6785:
6784:
6780:
6772:
6770:
6769:
6765:
6758:
6747:
6734:
6733:
6725:
6713:
6709:
6697:
6686:
6685:
6684:
6680:
6672:
6670:
6659:
6658:
6653:
6651:
6637:
6633:
6621:
6610:
6609:
6608:
6604:
6596:
6594:
6593:
6589:
6578:
6577:
6572:
6570:
6561:
6557:
6545:
6541:
6537:
6529:
6527:
6526:
6522:
6516:
6505:
6493:
6489:
6488:
6481:
6477:
6476:
6465:
6457:
6442:
6438:
6437:
6430:
6426:
6425:
6417:
6415:
6412:
6411:
6409:
6396:
6382:
6372:
6328:
6327:
6317:
6313:
6302:
6301:
6299:
6285:
6284:
6277:
6273:
6268:
6267:
6265:
6253:
6242:
6241:
6240:
6222:
6211:
6210:
6209:
6197:
6186:
6185:
6184:
6180:
6172:
6170:
6161:
6157:
6145:
6141:
6137:
6129:
6127:
6126:
6122:
6116:
6105:
6090:
6087:
6086:
6042:
6031:
6030:
6029:
6025:
6017:
6015:
6004:
6003:
5998:
5996:
5984:
5980:
5976:
5968:
5966:
5964:
5961:
5960:
5938:
5915:
5911:
5907:
5899:
5897:
5885:
5874:
5873:
5872:
5868:
5860:
5858:
5847:
5846:
5841:
5839:
5830:
5826:
5824:
5821:
5820:
5814:
5772:
5741:
5737:
5733:
5725:
5723:
5711:
5700:
5699:
5698:
5694:
5686:
5684:
5673:
5672:
5667:
5665:
5656:
5652:
5650:
5647:
5646:
5636:
5603:
5599:
5577:
5573:
5569:
5561:
5559:
5547:
5536:
5535:
5534:
5530:
5522:
5520:
5509:
5508:
5503:
5501:
5500:
5496:
5487:
5483:
5482:
5478:
5472:
5461:
5455:
5452:
5451:
5426:
5422:
5418:
5410:
5408:
5396:
5385:
5384:
5383:
5379:
5371:
5369:
5358:
5357:
5352:
5350:
5338:
5334:
5330:
5323:
5318:
5317:
5313:
5311:
5302:
5297:
5296:
5290:
5286:
5280:
5269:
5263:
5260:
5259:
5236:
5232:
5223:
5219:
5213:
5202:
5189:
5185:
5173:
5169:
5165:
5158:
5153:
5152:
5148:
5146:
5140:
5129:
5116:
5111:
5110:
5104:
5093:
5080:
5075:
5074:
5062:
5057:
5056:
5050:
5039:
5033:
5030:
5029:
5006:
5002:
4998:
4991:
4986:
4985:
4981:
4979:
4970:
4965:
4964:
4958:
4947:
4934:
4930:
4928:
4925:
4924:
4919:
4907:
4871:
4867:
4855:
4851:
4847:
4840:
4835:
4834:
4830:
4828:
4822:
4811:
4798:
4793:
4792:
4787:
4784:
4783:
4777:
4763:
4754:
4745:
4736:
4725:
4718:
4709:
4700:
4691:
4683:
4676:
4664:
4628:
4623:
4622:
4607:
4602:
4601:
4595:
4591:
4582:
4577:
4576:
4567:
4556:
4550:
4547:
4546:
4520:
4515:
4514:
4502:
4497:
4496:
4490:
4479:
4473:
4470:
4469:
4456:
4415:
4410:
4409:
4394:
4389:
4388:
4382:
4378:
4369:
4364:
4363:
4354:
4349:
4348:
4339:
4328:
4322:
4319:
4318:
4316:
4279:
4278:
4277:
4276:
4253:
4252:
4251:
4248:
4240:
4239:
4236:
4221:Jean d'Alembert
4214:
4184:
4176:
4168:
4166:
4163:
4162:
4124:
4107:
4078:
4052:
4051:
4044:
4040:
4035:
4034:
4032:
4021:
4020:
4013:
4009:
4004:
4003:
4001:
3992:
3988:
3975:
3967:
3964:
3963:
3957:
3916:
3915:
3908:
3904:
3899:
3898:
3896:
3887:
3876:
3875:
3874:
3856:
3852:
3848:
3840:
3838:
3826:
3815:
3814:
3813:
3809:
3801:
3799:
3788:
3787:
3782:
3780:
3779:
3775:
3766:
3762:
3743:
3742:
3735:
3731:
3726:
3725:
3723:
3712:
3711:
3704:
3700:
3695:
3694:
3692:
3683:
3681:
3680:
3674:
3670:
3658:
3654:
3649:
3648:
3641:
3637:
3631:
3626:
3625:
3624:
3622:
3621:
3617:
3605:
3601:
3599:
3596:
3595:
3567:
3554:
3499:
3488:
3486:
3483:
3482:
3452:
3447:
3437:often involves
3405:
3374:
3367:
3347:
3343:
3339:
3331:
3329:
3313:
3302:
3301:
3300:
3296:
3288:
3286:
3282:
3271:
3270:
3265:
3263:
3261:
3258:
3257:
3202:
3200:
3199:
3191:
3177:
3174:
3173:
3167:
3134:
3127:
3122:
3121:
3117:
3115:
3106:
3095:
3094:
3093:
3084:
3080:
3076:
3069:
3064:
3063:
3059:
3057:
3051:
3040:
3027:
3022:
3021:
3006:
3005:
2998:
2994:
2989:
2988:
2986:
2977:
2966:
2965:
2964:
2962:
2959:
2958:
2924:
2923:
2903:
2894:
2890:
2873:
2864:
2860:
2843:
2834:
2830:
2824:
2823:
2806:
2797:
2792:
2791:
2782:
2777:
2776:
2774:
2771:
2770:
2761:
2755:
2748:
2738:
2713:
2704:
2692:
2678:
2671:
2664:
2655:
2616:
2609:time derivative
2604:
2576:
2565:
2564:
2563:
2559:
2554:
2542:
2531:
2530:
2529:
2525:
2520:
2508:
2497:
2496:
2495:
2491:
2486:
2485:
2481:
2469:
2458:
2456:
2455:
2454:
2450:
2445:
2427:
2423:
2419:
2414:
2402:
2398:
2394:
2389:
2377:
2373:
2369:
2364:
2363:
2359:
2347:
2342:
2341:
2337:
2332:
2330:
2327:
2326:
2324:
2314:
2296:
2267:
2262:
2261:
2257:
2250:
2246:
2242:
2240:
2234:
2230:
2224:
2213:
2197:
2186:
2184:
2183:
2182:
2178:
2170:
2168:
2157:
2156:
2151:
2149:
2137:
2132:
2131:
2127:
2119:
2117:
2115:
2112:
2111:
2086:
2079:
2072:
2065:
2055:
2043:
2036:
2029:
2022:
2012:
1976:
1966:
1961:
1940:
1935:
1923:
1917:
1906:
1900:
1881:
1851:
1844:
1837:
1830:
1816:
1812:
1805:
1798:
1791:
1777:
1769:
1762:
1748:
1727:
1720:
1706:
1700:
1691:
1678:
1677:Kinetic energy
1668:
1659:
1650:
1642:
1622:
1618:
1616:
1613:
1612:
1596:
1593:
1592:
1576:
1573:
1572:
1547:
1542:
1532:
1528:
1522:
1511:
1497:
1489:
1486:
1485:
1454:
1451:
1450:
1435:
1395:
1391:
1387:
1381:
1375:
1371:
1370:
1368:
1357:
1352:
1349:
1348:
1308:
1301:
1296:
1295:
1291:
1289:
1280:
1275:
1274:
1255:
1248:
1243:
1242:
1238:
1236:
1227:
1222:
1221:
1208:
1201:
1196:
1195:
1191:
1189:
1180:
1175:
1174:
1172:
1169:
1168:
1165:time derivative
1141:
1125:
1118:
1111:
1104:
1098:
1094:
1087:
1080:
1073:
1067:
1061:
1052:
1045:
1037:position vector
1033:
1027:
1020:
952:
929:
871:
849:
846:
845:
826:
808:founded on the
779:
738:
725:
724:
717:
716:
715:
590:
582:
581:
561:
515:Circular motion
509:
499:
498:
497:
454:
424:
421:
400:
379:
371:
370:
367:
366:
324:
314:
306:
305:
304:
263:
259:Mechanical work
252:
236:
174:
166:
165:
164:
119:
111:
88:
64:
58:
54:
52:
43:
42:
40:
37:
36:
17:
12:
11:
5:
29165:
29155:
29154:
29149:
29144:
29127:
29126:
29124:
29123:
29118:
29113:
29108:
29103:
29097:
29095:
29091:
29090:
29088:
29087:
29082:
29077:
29072:
29067:
29062:
29057:
29052:
29047:
29042:
29036:
29034:
29028:
29027:
29025:
29024:
29019:
29018:
29017:
29012:
29007:
28997:
28992:
28987:
28982:
28981:
28980:
28975:
28964:
28962:
28956:
28955:
28953:
28952:
28951:
28950:
28945:
28938:Thermodynamics
28935:
28934:
28933:
28928:
28918:
28913:
28908:
28907:
28906:
28901:
28896:
28891:
28880:
28878:
28872:
28871:
28869:
28868:
28867:
28866:
28856:
28850:
28848:
28844:
28843:
28841:
28840:
28839:
28838:
28828:
28822:
28820:
28816:
28815:
28807:
28806:
28799:
28792:
28784:
28775:
28774:
28772:
28771:
28758:
28750:
28747:
28746:
28744:
28743:
28738:
28733:
28728:
28727:
28726:
28715:
28713:
28709:
28708:
28706:
28705:
28699:
28697:
28693:
28692:
28690:
28689:
28682:
28677:
28672:
28667:
28661:
28659:
28655:
28654:
28652:
28651:
28646:
28641:
28636:
28631:
28626:
28621:
28615:
28613:
28607:
28606:
28603:
28602:
28600:
28599:
28594:
28589:
28584:
28579:
28574:
28569:
28564:
28559:
28553:
28551:
28542:
28541:
28540:
28539:
28534:
28524:
28523:
28522:
28517:
28507:
28502:
28501:
28500:
28490:
28489:
28488:
28483:
28478:
28473:
28468:
28463:
28458:
28448:
28447:
28446:
28441:
28430:
28428:
28420:
28419:
28417:
28416:
28411:
28406:
28397:
28386:
28384:
28378:
28377:
28375:
28374:
28369:
28368:
28367:
28362:
28357:
28352:
28342:
28341:
28340:
28335:
28330:
28325:
28315:
28314:
28313:
28308:
28303:
28298:
28288:
28287:
28286:
28276:
28275:
28274:
28269:
28259:
28258:
28257:
28255:Control theory
28252:
28242:
28241:
28240:
28235:
28230:
28220:
28219:
28218:
28213:
28208:
28198:
28197:
28196:
28186:
28180:
28178:
28172:
28171:
28169:
28168:
28163:
28158:
28153:
28148:
28142:
28140:
28134:
28133:
28130:
28129:
28127:
28126:
28121:
28116:
28111:
28106:
28101:
28096:
28091:
28086:
28080:
28078:
28069:
28068:
28063:
28058:
28053:
28048:
28047:
28046:
28036:
28031:
28026:
28021:
28016:
28015:
28014:
28009:
27998:
27996:
27990:
27989:
27982:
27981:
27974:
27967:
27959:
27950:
27949:
27947:
27946:
27944:Lagrange point
27941:
27936:
27931:
27926:
27921:
27916:
27911:
27906:
27901:
27896:
27891:
27885:
27882:
27881:
27874:
27873:
27866:
27859:
27851:
27845:
27844:
27838:
27837:(Gallica-Math)
27832:
27826:
27805:
27804:External links
27802:
27801:
27800:
27788:
27782:
27769:
27768:(Wiley, 1988).
27760:
27757:
27755:
27754:
27748:
27733:
27727:
27710:
27704:
27689:
27683:
27670:
27664:
27651:
27645:
27630:
27624:
27609:
27573:(10): 104037.
27562:
27515:
27471:(17): 174301.
27460:
27454:
27431:
27425:
27412:
27406:
27389:
27383:
27368:
27362:
27347:
27341:
27326:
27320:
27305:
27299:
27282:
27276:
27261:
27255:
27238:
27232:
27226:
27211:
27205:
27188:
27182:
27160:
27154:
27138:
27132:
27119:
27113:
27098:
27092:
27077:
27071:
27051:
27045:
27025:
27019:
27006:
27004:. Vol. 2.
26995:
26993:. Vol. 1.
26983:
26981:
26978:
26976:
26975:
26960:
26949:
26938:
26927:
26916:
26904:
26892:
26890:, pp. 202
26880:
26868:
26856:
26844:
26832:
26825:
26797:
26785:
26773:
26769:Goldstein 1980
26761:
26749:
26737:
26722:
26710:
26707:Goldstein 1980
26699:
26684:
26657:(7): 603–610.
26637:
26625:
26613:
26609:Goldstein 1980
26601:
26594:
26574:
26562:
26550:
26538:
26526:
26514:
26502:
26490:
26486:Goldstein 1980
26478:
26466:
26454:
26442:
26430:
26414:
26403:
26391:
26379:
26367:
26337:
26325:
26312:
26273:
26254:
26242:
26235:
26210:
26179:
26143:
26131:
26119:
26107:
26093:
26091:
26088:
26085:
26084:
26075:
26068:
26053:
26050:
26045:
26041:
26037:
26032:
26028:
26024:
26019:
26016:
26013:
26008:
26004:
26000:
25995:
25991:
25987:
25982:
25979:
25976:
25971:
25967:
25963:
25958:
25954:
25950:
25925:
25922:
25917:
25912:
25907:
25904:
25899:
25894:
25876:
25863:
25845:
25838:
25834:
25826:
25822:
25814:
25810:
25804:
25798:
25790:
25785:
25780:
25776:
25771:
25763:
25758:
25753:
25749:
25726:
25725:
25723:
25720:
25718:
25717:
25712:
25707:
25702:
25697:
25692:
25687:
25681:
25676:
25671:
25666:
25661:
25656:
25650:
25649:
25648:
25632:
25629:
25611:to continuous
25600:
25597:
25593:volume element
25572:
25566:
25561:
25553:
25548:
25545:
25542:
25539:
25536:
25533:
25501:
25498:
25495:
25491:
25487:
25481:
25478:
25472:
25469:
25466:
25463:
25460:
25457:
25452:
25416:
25413:
25387:extending the
25374:wave functions
25345:
25342:
25322:Main article:
25319:
25316:
25304:Main article:
25301:
25298:
25284:
25281:
25225:Main article:
25222:
25219:
25180:
25177:
25169:
25162:
25159:
25152:
25147:
25144:
25136:
25129:
25126:
25117:
25112:
25109:
25106:
25102:
25098:
25095:
25079:
25076:
25071:
25068:
24993:
24989:
24985:
24982:
24976:
24972:
24965:
24962:
24953:
24949:
24942:
24937:
24934:
24928:
24924:
24894:Note that the
24880:
24874:
24871:
24866:
24863:
24857:
24854:
24851:
24848:
24845:
24841:
24834:
24831:
24827:
24822:
24818:
24814:
24808:
24804:
24796:
24792:
24789:
24786:
24783:
24780:
24745:
24739:
24736:
24730:
24727:
24724:
24721:
24717:
24714:
24710:
24706:
24702:
24698:
24694:
24666:
24662:
24656:
24652:
24645:
24642:
24638:
24634:
24631:
24625:
24621:
24614:
24578:
24575:
24572:
24569:
24565:
24561:
24555:
24551:
24543:
24540:
24535:
24528:
24524:
24516:
24510:
24507:
24501:
24498:
24476:
24472:
24468:
24464:
24460:
24456:
24451:
24445:
24442:
24436:
24432:
24426:
24423:
24419:
24415:
24412:
24408:
24378:magnetic field
24356:electric field
24271:
24268:
24256:
24253:
24245:
24238:
24235:
24228:
24223:
24220:
24214:
24206:
24202:
24198:
24193:
24190:
24184:
24180:
24172:
24165:
24162:
24155:
24150:
24147:
24141:
24134:
24130:
24124:
24096:
24089:
24086:
24079:
24074:
24071:
24065:
24057:
24053:
24049:
24044:
24041:
24035:
24032:
24027:
24023:
24004:
23989:
23984:
23977:
23974:
23965:
23958:
23955:
23946:
23943:
23939:
23933:
23928:
23925:
23922:
23918:
23912:
23907:
23904:
23901:
23897:
23891:
23888:
23883:
23880:
23854:
23837:
23834:
23832:
23829:
23793:inertia forces
23748:
23740:
23736:
23732:
23726:
23722:
23716:
23711:
23704:
23701:
23694:
23691:
23688:
23682:
23679:
23674:
23628:
23620:
23616:
23612:
23606:
23602:
23596:
23590:
23586:
23580:
23576:
23569:
23566:
23560:
23557:
23551:
23514:
23508:
23505:
23499:
23496:
23490:
23487:
23482:
23479:
23473:
23468:
23461:
23458:
23451:
23448:
23397:
23394:
23391:
23385:
23382:
23374:
23370:
23366:
23363:
23354:
23351:
23345:
23334:
23330:
23324:
23319:
23315:
23288:
23285:
23282:
23279:
23276:
23273:
23269:
23263:
23256:
23253:
23244:
23240:
23236:
23231:
23224:
23221:
23213:
23209:
23204:
23201:
23196:
23187:
23138:
23135:
23132:
23126:
23122:
23115:
23093:
23080:
23073:center-of-mass
23065:
23058:
23026:
23019:
23012:
23006:
22993:
22986:
22958:
22951:
22947:
22943:
22938:
22933:
22929:
22926:
22923:
22918:
22911:
22907:
22899:
22894:
22891:
22882:
22871:
22864:
22858:
22851:
22847:
22839:
22834:
22831:
22822:
22819:
22805:
22798:
22791:
22785:
22778:
22772:
22762:center of mass
22756:
22749:
22723:
22714:
22705:
22696:
22678:
22675:
22657:
22654:
22648:
22645:
22631:inertial frame
22614:
22611:
22605:
22602:
22576:
22573:
22570:
22567:
22564:
22559:
22556:
22551:
22548:
22545:
22542:
22537:
22532:
22529:
22522:
22516:
22513:
22490:
22487:
22484:
22481:
22478:
22475:
22472:
22469:
22466:
22460:
22457:
22448:
22445:
22439:
22436:
22433:
22430:
22426:
22422:
22416:
22413:
22405:
22401:
22397:
22394:
22391:
22388:
22385:
22379:
22376:
22370:
22367:
22363:
22356:
22352:
22346:
22317:
22314:
22311:
22308:
22305:
22300:
22293:
22290:
22283:
22280:
22277:
22274:
22271:
22268:
22262:
22259:
22253:
22250:
22247:
22241:
22238:
22232:
22229:
22226:
22223:
22220:
22200:
22180:
22177:
22174:
22171:
22165:
22162:
22156:
22153:
22150:
22144:
22141:
22135:
22132:
22129:
22126:
22123:
22120:
22111:
22108:
22102:
22097:
22094:
22088:
22083:
22079:
22048:
22045:
22042:
22039:
22036:
22033:
22030:
22027:
22022:
22015:
22012:
22003:
21999:
21995:
21990:
21987:
21982:
21979:
21976:
21973:
21967:
21964:
21958:
21952:
21949:
21943:
21940:
21935:
21928:
21925:
21917:
21913:
21910:
21907:
21903:
21897:
21894:
21889:
21887:
21884:
21882:
21879:
21876:
21873:
21870:
21867:
21864:
21861:
21857:
21851:
21846:
21842:
21839:
21836:
21830:
21827:
21821:
21817:
21812:
21807:
21802:
21798:
21795:
21792:
21786:
21783:
21777:
21774:
21768:
21765:
21758:
21752:
21748:
21743:
21740:
21735:
21730:
21723:
21720:
21713:
21708:
21705:
21700:
21698:
21695:
21693:
21690:
21687:
21684:
21681:
21679:
21676:
21674:
21671:
21670:
21647:
21644:
21641:
21638:
21633:
21629:
21626:
21623:
21620:
21599:
21593:
21587:
21584:
21581:
21578:
21570:
21567:
21560:
21555:
21549:
21546:
21543:
21540:
21532:
21529:
21521:
21517:
21512:
21509:
21504:
21499:
21492:
21489:
21482:
21477:
21474:
21469:
21466:
21446:
21426:
21400:
21397:
21394:
21391:
21385:
21382:
21376:
21373:
21367:
21364:
21361:
21358:
21350:
21347:
21339:
21335:
21333:
21330:
21327:
21324:
21321:
21318:
21312:
21309:
21306:
21303:
21298:
21294:
21292:
21289:
21286:
21283:
21277:
21274:
21268:
21265:
21259:
21256:
21250:
21244:
21241:
21238:
21235:
21227:
21224:
21216:
21212:
21210:
21207:
21204:
21201:
21198:
21195:
21192:
21186:
21183:
21180:
21177:
21172:
21168:
21166:
21146:
21126:
21106:
21074:
21071:
21053:
21022:
21016:
21013:
21007:
21004:
20999:
20995:
20989:
20985:
20981:
20978:
20969:
20966:
20960:
20955:
20952:
20946:
20941:
20937:
20912:
20909:
20906:
20900:
20897:
20890:
20887:
20882:
20878:
20872:
20868:
20864:
20861:
20855:
20851:
20845:
20823:
20820:
20817:
20812:
20805:
20802:
20794:
20791:
20788:
20785:
20782:
20779:
20774:
20770:
20766:
20763:
20760:
20754:
20751:
20743:
20739:
20735:
20732:
20726:
20722:
20716:
20694:
20691:
20688:
20682:
20679:
20674:
20671:
20665:
20662:
20657:
20650:
20647:
20639:
20636:
20631:
20627:
20623:
20618:
20611:
20608:
20601:
20598:
20595:
20592:
20586:
20583:
20577:
20557:
20554:
20551:
20548:
20545:
20542:
20539:
20534:
20527:
20524:
20516:
20513:
20508:
20504:
20498:
20494:
20490:
20485:
20478:
20475:
20466:
20462:
20458:
20453:
20446:
20443:
20436:
20431:
20428:
20423:
20420:
20380:
20377:
20360:
20356:
20352:
20349:
20343:
20339:
20332:
20304:
20298:
20295:
20290:
20287:
20281:
20278:
20272:
20269:
20263:
20243:
20237:
20234:
20228:
20225:
20221:
20212:
20209:
20203:
20198:
20195:
20189:
20182:
20178:
20172:
20165:
20159:
20156:
20150:
20147:
20138:
20135:
20129:
20124:
20121:
20114:
20108:
20105:
20100:
20097:
20091:
20088:
20082:
20079:
20074:
20071:
20048:
20042:
20039:
20034:
20031:
20025:
20021:
20012:
20009:
20003:
19998:
19995:
19989:
19982:
19978:
19972:
19939:
19936:
19933:
19930:
19927:
19924:
19921:
19918:
19915:
19912:
19909:
19904:
19897:
19894:
19887:
19882:
19875:
19872:
19865:
19860:
19853:
19850:
19843:
19840:
19835:
19832:
19827:
19824:
19818:
19815:
19809:
19803:
19800:
19794:
19788:
19785:
19779:
19776:
19773:
19770:
19767:
19764:
19761:
19758:
19746:
19743:
19728:
19725:
19721:
19717:
19714:
19710:
19706:
19703:
19699:
19670:
19667:
19662:
19659:
19646:
19643:
19640:
19637:
19634:
19628:
19625:
19619:
19596:
19593:
19570:
19567:
19564:
19561:
19558:
19552:
19549:
19543:
19520:
19517:
19494:
19469:
19462:
19459:
19452:
19447:
19440:
19437:
19430:
19425:
19420:
19416:
19409:
19404:
19401:
19397:
19370:
19365:
19358:
19355:
19343:
19339:
19335:
19330:
19323:
19320:
19313:
19308:
19303:
19299:
19290:
19286:
19282:
19276:
19273:
19268:
19265:
19259:
19251:
19247:
19243:
19238:
19235:
19229:
19224:
19217:
19214:
19207:
19204:
19201:
19195:
19192:
19186:
19183:
19180:
19175:
19172:
19168:
19162:
19158:
19154:
19152:
19150:
19147:
19144:
19136:
19132:
19128:
19123:
19120:
19114:
19108:
19105:
19100:
19097:
19091:
19085:
19082:
19075:
19068:
19065:
19058:
19047:
19040:
19037:
19030:
19025:
19018:
19015:
19008:
19003:
18998:
18994:
18987:
18981:
18978:
18971:
18967:
18963:
18952:
18945:
18942:
18935:
18930:
18926:
18922:
18917:
18912:
18908:
18901:
18899:
18896:
18893:
18885:
18881:
18877:
18872:
18869:
18863:
18855:
18848:
18845:
18838:
18833:
18830:
18821:
18818:
18814:
18809:
18807:
18795:
18792:
18781:
18763:
18758:
18755:
18751:
18747:
18742:
18738:
18734:
18729:
18725:
18721:
18718:
18704:
18683:
18678:
18674:
18670:
18665:
18661:
18657:
18654:
18640:
18631:
18612:
18609:
18597:
18590:
18587:
18582:
18579:
18574:
18569:
18565:
18562:
18556:
18551:
18546:
18542:
18539:
18498:
18491:
18488:
18483:
18480:
18476:
18472:
18469:
18465:
18459:
18455:
18451:
18444:
18440:
18434:
18430:
18393:
18366:
18361:
18356:
18351:
18348:
18345:
18340:
18335:
18330:
18325:
18320:
18315:
18312:
18307:
18303:
18299:
18296:
18291:
18286:
18281:
18278:
18275:
18272:
18267:
18262:
18257:
18254:
18249:
18244:
18239:
18236:
18233:
18218:
18206:
18187:Main article:
18184:
18181:
18163:
18160:
18157:
18154:
18151:
18139:
18136:
18123:
18115:
18112:
18109:
18106:
18103:
18100:
18097:
18091:
18087:
18080:
18077:
18074:
18071:
18067:
18063:
18060:
18011:
18008:
18003:
18000:
17994:
17990:
17986:
17983:
17977:
17974:
17962:
17959:
17953:
17948:
17945:
17938:
17931:
17928:
17924:
17919:
17916:
17914:
17909:
17906:
17901:
17898:
17892:
17889:
17888:
17882:
17879:
17873:
17864:
17861:
17855:
17850:
17847:
17838:
17834:
17827:
17821:
17817:
17809:
17800:
17797:
17791:
17786:
17783:
17777:
17770:
17767:
17763:
17758:
17755:
17753:
17748:
17745:
17740:
17737:
17731:
17728:
17727:
17721:
17718:
17713:
17710:
17704:
17695:
17692:
17686:
17681:
17678:
17669:
17665:
17658:
17651:
17647:
17642:
17639:
17630:
17626:
17619:
17616:
17614:
17609:
17606:
17601:
17598:
17592:
17591:
17535:
17511:
17495:
17492:
17478:
17458:
17449:
17445:
17438:
17433:
17430:
17421:
17417:
17410:
17401:
17397:
17390:
17385:
17382:
17373:
17369:
17342:
17338:
17311:
17307:
17277:
17270:
17267:
17260:
17255:
17252:
17225:
17218:
17215:
17208:
17200:
17193:
17190:
17183:
17178:
17175:
17167:
17162:
17159:
17156:
17152:
17131:
17128:
17108:
17102:
17098:
17075:
17071:
17050:
17041:
17037:
17030:
17025:
17022:
17016:
17013:
17009:
17005:
17000:
16996:
16986:
16982:
16975:
16970:
16967:
16944:
16941:
16938:
16935:
16929:
16925:
16918:
16914:
16910:
16907:
16904:
16901:
16898:
16895:
16892:
16886:
16882:
16875:
16871:
16867:
16864:
16861:
16841:
16838:
16835:
16832:
16826:
16822:
16815:
16811:
16807:
16804:
16801:
16798:
16795:
16792:
16786:
16782:
16775:
16771:
16767:
16764:
16742:
16739:
16736:
16730:
16727:
16722:
16719:
16713:
16707:
16703:
16696:
16687:
16683:
16676:
16671:
16668:
16662:
16658:
16654:
16647:
16643:
16638:
16635:
16629:
16626:
16623:
16620:
16614:
16610:
16603:
16599:
16595:
16592:
16589:
16563:
16557:
16554:
16549:
16546:
16540:
16534:
16530:
16523:
16520:
16516:
16512:
16507:
16503:
16493:
16489:
16482:
16477:
16474:
16468:
16464:
16460:
16456:
16452:
16446:
16442:
16435:
16431:
16427:
16424:
16415:
16411:
16404:
16399:
16396:
16390:
16387:
16383:
16379:
16374:
16370:
16362:
16358:
16353:
16350:
16343:
16339:
16336:
16334:
16332:
16329:
16326:
16320:
16317:
16312:
16309:
16303:
16297:
16293:
16286:
16277:
16273:
16266:
16261:
16258:
16252:
16248:
16244:
16237:
16233:
16228:
16225:
16219:
16216:
16214:
16212:
16209:
16206:
16200:
16196:
16189:
16185:
16181:
16178:
16175:
16172:
16171:
16145:
16141:
16137:
16132:
16127:
16124:
16121:
16118:
16115:
16110:
16104:
16097:
16094:
16084:
16077:
16066:
16062:
16058:
16053:
16049:
16045:
16038:
16034:
16028:
16024:
16015:
16010:
16007:
16004:
16000:
15995:
15990:
15985:
15980:
15977:
15974:
15971:
15968:
15965:
15962:
15957:
15951:
15944:
15941:
15931:
15924:
15913:
15909:
15905:
15900:
15896:
15892:
15885:
15881:
15875:
15871:
15862:
15857:
15854:
15851:
15847:
15841:
15837:
15833:
15828:
15823:
15820:
15817:
15813:
15808:
15803:
15800:
15798:
15796:
15791:
15786:
15783:
15780:
15777:
15774:
15771:
15768:
15763:
15757:
15753:
15749:
15743:
15736:
15725:
15721:
15717:
15712:
15708:
15704:
15697:
15693:
15687:
15683:
15674:
15669:
15666:
15663:
15659:
15653:
15646:
15643:
15634:
15629:
15626:
15623:
15619:
15614:
15609:
15603:
15599:
15590:
15585:
15582:
15579:
15576:
15573:
15568:
15562:
15558:
15554:
15548:
15541:
15530:
15526:
15522:
15517:
15513:
15509:
15502:
15498:
15492:
15488:
15479:
15474:
15471:
15468:
15464:
15459:
15454:
15448:
15444:
15437:
15434:
15430:
15426:
15421:
15417:
15413:
15410:
15397:
15389:
15387:
15384:
15380:
15377:
15371:
15367:
15360:
15356:
15352:
15349:
15346:
15345:
15325:
15319:
15315:
15308:
15305:
15301:
15297:
15292:
15288:
15284:
15280:
15276:
15273:
15267:
15263:
15256:
15252:
15248:
15245:
15242:
15236:
15232:
15225:
15205:
15199:
15195:
15188:
15184:
15180:
15175:
15171:
15167:
15161:
15157:
15132:
15128:
15108:
15103:
15096:
15093:
15067:
15060:
15057:
15030:
15025:
15022:
15019:
15016:
15013:
15008:
15002:
14996:
14988:
14984:
14980:
14973:
14969:
14965:
14959:
14955:
14950:
14947:
14943:
14939:
14934:
14930:
14908:
14904:
14899:
14895:
14890:
14883:
14880:
14868:
14859:
14854:
14850:
14846:
14843:
14838:
14830:
14826:
14822:
14818:
14813:
14809:
14805:
14800:
14793:
14790:
14781:
14776:
14773:
14770:
14766:
14761:
14739:
14734:
14730:
14725:
14718:
14715:
14703:
14694:
14688:
14682:
14674:
14670:
14666:
14662:
14657:
14653:
14649:
14644:
14637:
14634:
14625:
14620:
14617:
14614:
14610:
14605:
14580:
14576:
14572:
14567:
14563:
14542:
14538:
14534:
14531:
14527:
14523:
14518:
14514:
14510:
14506:
14502:
14473:
14458:
14451:
14448:
14441:
14437:
14433:
14430:
14410:
14407:
14404:
14401:
14395:
14391:
14384:
14380:
14376:
14373:
14370:
14367:
14364:
14361:
14355:
14351:
14344:
14340:
14336:
14333:
14303:
14300:
14282:
14278:
14274:
14268:
14264:
14258:
14250:
14246:
14242:
14236:
14232:
14226:
14221:
14218:
14214:
14193:
14190:
14187:
14184:
14181:
14178:
14175:
14172:
14169:
14166:
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14160:
14157:
14154:
14151:
14148:
14145:
14141:
14132:
14125:
14122:
14115:
14110:
14107:
14099:
14092:
14089:
14080:
14075:
14072:
14069:
14065:
14060:
14056:
14053:
14034:
14031:
14028:
14024:
14018:
14011:
14008:
13999:
13992:
13989:
13980:
13977:
13973:
13967:
13964:
13961:
13957:
13953:
13949:
13943:
13940:
13935:
13927:
13920:
13917:
13910:
13905:
13902:
13894:
13887:
13884:
13875:
13870:
13867:
13864:
13860:
13856:
13848:
13841:
13838:
13831:
13826:
13823:
13815:
13808:
13805:
13796:
13791:
13788:
13785:
13781:
13759:
13752:
13749:
13740:
13733:
13730:
13721:
13718:
13714:
13708:
13705:
13702:
13698:
13692:
13689:
13684:
13680:
13674:
13667:
13664:
13652:
13648:
13644:
13636:
13633:
13627:
13620:
13616:
13612:
13606:
13599:
13596:
13584:
13580:
13576:
13568:
13565:
13559:
13552:
13546:
13543:
13540:
13536:
13530:
13527:
13522:
13517:
13513:
13507:
13504:
13499:
13496:
13476:
13471:
13467:
13463:
13460:
13457:
13452:
13448:
13444:
13440:
13436:
13432:
13411:
13408:
13405:
13400:
13390:
13383:
13380:
13373:
13368:
13365:
13357:
13350:
13347:
13338:
13333:
13330:
13327:
13323:
13317:
13312:
13309:
13285:
13282:
13270:
13267:
13256:
13243:
13230:
13215:
13206:
13203:
13197:
13192:
13189:
13183:
13178:
13174:
13169:
13160:
13157:
13151:
13146:
13143:
13137:
13132:
13128:
13123:
13114:
13111:
13105:
13100:
13097:
13091:
13086:
13082:
13029:
13026:
13023:
13020:
13014:
13011:
13005:
12999:
12996:
12990:
12984:
12981:
12975:
12969:
12966:
12960:
12954:
12951:
12945:
12942:
12939:
12936:
12933:
12930:
12903:
12900:
12892:
12888:
12884:
12879:
12876:
12870:
12862:
12855:
12852:
12845:
12840:
12837:
12828:
12824:
12818:
12812:
12807:
12800:
12797:
12780:
12756:
12748:
12741:
12738:
12731:
12726:
12723:
12717:
12712:
12708:
12698:is defined by
12694:
12680:
12677:
12664:
12661:
12653:
12649:
12645:
12640:
12636:
12629:
12621:
12614:
12611:
12604:
12599:
12596:
12587:
12584:
12580:
12558:
12555:
12552:
12526:
12523:
12515:
12511:
12507:
12500:
12496:
12492:
12485:
12476:
12469:
12466:
12459:
12454:
12451:
12445:
12441:
12433:
12426:
12423:
12416:
12411:
12408:
12402:
12395:
12392:
12388:
12382:
12376:
12372:
12368:
12367:
12364:
12361:
12357:
12352:
12344:
12340:
12336:
12329:
12325:
12321:
12315:
12308:
12305:
12301:
12291:
12284:
12281:
12274:
12269:
12265:
12258:
12250:
12243:
12240:
12233:
12226:
12219:
12216:
12209:
12198:
12194:
12190:
12185:
12181:
12174:
12170:
12162:
12155:
12152:
12145:
12138:
12131:
12128:
12121:
12115:
12108:
12105:
12101:
12091:
12084:
12081:
12074:
12069:
12066:
12060:
12052:
12045:
12042:
12035:
12028:
12021:
12018:
12011:
12004:
11996:
11989:
11986:
11979:
11974:
11971:
11965:
11958:
11955:
11951:
11945:
11939:
11935:
11931:
11930:
11927:
11924:
11920:
11911:
11906:
11901:
11894:
11887:
11884:
11877:
11866:
11859:
11856:
11849:
11844:
11840:
11833:
11825:
11820:
11815:
11808:
11804:
11800:
11789:
11785:
11781:
11776:
11772:
11765:
11761:
11752:
11745:
11742:
11735:
11728:
11721:
11718:
11711:
11700:
11693:
11690:
11683:
11678:
11675:
11668:
11661:
11658:
11654:
11648:
11642:
11638:
11634:
11633:
11630:
11627:
11619:
11615:
11611:
11606:
11602:
11595:
11587:
11580:
11577:
11570:
11565:
11562:
11553:
11550:
11546:
11541:
11540:
11512:
11508:
11504:
11497:
11490:
11487:
11480:
11474:
11470:
11462:
11458:
11454:
11447:
11443:
11439:
11433:
11426:
11423:
11419:
11397:
11394:
11391:
11387:
11383:
11375:
11371:
11367:
11364:
11361:
11354:
11350:
11344:
11340:
11333:
11328:
11321:
11318:
11310:
11307:
11304:
11300:
11296:
11288:
11284:
11280:
11275:
11271:
11267:
11260:
11256:
11250:
11246:
11237:
11233:
11229:
11225:
11217:
11213:
11209:
11201:
11197:
11192:
11186:
11179:
11176:
11172:
11150:
11147:
11144:
11140:
11136:
11130:
11127:
11122:
11118:
11114:
11107:
11103:
11097:
11093:
11086:
11081:
11074:
11071:
11063:
11060:
11057:
11053:
11049:
11041:
11037:
11033:
11028:
11024:
11020:
11013:
11009:
11003:
10999:
10990:
10986:
10982:
10974:
10970:
10966:
10958:
10953:
10949:
10942:
10911:
10907:
10903:
10896:
10892:
10888:
10882:
10874:
10867:
10864:
10857:
10849:
10844:
10840:
10833:
10810:
10807:
10804:
10800:
10796:
10790:
10787:
10780:
10776:
10772:
10766:
10761:
10754:
10751:
10743:
10740:
10737:
10733:
10729:
10721:
10717:
10713:
10706:
10702:
10698:
10690:
10686:
10682:
10676:
10671:
10667:
10643:
10640:
10637:
10631:
10628:
10621:
10617:
10613:
10607:
10601:
10597:
10592:
10584:
10580:
10576:
10569:
10565:
10561:
10553:
10549:
10545:
10539:
10535:
10530:
10510:
10507:
10504:
10500:
10496:
10491:
10487:
10483:
10478:
10474:
10459:
10444:
10436:
10432:
10428:
10422:
10419:
10415:
10409:
10401:
10394:
10391:
10384:
10378:
10375:
10371:
10362:
10358:
10352:
10325:
10322:
10319:
10316:
10313:
10310:
10307:
10301:
10297:
10290:
10286:
10282:
10276:
10272:
10265:
10262:
10259:
10256:
10252:
10248:
10244:
10240:
10237:
10234:
10231:
10228:
10225:
10219:
10215:
10208:
10204:
10200:
10196:
10193:
10127:
10124:
10113:
10106:
10092:
10085:
10061:
10058:
10049:
10045:
10036:
10032:
10029:
10026:
10023:
10014:
10010:
10001:
9997:
9994:
9991:
9988:
9984:
9980:
9977:
9974:
9971:
9969:
9967:
9964:
9961:
9954:
9950:
9944:
9941:
9938:
9935:
9932:
9929:
9925:
9921:
9918:
9914:
9899:
9887:
9882:
9878:
9875:
9872:
9868:
9865:
9862:
9859:
9856:
9853:
9847:
9843:
9836:
9833:
9830:
9827:
9823:
9819:
9816:
9805:
9793:
9788:
9784:
9781:
9779:
9777:
9774:
9771:
9767:
9764:
9761:
9758:
9755:
9752:
9746:
9742:
9735:
9732:
9729:
9726:
9722:
9718:
9714:
9711:
9699:
9687:
9682:
9678:
9675:
9673:
9671:
9667:
9663:
9659:
9656:
9652:
9651:
9613:
9608:
9601:
9598:
9586:
9582:
9578:
9573:
9570:
9567:
9563:
9559:
9556:
9553:
9545:
9541:
9537:
9531:
9528:
9523:
9520:
9517:
9513:
9509:
9506:
9503:
9477:
9473:
9467:
9464:
9461:
9457:
9453:
9450:
9446:
9422:
9416:
9412:
9406:
9403:
9400:
9396:
9392:
9389:
9385:
9378:
9375:
9372:
9369:
9363:
9359:
9352:
9348:
9344:
9341:
9338:
9335:
9332:
9329:
9323:
9319:
9312:
9308:
9304:
9300:
9297:
9267:
9256:
9246:
9235:
9225:
9218:
9175:
9174:Non-uniqueness
9172:
9170:
9167:
9154:
9146:
9141:
9136:
9129:
9125:
9121:
9113:
9109:
9103:
9098:
9095:
9092:
9088:
9084:
9079:
9074:
9052:
9049:
9046:
9038:
9033:
9028:
9021:
9017:
9013:
9005:
9001:
8995:
8990:
8987:
8984:
8980:
8976:
8969:
8964:
8956:
8947:
8942:
8937:
8932:
8929:
8923:
8916:
8909:
8904:
8899:
8893:
8884:
8877:
8873:
8865:
8860:
8857:
8848:
8844:
8838:
8832:
8824:
8819:
8814:
8809:
8806:
8773:
8752:
8749:
8746:
8743:
8740:
8735:
8730:
8725:
8720:
8716:
8711:
8707:
8704:
8696:
8689:
8686:
8679:
8673:
8670:
8666:
8657:
8653:
8647:
8641:
8633:
8629:
8625:
8619:
8616:
8612:
8595:
8576:
8573:
8570:
8562:
8557:
8552:
8545:
8541:
8537:
8529:
8525:
8519:
8514:
8511:
8508:
8504:
8500:
8492:
8485:
8481:
8473:
8468:
8465:
8456:
8452:
8446:
8440:
8432:
8427:
8422:
8417:
8414:
8407:
8403:
8400:
8392:
8385:
8381:
8373:
8367:
8364:
8360:
8351:
8347:
8341:
8335:
8327:
8322:
8317:
8311:
8308:
8304:
8287:
8278:
8258:
8255:
8252:
8248:
8241:
8236:
8231:
8228:
8224:
8215:
8210:
8205:
8198:
8194:
8190:
8182:
8178:
8172:
8167:
8164:
8161:
8157:
8153:
8145:
8138:
8134:
8126:
8121:
8118:
8109:
8105:
8099:
8093:
8085:
8080:
8075:
8070:
8067:
8060:
8054:
8049:
8046:
8043:
8039:
8031:
8027:
8019:
8015:
8010:
8006:
8003:
7999:
7994:
7991:
7987:
7980:
7976:
7968:
7964:
7959:
7944:
7923:
7920:
7917:
7914:
7909:
7904:
7899:
7894:
7890:
7886:
7883:
7880:
7875:
7871:
7865:
7860:
7857:
7854:
7850:
7846:
7843:
7840:
7837:
7834:
7831:
7826:
7819:
7815:
7807:
7802:
7795:
7791:
7783:
7780:
7777:
7772:
7767:
7762:
7757:
7752:
7747:
7744:
7741:
7737:
7734:
7711:
7696:
7687:
7673:
7661:
7642:
7633:
7623:
7614:
7601:
7580:
7577:
7574:
7570:
7563:
7558:
7553:
7550:
7546:
7537:
7530:
7526:
7518:
7513:
7510:
7501:
7497:
7491:
7485:
7477:
7472:
7467:
7462:
7459:
7452:
7446:
7441:
7438:
7435:
7431:
7423:
7419:
7411:
7407:
7402:
7383:
7372:
7369:
7356:, and others.
7318:Jean Bernoulli
7287:
7284:
7281:
7278:
7262:
7255:
7234:
7231:
7227:
7222:
7215:
7211:
7203:
7199:
7194:
7190:
7187:
7161:
7158:
7155:
7151:
7146:
7143:
7136:
7132:
7124:
7120:
7115:
7094:
7085:
7073:holds for all
7068:
7059:
7052:
7043:
7021:
7018:
7014:
7007:
7003:
6999:
6995:
6986:
6979:
6976:
6969:
6964:
6961:
6952:
6948:
6942:
6936:
6928:
6924:
6920:
6915:
6912:
6905:
6899:
6894:
6891:
6888:
6884:
6876:
6872:
6864:
6860:
6855:
6851:
6844:
6840:
6832:
6828:
6822:
6816:
6812:
6808:
6800:
6793:
6790:
6783:
6778:
6775:
6768:
6761:
6756:
6753:
6750:
6746:
6742:
6739:
6737:
6735:
6732:
6728:
6722:
6716:
6712:
6708:
6700:
6693:
6690:
6683:
6678:
6675:
6666:
6662:
6656:
6650:
6646:
6640:
6636:
6632:
6624:
6617:
6614:
6607:
6602:
6599:
6592:
6585:
6581:
6575:
6569:
6564:
6560:
6556:
6548:
6544:
6540:
6535:
6532:
6525:
6519:
6514:
6511:
6508:
6504:
6496:
6492:
6484:
6480:
6475:
6471:
6468:
6466:
6464:
6460:
6455:
6452:
6445:
6441:
6433:
6429:
6424:
6420:
6419:
6405:
6392:
6380:
6368:
6341:
6335:
6331:
6325:
6320:
6316:
6312:
6309:
6305:
6298:
6292:
6288:
6280:
6276:
6271:
6264:
6261:
6256:
6249:
6246:
6239:
6235:
6231:
6225:
6218:
6215:
6208:
6200:
6193:
6190:
6183:
6178:
6175:
6169:
6164:
6160:
6156:
6148:
6144:
6140:
6135:
6132:
6125:
6119:
6114:
6111:
6108:
6104:
6100:
6097:
6094:
6056:
6053:
6045:
6038:
6035:
6028:
6023:
6020:
6011:
6007:
6001:
5995:
5987:
5983:
5979:
5974:
5971:
5926:
5918:
5914:
5910:
5905:
5902:
5896:
5888:
5881:
5878:
5871:
5866:
5863:
5854:
5850:
5844:
5838:
5833:
5829:
5810:
5771:
5768:
5744:
5740:
5736:
5731:
5728:
5722:
5714:
5707:
5704:
5697:
5692:
5689:
5680:
5676:
5670:
5664:
5659:
5655:
5632:
5617:
5614:
5611:
5606:
5602:
5598:
5594:
5589:
5580:
5576:
5572:
5567:
5564:
5558:
5550:
5543:
5540:
5533:
5528:
5525:
5516:
5512:
5506:
5499:
5495:
5490:
5486:
5481:
5475:
5470:
5467:
5464:
5460:
5437:
5429:
5425:
5421:
5416:
5413:
5407:
5399:
5392:
5389:
5382:
5377:
5374:
5365:
5361:
5355:
5349:
5341:
5337:
5333:
5326:
5321:
5316:
5310:
5305:
5300:
5293:
5289:
5283:
5278:
5275:
5272:
5268:
5244:
5239:
5235:
5231:
5226:
5222:
5216:
5211:
5208:
5205:
5201:
5197:
5192:
5188:
5184:
5176:
5172:
5168:
5161:
5156:
5151:
5143:
5138:
5135:
5132:
5128:
5124:
5119:
5114:
5107:
5102:
5099:
5096:
5092:
5088:
5083:
5078:
5073:
5070:
5065:
5060:
5053:
5048:
5045:
5042:
5038:
5017:
5009:
5005:
5001:
4994:
4989:
4984:
4978:
4973:
4968:
4961:
4956:
4953:
4950:
4946:
4942:
4937:
4933:
4915:
4903:
4879:
4874:
4870:
4866:
4858:
4854:
4850:
4843:
4838:
4833:
4825:
4820:
4817:
4814:
4810:
4806:
4801:
4796:
4791:
4773:
4759:
4750:
4741:
4732:
4714:
4705:
4696:
4687:
4675:
4672:
4660:
4639:
4636:
4631:
4626:
4621:
4618:
4615:
4610:
4605:
4598:
4594:
4590:
4585:
4580:
4575:
4570:
4565:
4562:
4559:
4555:
4534:
4531:
4528:
4523:
4518:
4513:
4510:
4505:
4500:
4493:
4488:
4485:
4482:
4478:
4452:
4426:
4423:
4418:
4413:
4408:
4405:
4402:
4397:
4392:
4385:
4381:
4377:
4372:
4367:
4362:
4357:
4352:
4347:
4342:
4337:
4334:
4331:
4327:
4312:
4293:to understand
4255:
4254:
4249:
4242:
4241:
4237:
4230:
4229:
4228:
4227:
4226:
4213:
4210:
4191:
4187:
4183:
4179:
4175:
4171:
4076:
4059:
4055:
4047:
4043:
4038:
4028:
4024:
4016:
4012:
4007:
3998:
3995:
3991:
3987:
3982:
3979:
3974:
3971:
3953:
3929:
3923:
3919:
3911:
3907:
3902:
3895:
3890:
3883:
3880:
3872:
3868:
3859:
3855:
3851:
3846:
3843:
3837:
3829:
3822:
3819:
3812:
3807:
3804:
3795:
3791:
3785:
3778:
3772:
3769:
3765:
3761:
3757:
3750:
3746:
3738:
3734:
3729:
3719:
3715:
3707:
3703:
3698:
3689:
3686:
3677:
3673:
3669:
3661:
3657:
3652:
3644:
3640:
3634:
3629:
3620:
3616:
3613:
3608:
3604:
3506:
3502:
3498:
3495:
3491:
3451:
3448:
3446:
3443:
3350:
3346:
3342:
3337:
3334:
3328:
3324:
3316:
3309:
3306:
3299:
3294:
3291:
3285:
3278:
3274:
3268:
3248:
3224:
3221:
3218:
3215:
3209:
3205:
3198:
3194:
3190:
3187:
3184:
3181:
3163:
3146:
3140:
3137:
3130:
3125:
3120:
3114:
3109:
3102:
3099:
3087:
3083:
3079:
3072:
3067:
3062:
3054:
3049:
3046:
3043:
3039:
3035:
3030:
3025:
3019:
3013:
3009:
3001:
2997:
2992:
2985:
2980:
2973:
2970:
2932:
2927:
2922:
2919:
2916:
2913:
2910:
2906:
2902:
2897:
2893:
2889:
2886:
2883:
2880:
2876:
2872:
2867:
2863:
2859:
2856:
2853:
2850:
2846:
2842:
2837:
2833:
2827:
2822:
2819:
2816:
2813:
2809:
2805:
2800:
2795:
2790:
2785:
2780:
2759:
2753:
2746:
2702:
2687:
2669:
2659:
2588:
2579:
2572:
2569:
2562:
2558:
2553:
2545:
2538:
2535:
2528:
2524:
2519:
2511:
2504:
2501:
2494:
2490:
2484:
2480:
2472:
2465:
2461:
2453:
2449:
2443:
2439:
2430:
2426:
2422:
2418:
2413:
2405:
2401:
2397:
2393:
2388:
2380:
2376:
2372:
2368:
2362:
2358:
2350:
2345:
2340:
2336:
2320:
2312:
2284:
2281:
2278:
2270:
2265:
2260:
2253:
2249:
2245:
2237:
2233:
2227:
2222:
2219:
2216:
2212:
2208:
2200:
2193:
2189:
2181:
2176:
2173:
2164:
2160:
2154:
2148:
2140:
2135:
2130:
2125:
2122:
2102:
2084:
2077:
2070:
2063:
2041:
2034:
2027:
2020:
1972:
1964:
1938:
1921:
1904:
1849:
1842:
1835:
1828:
1810:
1803:
1796:
1789:
1767:
1760:
1725:
1718:
1696:
1687:
1664:
1655:
1646:
1630:
1625:
1621:
1600:
1580:
1565:kinetic energy
1550:
1545:
1541:
1535:
1531:
1525:
1520:
1517:
1514:
1510:
1504:
1501:
1496:
1493:
1473:
1470:
1467:
1464:
1461:
1458:
1434:
1431:
1406:
1398:
1394:
1390:
1384:
1378:
1374:
1367:
1364:
1360:
1356:
1320:
1314:
1311:
1304:
1299:
1294:
1288:
1283:
1278:
1273:
1270:
1267:
1261:
1258:
1251:
1246:
1241:
1235:
1230:
1225:
1220:
1214:
1211:
1204:
1199:
1194:
1188:
1183:
1178:
1128:and so on. In
1123:
1116:
1109:
1102:
1092:
1085:
1078:
1071:
1057:
1050:
1043:
1031:
1025:
1018:
1005:point particle
928:
925:
857:{\textstyle L}
853:
781:
780:
778:
777:
770:
763:
755:
752:
751:
750:
749:
736:
719:
718:
714:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
598:
592:
591:
588:
587:
584:
583:
580:
579:
560:
559:
554:
549:
544:
542:Coriolis force
539:
538:
537:
527:
522:
517:
511:
510:
505:
504:
501:
500:
496:
495:
490:
485:
484:
483:
478:
468:
463:
458:
451:
440:
439:
438:
433:
420:
419:
414:
409:
404:
397:
392:
387:
381:
380:
377:
376:
373:
372:
369:
368:
365:
364:
359:
354:
349:
344:
339:
333:
327:
325:
318:
315:
312:
311:
308:
307:
303:
302:
297:
292:
287:
282:
277:
272:
267:
261:
256:
250:
245:
234:
229:
224:
219:
214:
213:
212:
207:
197:
192:
187:
182:
176:
175:
172:
171:
168:
167:
163:
162:
157:
152:
147:
142:
137:
132:
127:
121:
120:
117:
116:
113:
112:
110:
109:
104:
99:
93:
90:
89:
84:
70:
67:
61:
57:
51:
33:
32:
26:
25:
15:
9:
6:
4:
3:
2:
29164:
29153:
29150:
29148:
29145:
29143:
29140:
29139:
29137:
29122:
29119:
29117:
29114:
29112:
29109:
29107:
29104:
29102:
29099:
29098:
29096:
29092:
29086:
29083:
29081:
29080:Ocean physics
29078:
29076:
29073:
29071:
29068:
29066:
29063:
29061:
29058:
29056:
29053:
29051:
29048:
29046:
29043:
29041:
29038:
29037:
29035:
29033:
29029:
29023:
29020:
29016:
29015:Modern optics
29013:
29011:
29008:
29006:
29003:
29002:
29001:
28998:
28996:
28993:
28991:
28988:
28986:
28983:
28979:
28976:
28974:
28971:
28970:
28969:
28966:
28965:
28963:
28961:
28957:
28949:
28946:
28944:
28941:
28940:
28939:
28936:
28932:
28929:
28927:
28924:
28923:
28922:
28919:
28917:
28914:
28912:
28909:
28905:
28902:
28900:
28897:
28895:
28892:
28890:
28887:
28886:
28885:
28882:
28881:
28879:
28877:
28873:
28865:
28864:Computational
28862:
28861:
28860:
28857:
28855:
28852:
28851:
28849:
28845:
28837:
28834:
28833:
28832:
28829:
28827:
28824:
28823:
28821:
28817:
28813:
28805:
28800:
28798:
28793:
28791:
28786:
28785:
28782:
28770:
28766:
28762:
28759:
28757:
28756:
28752:
28751:
28748:
28742:
28739:
28737:
28734:
28732:
28729:
28725:
28722:
28721:
28720:
28717:
28716:
28714:
28712:Organizations
28710:
28704:
28701:
28700:
28698:
28694:
28687:
28683:
28681:
28678:
28676:
28673:
28671:
28668:
28666:
28663:
28662:
28660:
28656:
28650:
28647:
28645:
28642:
28640:
28637:
28635:
28632:
28630:
28627:
28625:
28622:
28620:
28617:
28616:
28614:
28612:
28608:
28598:
28595:
28593:
28590:
28588:
28585:
28583:
28580:
28578:
28575:
28573:
28572:Quantum group
28570:
28568:
28565:
28563:
28560:
28558:
28555:
28554:
28552:
28550:
28546:
28538:
28535:
28533:
28530:
28529:
28528:
28527:Supersymmetry
28525:
28521:
28518:
28516:
28513:
28512:
28511:
28510:String theory
28508:
28506:
28503:
28499:
28496:
28495:
28494:
28491:
28487:
28484:
28482:
28479:
28477:
28474:
28472:
28469:
28467:
28464:
28462:
28459:
28457:
28454:
28453:
28452:
28449:
28445:
28442:
28440:
28437:
28436:
28435:
28432:
28431:
28429:
28427:
28421:
28415:
28412:
28410:
28409:Path integral
28407:
28405:
28401:
28398:
28395:
28391:
28390:Distributions
28388:
28387:
28385:
28383:
28379:
28373:
28370:
28366:
28363:
28361:
28358:
28356:
28353:
28351:
28348:
28347:
28346:
28343:
28339:
28336:
28334:
28331:
28329:
28326:
28324:
28321:
28320:
28319:
28316:
28312:
28309:
28307:
28304:
28302:
28299:
28297:
28294:
28293:
28292:
28289:
28285:
28282:
28281:
28280:
28277:
28273:
28270:
28268:
28265:
28264:
28263:
28260:
28256:
28253:
28251:
28248:
28247:
28246:
28243:
28239:
28236:
28234:
28231:
28229:
28226:
28225:
28224:
28221:
28217:
28214:
28212:
28209:
28207:
28204:
28203:
28202:
28199:
28195:
28192:
28191:
28190:
28187:
28185:
28182:
28181:
28179:
28177:
28173:
28167:
28164:
28162:
28159:
28157:
28156:Combinatorics
28154:
28152:
28149:
28147:
28144:
28143:
28141:
28139:
28135:
28125:
28122:
28120:
28117:
28115:
28112:
28110:
28107:
28105:
28102:
28100:
28097:
28095:
28092:
28090:
28087:
28085:
28082:
28081:
28079:
28077:
28073:
28067:
28064:
28062:
28059:
28057:
28054:
28052:
28049:
28045:
28042:
28041:
28040:
28037:
28035:
28032:
28030:
28029:Coding theory
28027:
28025:
28022:
28020:
28017:
28013:
28010:
28008:
28005:
28004:
28003:
28000:
27999:
27997:
27995:
27994:Computational
27991:
27987:
27980:
27975:
27973:
27968:
27966:
27961:
27960:
27957:
27945:
27942:
27940:
27937:
27935:
27932:
27930:
27927:
27925:
27922:
27920:
27917:
27915:
27912:
27910:
27907:
27905:
27902:
27900:
27897:
27895:
27892:
27890:
27887:
27886:
27883:
27879:
27872:
27867:
27865:
27860:
27858:
27853:
27852:
27849:
27842:
27839:
27836:
27833:
27830:
27827:
27817:
27813:
27808:
27807:
27798:
27797:
27792:
27789:
27785:
27779:
27775:
27770:
27767:
27763:
27762:
27751:
27745:
27741:
27740:
27734:
27730:
27724:
27719:
27718:
27711:
27707:
27705:9780521829519
27701:
27697:
27696:
27690:
27686:
27684:0-03-063366-4
27680:
27676:
27671:
27667:
27665:0-03-063366-4
27661:
27657:
27652:
27648:
27646:0-521-83531-3
27642:
27638:
27637:
27631:
27627:
27621:
27617:
27616:
27610:
27606:
27602:
27598:
27594:
27590:
27586:
27581:
27576:
27572:
27568:
27563:
27559:
27555:
27551:
27547:
27543:
27539:
27534:
27529:
27525:
27521:
27516:
27512:
27508:
27504:
27500:
27496:
27492:
27488:
27484:
27479:
27474:
27470:
27466:
27461:
27457:
27451:
27447:
27443:
27439:
27438:
27432:
27428:
27426:0-201-41625-5
27422:
27418:
27413:
27409:
27407:0-521-56632-0
27403:
27399:
27395:
27390:
27386:
27384:1-891389-22-X
27380:
27376:
27375:
27369:
27365:
27359:
27355:
27354:
27348:
27344:
27342:3-540-63055-4
27338:
27334:
27333:
27327:
27323:
27321:0-486-67002-3
27317:
27313:
27312:
27306:
27302:
27300:0-486-60595-7
27296:
27292:
27288:
27283:
27279:
27277:3-540-33021-6
27273:
27269:
27268:
27262:
27258:
27256:3-540-28208-4
27252:
27248:
27244:
27239:
27236:
27233:
27229:
27223:
27219:
27218:
27212:
27208:
27206:0-486-65067-7
27202:
27198:
27194:
27189:
27185:
27183:0-201-65702-3
27179:
27175:
27174:
27169:
27165:
27161:
27157:
27151:
27147:
27143:
27139:
27135:
27133:9781860944352
27129:
27125:
27120:
27116:
27114:9780521636360
27110:
27106:
27105:
27099:
27095:
27093:9780521575720
27089:
27085:
27084:
27078:
27074:
27068:
27064:
27060:
27056:
27052:
27048:
27046:9780750628969
27042:
27038:
27034:
27030:
27029:Landau, L. D.
27026:
27022:
27016:
27012:
27007:
27003:
27002:
26996:
26992:
26991:
26985:
26984:
26973:, p. 271
26972:
26967:
26965:
26958:
26953:
26947:
26942:
26936:
26931:
26925:
26924:Kosyakov 2007
26920:
26914:, p. 267
26913:
26908:
26901:
26896:
26889:
26884:
26878:, p. 156
26877:
26872:
26865:
26860:
26853:
26848:
26842:, p. 297
26841:
26836:
26828:
26822:
26818:
26814:
26810:
26809:
26801:
26794:
26789:
26782:
26777:
26770:
26765:
26758:
26753:
26746:
26741:
26734:
26729:
26727:
26719:
26714:
26708:
26703:
26696:
26691:
26689:
26680:
26676:
26672:
26668:
26664:
26660:
26656:
26652:
26648:
26641:
26634:
26629:
26622:
26617:
26610:
26605:
26597:
26591:
26587:
26586:
26578:
26571:
26566:
26559:
26554:
26547:
26542:
26536:, p. 234
26535:
26530:
26523:
26518:
26511:
26506:
26499:
26494:
26487:
26482:
26475:
26470:
26463:
26458:
26451:
26446:
26440:, p. 156
26439:
26434:
26428:
26424:
26418:
26412:
26407:
26400:
26395:
26388:
26383:
26370:
26364:
26360:
26356:
26352:
26348:
26341:
26334:
26329:
26322:
26316:
26308:
26304:
26300:
26296:
26292:
26288:
26284:
26277:
26271:, p. 269
26270:
26265:
26263:
26261:
26259:
26252:, p. 270
26251:
26246:
26238:
26236:9781784296148
26232:
26228:
26224:
26217:
26215:
26200:
26193:
26190:Tatum, J. B.
26186:
26184:
26169:
26165:
26161:
26157:
26150:
26148:
26140:
26135:
26128:
26123:
26116:
26111:
26104:
26098:
26094:
26079:
26071:
26067:
26051:
26048:
26043:
26039:
26035:
26030:
26026:
26022:
26017:
26014:
26011:
26006:
26002:
25998:
25993:
25989:
25985:
25980:
25977:
25974:
25969:
25965:
25961:
25956:
25952:
25948:
25939:
25936:for particle
25923:
25920:
25915:
25905:
25902:
25897:
25880:
25873:
25867:
25843:
25836:
25812:
25796:
25788:
25769:
25761:
25751:
25747:
25737:
25731:
25727:
25716:
25713:
25711:
25708:
25706:
25703:
25701:
25698:
25696:
25693:
25691:
25688:
25685:
25682:
25680:
25677:
25675:
25672:
25670:
25667:
25665:
25662:
25660:
25657:
25655:
25652:
25651:
25646:
25635:
25628:
25626:
25622:
25616:
25614:
25610:
25606:
25596:
25594:
25591:
25587:
25564:
25546:
25543:
25537:
25531:
25523:
25519:
25515:
25496:
25493:
25485:
25479:
25476:
25470:
25467:
25461:
25458:
25441:
25435:
25431:
25427:
25422:
25412:
25410:
25406:
25402:
25398:
25394:
25390:
25386:
25382:
25377:
25375:
25371:
25367:
25363:
25359:
25355:
25351:
25341:
25339:
25335:
25331:
25325:
25315:
25313:
25307:
25297:
25296:
25291:
25280:
25275:
25271:
25267:
25263:
25259:
25252:
25248:
25244:
25240:
25236:
25228:
25218:
25216:
25212:
25210:
25206:
25202:
25198:
25194:
25178:
25175:
25167:
25160:
25157:
25145:
25134:
25127:
25124:
25115:
25110:
25107:
25104:
25100:
25096:
25093:
25085:
25075:
25067:
25064:
25061:
25055:
25049:
25043:
25037:
25031:
25025:
25021:
25017:
25013:
25009:
25004:
24991:
24983:
24980:
24974:
24963:
24960:
24951:
24935:
24926:
24912:
24907:
24902:
24897:
24892:
24878:
24872:
24869:
24864:
24861:
24855:
24852:
24849:
24846:
24843:
24839:
24832:
24820:
24812:
24806:
24794:
24790:
24787:
24784:
24778:
24768:
24764:
24760:
24743:
24737:
24734:
24728:
24725:
24719:
24715:
24712:
24704:
24683:
24678:
24660:
24654:
24643:
24640:
24632:
24629:
24623:
24612:
24604:
24603:Lorentz force
24600:
24596:
24595:rule of thumb
24592:
24576:
24573:
24570:
24567:
24559:
24553:
24541:
24538:
24533:
24526:
24514:
24508:
24505:
24499:
24496:
24487:
24474:
24466:
24458:
24449:
24443:
24424:
24421:
24413:
24410:
24395:
24391:
24387:
24383:
24379:
24373:
24369:
24365:
24361:
24357:
24351:
24347:
24343:
24339:
24335:
24329:
24325:
24321:
24317:
24313:
24309:
24304:
24300:
24296:
24291:
24289:
24285:
24281:
24277:
24267:
24254:
24251:
24243:
24236:
24233:
24221:
24212:
24204:
24200:
24191:
24182:
24178:
24170:
24163:
24160:
24148:
24139:
24132:
24094:
24087:
24084:
24072:
24063:
24055:
24051:
24042:
24033:
24030:
24025:
24021:
24012:
24007:
24003:
23987:
23982:
23975:
23972:
23963:
23956:
23953:
23944:
23941:
23937:
23931:
23926:
23923:
23920:
23916:
23910:
23905:
23902:
23899:
23895:
23889:
23886:
23881:
23878:
23870:
23866:
23862:
23857:
23853:
23849:
23844:
23842:
23828:
23824:
23822:
23818:
23814:
23810:
23804:
23802:
23798:
23794:
23789:
23787:
23781:
23777:
23769:
23765:
23759:
23746:
23738:
23734:
23730:
23724:
23720:
23714:
23709:
23702:
23699:
23692:
23689:
23686:
23672:
23663:
23659:
23654:
23650:
23643:
23626:
23618:
23614:
23610:
23604:
23600:
23594:
23588:
23578:
23567:
23564:
23558:
23555:
23549:
23540:
23536:
23530:
23525:
23512:
23506:
23503:
23497:
23494:
23488:
23485:
23480:
23477:
23471:
23466:
23459:
23456:
23449:
23446:
23437:
23431:
23425:
23421:
23414:
23408:
23395:
23392:
23389:
23383:
23380:
23372:
23368:
23364:
23361:
23352:
23349:
23332:
23322:
23317:
23313:
23303:
23286:
23280:
23274:
23271:
23267:
23261:
23254:
23251:
23242:
23238:
23234:
23229:
23222:
23219:
23211:
23207:
23202:
23199:
23194:
23185:
23173:
23168:
23161:
23157:
23150:
23136:
23133:
23130:
23124:
23113:
23104:
23098:
23092:
23087:
23079:
23074:
23064:
23057:
23051:
23044:
23039:
23034:
23025:
23018:
23011:
23005:
23001:
22992:
22985:
22981:
22956:
22949:
22924:
22921:
22916:
22909:
22897:
22892:
22889:
22880:
22869:
22862:
22856:
22849:
22837:
22832:
22829:
22820:
22817:
22804:
22797:
22790:
22784:
22777:
22771:
22767:
22763:
22755:
22748:
22744:
22739:
22734:
22730:
22722:
22713:
22704:
22695:
22688:
22687:Central force
22684:
22674:
22672:
22655:
22646:
22643:
22632:
22628:
22612:
22603:
22600:
22587:
22574:
22571:
22568:
22565:
22562:
22557:
22554:
22549:
22546:
22543:
22540:
22535:
22530:
22527:
22520:
22514:
22511:
22488:
22485:
22482:
22479:
22476:
22473:
22467:
22464:
22458:
22455:
22446:
22443:
22434:
22431:
22428:
22424:
22414:
22411:
22403:
22399:
22395:
22392:
22389:
22386:
22383:
22377:
22374:
22365:
22361:
22354:
22333:
22328:
22315:
22312:
22309:
22306:
22303:
22298:
22291:
22288:
22281:
22278:
22275:
22272:
22269:
22266:
22260:
22257:
22251:
22248:
22245:
22239:
22236:
22227:
22224:
22221:
22198:
22178:
22175:
22172:
22169:
22163:
22160:
22154:
22151:
22148:
22142:
22139:
22130:
22127:
22124:
22118:
22109:
22106:
22095:
22086:
22081:
22077:
22068:
22063:
22046:
22043:
22040:
22037:
22034:
22031:
22028:
22025:
22020:
22013:
22010:
22001:
21997:
21993:
21988:
21985:
21980:
21977:
21974:
21971:
21965:
21962:
21956:
21950:
21947:
21941:
21938:
21933:
21926:
21923:
21915:
21911:
21908:
21905:
21901:
21895:
21892:
21885:
21877:
21874:
21871:
21868:
21865:
21862:
21859:
21855:
21849:
21844:
21840:
21837:
21834:
21828:
21825:
21819:
21815:
21810:
21805:
21800:
21796:
21793:
21790:
21784:
21781:
21775:
21772:
21766:
21763:
21756:
21750:
21746:
21741:
21738:
21733:
21728:
21721:
21718:
21711:
21706:
21703:
21696:
21688:
21685:
21682:
21677:
21672:
21631:
21627:
21624:
21621:
21618:
21597:
21591:
21568:
21565:
21558:
21553:
21530:
21527:
21519:
21515:
21510:
21507:
21502:
21497:
21490:
21487:
21480:
21475:
21472:
21467:
21464:
21444:
21424:
21415:
21398:
21395:
21392:
21389:
21383:
21380:
21374:
21371:
21348:
21345:
21331:
21328:
21325:
21322:
21319:
21316:
21296:
21287:
21284:
21281:
21275:
21272:
21266:
21263:
21257:
21254:
21248:
21225:
21222:
21208:
21205:
21202:
21199:
21196:
21193:
21190:
21170:
21144:
21124:
21104:
21096:
21092:
21088:
21079:
21070:
21068:
21064:
21059:
21058:is constant.
21056:
21052:
21048:
21044:
21040:
21036:
21020:
21014:
21011:
21005:
21002:
20997:
20993:
20987:
20983:
20979:
20976:
20967:
20964:
20953:
20944:
20939:
20935:
20926:
20910:
20907:
20898:
20895:
20888:
20885:
20880:
20876:
20870:
20866:
20862:
20853:
20821:
20818:
20815:
20810:
20803:
20800:
20792:
20789:
20786:
20783:
20780:
20777:
20772:
20768:
20764:
20761:
20752:
20749:
20741:
20737:
20733:
20724:
20692:
20689:
20686:
20680:
20672:
20663:
20655:
20648:
20645:
20637:
20634:
20629:
20625:
20621:
20616:
20609:
20606:
20596:
20593:
20590:
20584:
20581:
20575:
20555:
20549:
20543:
20540:
20532:
20525:
20522:
20514:
20511:
20506:
20502:
20496:
20492:
20488:
20483:
20476:
20473:
20464:
20460:
20456:
20451:
20444:
20441:
20429:
20426:
20421:
20418:
20410:
20406:
20402:
20396:
20392:
20388:
20376:
20374:
20358:
20350:
20347:
20341:
20330:
20322:
20318:
20302:
20296:
20288:
20279:
20276:
20270:
20267:
20261:
20241:
20235:
20232:
20226:
20223:
20219:
20210:
20207:
20196:
20187:
20180:
20163:
20157:
20154:
20148:
20145:
20136:
20133:
20122:
20112:
20106:
20098:
20089:
20086:
20080:
20072:
20046:
20040:
20032:
20023:
20019:
20010:
20007:
19996:
19987:
19980:
19959:
19955:
19950:
19937:
19931:
19928:
19925:
19922:
19919:
19913:
19910:
19902:
19895:
19892:
19885:
19880:
19873:
19870:
19863:
19858:
19851:
19848:
19838:
19833:
19830:
19825:
19816:
19813:
19807:
19801:
19798:
19792:
19786:
19783:
19777:
19774:
19771:
19768:
19765:
19762:
19756:
19742:
19739:
19726:
19712:
19704:
19701:
19688:
19684:
19680:
19676:
19666:
19658:
19641:
19638:
19635:
19632:
19626:
19623:
19594:
19591:
19565:
19562:
19559:
19556:
19550:
19547:
19518:
19515:
19492:
19467:
19460:
19457:
19445:
19438:
19435:
19423:
19418:
19407:
19402:
19399:
19395:
19385:
19368:
19363:
19356:
19353:
19341:
19337:
19328:
19321:
19318:
19306:
19301:
19288:
19284:
19280:
19274:
19266:
19257:
19249:
19245:
19236:
19227:
19222:
19215:
19212:
19202:
19199:
19193:
19190:
19184:
19181:
19173:
19170:
19166:
19160:
19156:
19153:
19145:
19142:
19134:
19130:
19121:
19112:
19106:
19098:
19089:
19083:
19080:
19073:
19066:
19063:
19056:
19045:
19038:
19035:
19023:
19016:
19013:
19001:
18996:
18985:
18979:
18976:
18969:
18965:
18961:
18950:
18943:
18940:
18928:
18924:
18915:
18910:
18894:
18891:
18883:
18879:
18870:
18861:
18853:
18846:
18843:
18831:
18819:
18816:
18812:
18791:
18787:
18784:
18780:
18774:
18761:
18756:
18753:
18749:
18745:
18740:
18736:
18732:
18727:
18723:
18719:
18716:
18707:
18703:
18699:
18694:
18681:
18676:
18672:
18668:
18663:
18659:
18655:
18652:
18643:
18639:
18634:
18630:
18626:
18622:
18618:
18608:
18595:
18588:
18585:
18580:
18577:
18572:
18567:
18563:
18560:
18554:
18549:
18544:
18540:
18537:
18526:
18522:
18518:
18514:
18509:
18496:
18489:
18486:
18481:
18478:
18474:
18470:
18467:
18457:
18453:
18449:
18442:
18438:
18432:
18428:
18417:
18413:
18409:
18405:
18401:
18396:
18392:
18388:
18384:
18380:
18359:
18349:
18346:
18343:
18338:
18328:
18323:
18310:
18305:
18301:
18297:
18289:
18279:
18276:
18273:
18270:
18265:
18255:
18252:
18247:
18237:
18231:
18221:
18217:
18214:
18209:
18205:
18200:
18196:
18190:
18180:
18178:
18161:
18158:
18155:
18152:
18149:
18135:
18121:
18113:
18107:
18104:
18098:
18089:
18078:
18072:
18058:
18050:
18044:
18040:
18034:
18029:
18009:
18006:
18001:
17998:
17992:
17988:
17984:
17981:
17946:
17936:
17929:
17926:
17922:
17917:
17915:
17907:
17899:
17890:
17880:
17877:
17871:
17848:
17836:
17825:
17819:
17807:
17784:
17775:
17768:
17765:
17761:
17756:
17754:
17746:
17738:
17729:
17719:
17711:
17702:
17679:
17667:
17656:
17640:
17628:
17617:
17615:
17607:
17604:
17599:
17596:
17581:
17576:
17571:
17565:
17561:
17557:
17551:
17549:
17533:
17525:
17509:
17501:
17490:
17476:
17456:
17447:
17431:
17419:
17408:
17399:
17383:
17371:
17340:
17309:
17275:
17268:
17265:
17253:
17223:
17216:
17213:
17206:
17198:
17191:
17188:
17176:
17165:
17160:
17157:
17154:
17129:
17126:
17106:
17100:
17073:
17048:
17039:
17023:
17014:
16998:
16994:
16984:
16968:
16942:
16936:
16933:
16927:
16916:
16905:
16902:
16899:
16893:
16890:
16884:
16873:
16862:
16859:
16839:
16833:
16830:
16824:
16813:
16802:
16799:
16793:
16790:
16784:
16773:
16762:
16753:
16740:
16737:
16734:
16728:
16720:
16711:
16705:
16694:
16685:
16669:
16660:
16652:
16636:
16627:
16621:
16618:
16612:
16601:
16590:
16587:
16578:
16561:
16555:
16547:
16538:
16532:
16521:
16505:
16501:
16491:
16475:
16466:
16458:
16454:
16444:
16433:
16422:
16413:
16397:
16388:
16372:
16368:
16351:
16341:
16337:
16335:
16327:
16324:
16318:
16310:
16301:
16295:
16284:
16275:
16259:
16250:
16242:
16226:
16217:
16215:
16207:
16204:
16198:
16187:
16176:
16173:
16160:
16143:
16135:
16130:
16125:
16122:
16119:
16116:
16113:
16108:
16102:
16095:
16092:
16064:
16060:
16051:
16047:
16036:
16032:
16026:
16013:
16008:
16005:
16002:
15998:
15993:
15988:
15983:
15978:
15975:
15972:
15969:
15966:
15963:
15960:
15955:
15949:
15942:
15939:
15911:
15907:
15898:
15894:
15883:
15879:
15873:
15860:
15855:
15852:
15849:
15845:
15839:
15835:
15831:
15826:
15821:
15818:
15815:
15811:
15806:
15801:
15799:
15789:
15784:
15781:
15778:
15775:
15772:
15769:
15766:
15761:
15755:
15751:
15747:
15723:
15719:
15710:
15706:
15695:
15691:
15685:
15672:
15667:
15664:
15661:
15657:
15651:
15644:
15641:
15632:
15627:
15624:
15621:
15617:
15612:
15607:
15601:
15588:
15583:
15580:
15577:
15574:
15571:
15566:
15560:
15556:
15552:
15528:
15524:
15515:
15511:
15500:
15496:
15490:
15477:
15472:
15469:
15466:
15462:
15457:
15452:
15446:
15419:
15415:
15408:
15395:
15388:
15378:
15369:
15358:
15347:
15323:
15317:
15306:
15290:
15286:
15282:
15274:
15265:
15254:
15243:
15240:
15234:
15223:
15203:
15197:
15173:
15169:
15165:
15159:
15144:has the form
15126:
15106:
15101:
15094:
15091:
15065:
15058:
15055:
15028:
15023:
15020:
15017:
15014:
15011:
15006:
14994:
14986:
14982:
14971:
14967:
14953:
14948:
14932:
14928:
14906:
14902:
14893:
14888:
14881:
14878:
14857:
14841:
14836:
14828:
14824:
14807:
14803:
14798:
14791:
14788:
14779:
14774:
14771:
14768:
14764:
14759:
14737:
14728:
14723:
14716:
14713:
14692:
14680:
14672:
14668:
14651:
14647:
14642:
14635:
14632:
14623:
14618:
14615:
14612:
14608:
14603:
14594:
14565:
14561:
14540:
14532:
14516:
14512:
14508:
14500:
14490:
14486:
14482:
14472:
14456:
14449:
14446:
14435:
14431:
14408:
14402:
14399:
14393:
14382:
14371:
14368:
14362:
14359:
14353:
14342:
14331:
14322:
14318:
14313:
14309:
14299:
14280:
14276:
14256:
14248:
14244:
14224:
14219:
14216:
14212:
14191:
14188:
14185:
14182:
14179:
14176:
14170:
14167:
14164:
14158:
14155:
14152:
14149:
14146:
14143:
14139:
14130:
14123:
14120:
14108:
14097:
14090:
14087:
14078:
14073:
14070:
14067:
14063:
14058:
14054:
14051:
14032:
14029:
14026:
14022:
14016:
14009:
14006:
13997:
13990:
13987:
13978:
13975:
13971:
13965:
13962:
13959:
13955:
13951:
13947:
13941:
13938:
13933:
13925:
13918:
13915:
13903:
13892:
13885:
13882:
13873:
13868:
13865:
13862:
13858:
13854:
13846:
13839:
13836:
13824:
13813:
13806:
13803:
13794:
13789:
13786:
13783:
13779:
13757:
13750:
13747:
13738:
13731:
13728:
13719:
13716:
13712:
13706:
13703:
13700:
13696:
13690:
13687:
13682:
13678:
13672:
13665:
13662:
13650:
13646:
13631:
13618:
13614:
13610:
13604:
13597:
13594:
13582:
13578:
13563:
13550:
13544:
13541:
13538:
13534:
13528:
13525:
13520:
13515:
13511:
13505:
13502:
13497:
13494:
13469:
13465:
13461:
13458:
13455:
13450:
13446:
13434:
13409:
13406:
13403:
13388:
13381:
13378:
13366:
13355:
13348:
13345:
13336:
13331:
13328:
13325:
13321:
13310:
13307:
13299:
13283:
13280:
13266:
13264:
13259:
13255:
13251:
13246:
13242:
13238:
13233:
13229:
13213:
13204:
13201:
13190:
13181:
13176:
13172:
13167:
13158:
13155:
13144:
13135:
13130:
13126:
13121:
13112:
13109:
13098:
13089:
13084:
13080:
13071:
13067:
13063:
13059:
13055:
13051:
13047:
13043:
13027:
13021:
13018:
13012:
13009:
13003:
12997:
12994:
12988:
12982:
12979:
12973:
12967:
12964:
12958:
12952:
12949:
12943:
12940:
12937:
12934:
12928:
12919:
12917:
12901:
12898:
12890:
12886:
12877:
12868:
12860:
12853:
12850:
12838:
12826:
12810:
12805:
12798:
12795:
12783:
12776:
12772:
12767:
12754:
12746:
12739:
12736:
12724:
12715:
12710:
12706:
12697:
12690:
12686:
12675:
12662:
12659:
12651:
12647:
12638:
12627:
12619:
12612:
12609:
12597:
12585:
12582:
12578:
12556:
12550:
12541:
12524:
12521:
12513:
12509:
12498:
12494:
12483:
12474:
12467:
12464:
12452:
12443:
12439:
12431:
12424:
12421:
12409:
12400:
12393:
12390:
12386:
12380:
12374:
12370:
12362:
12359:
12355:
12350:
12342:
12338:
12327:
12323:
12313:
12306:
12303:
12299:
12289:
12282:
12279:
12267:
12256:
12248:
12241:
12238:
12224:
12217:
12214:
12196:
12192:
12183:
12172:
12168:
12160:
12153:
12150:
12136:
12129:
12126:
12113:
12106:
12103:
12099:
12089:
12082:
12079:
12067:
12058:
12050:
12043:
12040:
12026:
12019:
12016:
12002:
11994:
11987:
11984:
11972:
11963:
11956:
11953:
11949:
11943:
11937:
11933:
11925:
11922:
11918:
11909:
11904:
11892:
11885:
11882:
11864:
11857:
11854:
11842:
11831:
11823:
11818:
11806:
11802:
11787:
11783:
11774:
11763:
11759:
11750:
11743:
11740:
11726:
11719:
11716:
11698:
11691:
11688:
11676:
11666:
11659:
11656:
11652:
11646:
11640:
11636:
11628:
11625:
11617:
11613:
11604:
11593:
11585:
11578:
11575:
11563:
11551:
11548:
11544:
11529:
11510:
11506:
11495:
11488:
11485:
11472:
11468:
11460:
11456:
11445:
11441:
11431:
11424:
11421:
11417:
11392:
11389:
11373:
11369:
11362:
11352:
11348:
11342:
11331:
11326:
11319:
11316:
11305:
11302:
11286:
11282:
11273:
11269:
11258:
11254:
11248:
11235:
11231:
11227:
11223:
11215:
11211:
11199:
11195:
11184:
11177:
11174:
11170:
11145:
11142:
11128:
11120:
11116:
11105:
11101:
11095:
11084:
11079:
11072:
11069:
11058:
11055:
11039:
11035:
11026:
11022:
11011:
11007:
11001:
10988:
10984:
10980:
10972:
10968:
10956:
10951:
10947:
10928:
10909:
10905:
10894:
10890:
10880:
10872:
10865:
10862:
10847:
10842:
10838:
10805:
10802:
10788:
10778:
10774:
10764:
10759:
10752:
10749:
10738:
10735:
10719:
10715:
10704:
10700:
10688:
10684:
10680:
10674:
10669:
10665:
10641:
10638:
10635:
10629:
10619:
10615:
10605:
10599:
10595:
10590:
10582:
10578:
10567:
10563:
10551:
10547:
10543:
10537:
10533:
10528:
10505:
10502:
10489:
10485:
10481:
10476:
10472:
10458:
10455:
10442:
10434:
10430:
10420:
10417:
10407:
10399:
10392:
10389:
10376:
10373:
10360:
10339:
10323:
10317:
10314:
10308:
10305:
10299:
10288:
10274:
10263:
10257:
10254:
10235:
10232:
10226:
10223:
10217:
10206:
10194:
10191:
10182:
10176:
10172:
10168:
10164:
10157:
10153:
10149:
10145:
10141:
10137:
10133:
10123:
10121:
10112:
10105:
10101:
10091:
10084:
10080:
10059:
10047:
10043:
10034:
10027:
10024:
10012:
10008:
9999:
9992:
9989:
9975:
9972:
9970:
9962:
9959:
9952:
9939:
9936:
9930:
9916:
9897:
9885:
9880:
9876:
9873:
9870:
9863:
9860:
9854:
9845:
9834:
9828:
9814:
9803:
9791:
9786:
9782:
9780:
9772:
9769:
9762:
9759:
9753:
9744:
9733:
9727:
9712:
9709:
9697:
9685:
9680:
9676:
9674:
9657:
9654:
9641:
9637:
9633:
9629:
9624:
9611:
9606:
9599:
9596:
9584:
9580:
9568:
9565:
9554:
9543:
9539:
9535:
9529:
9518:
9515:
9504:
9475:
9462:
9459:
9448:
9420:
9414:
9401:
9398:
9387:
9376:
9370:
9367:
9361:
9350:
9339:
9336:
9330:
9327:
9321:
9310:
9298:
9295:
9284:
9280:
9276:
9266:
9262:
9255:
9245:
9241:
9234:
9224:
9217:
9211:
9207:
9202:
9198:
9194:
9189:
9185:
9181:
9166:
9152:
9144:
9127:
9123:
9111:
9107:
9101:
9096:
9093:
9090:
9086:
9082:
9077:
9050:
9047:
9044:
9036:
9019:
9015:
9003:
8999:
8993:
8988:
8985:
8982:
8978:
8974:
8967:
8954:
8945:
8930:
8921:
8914:
8907:
8897:
8891:
8882:
8875:
8858:
8846:
8830:
8822:
8807:
8788:
8784:
8780:
8772:
8768:
8763:
8750:
8747:
8741:
8738:
8733:
8718:
8714:
8705:
8702:
8694:
8687:
8684:
8671:
8668:
8655:
8639:
8631:
8627:
8617:
8614:
8598:
8594:
8590:
8574:
8571:
8568:
8560:
8543:
8539:
8527:
8523:
8517:
8512:
8509:
8506:
8502:
8498:
8490:
8483:
8466:
8454:
8438:
8430:
8415:
8401:
8398:
8390:
8383:
8365:
8362:
8349:
8333:
8325:
8309:
8306:
8290:
8286:
8281:
8277:
8274:
8269:
8256:
8253:
8250:
8239:
8229:
8226:
8222:
8213:
8196:
8192:
8180:
8176:
8170:
8165:
8162:
8159:
8155:
8151:
8143:
8136:
8119:
8107:
8091:
8083:
8068:
8058:
8052:
8047:
8044:
8041:
8037:
8029:
8025:
8017:
8013:
8008:
8004:
8001:
7992:
7989:
7985:
7978:
7974:
7966:
7962:
7957:
7947:
7943:
7939:
7934:
7921:
7915:
7912:
7907:
7892:
7888:
7881:
7873:
7869:
7863:
7858:
7855:
7852:
7848:
7844:
7838:
7835:
7832:
7829:
7824:
7817:
7805:
7800:
7793:
7781:
7778:
7775:
7770:
7760:
7755:
7742:
7739:
7735:
7732:
7723:
7720:= 1, 2, ...,
7719:
7714:
7710:
7704:
7699:
7695:
7690:
7686:
7681:
7676:
7672:
7669:
7664:
7660:
7657:
7653:
7648:
7641:
7636:
7632:
7629:
7622:
7617:
7613:
7610:
7604:
7600:
7596:
7591:
7578:
7575:
7572:
7561:
7551:
7548:
7544:
7535:
7528:
7511:
7499:
7483:
7475:
7460:
7450:
7444:
7439:
7436:
7433:
7429:
7421:
7417:
7409:
7405:
7400:
7391:
7386:
7382:
7378:
7368:
7366:
7362:
7357:
7355:
7351:
7347:
7343:
7339:
7335:
7331:
7327:
7323:
7319:
7315:
7311:
7306:
7304:
7298:
7285:
7282:
7279:
7276:
7268:
7261:
7254:
7250:
7249:
7232:
7229:
7220:
7213:
7209:
7201:
7197:
7192:
7188:
7185:
7178:, defined as
7177:
7172:
7159:
7156:
7153:
7144:
7141:
7134:
7130:
7122:
7118:
7113:
7104:
7103:
7097:
7093:
7088:
7084:
7080:
7076:
7067:
7062:
7058:
7051:
7046:
7042:
7036:
7019:
7016:
7005:
7001:
6997:
6993:
6984:
6977:
6974:
6962:
6950:
6934:
6926:
6922:
6913:
6903:
6897:
6892:
6889:
6886:
6882:
6874:
6870:
6862:
6858:
6853:
6849:
6842:
6838:
6830:
6826:
6820:
6814:
6810:
6806:
6798:
6791:
6788:
6776:
6766:
6759:
6754:
6751:
6748:
6744:
6740:
6738:
6730:
6720:
6714:
6710:
6706:
6698:
6691:
6688:
6676:
6664:
6648:
6644:
6638:
6634:
6630:
6622:
6615:
6612:
6600:
6590:
6583:
6567:
6562:
6558:
6554:
6546:
6542:
6533:
6523:
6517:
6512:
6509:
6506:
6502:
6494:
6490:
6482:
6478:
6473:
6469:
6467:
6462:
6453:
6450:
6443:
6439:
6431:
6427:
6422:
6408:
6404:
6400:
6395:
6391:
6387:
6383:
6376:
6371:
6367:
6363:
6359:
6355:
6339:
6333:
6318:
6314:
6310:
6296:
6290:
6278:
6274:
6262:
6259:
6254:
6247:
6244:
6237:
6233:
6229:
6223:
6216:
6213:
6206:
6198:
6191:
6188:
6176:
6167:
6162:
6158:
6154:
6146:
6142:
6133:
6123:
6117:
6112:
6109:
6106:
6102:
6098:
6095:
6092:
6084:
6080:
6075:
6072:
6067:
6054:
6051:
6043:
6036:
6033:
6021:
6009:
5993:
5985:
5981:
5972:
5958:
5954:
5949:
5945:
5941:
5924:
5916:
5912:
5903:
5894:
5886:
5879:
5876:
5864:
5852:
5836:
5831:
5827:
5818:
5813:
5809:
5805:
5801:
5793:
5789:
5785:
5781:
5776:
5767:
5765:
5760:
5742:
5738:
5729:
5720:
5712:
5705:
5702:
5690:
5678:
5662:
5657:
5653:
5644:
5640:
5635:
5631:
5615:
5612:
5609:
5604:
5600:
5596:
5592:
5587:
5578:
5574:
5565:
5556:
5548:
5541:
5538:
5526:
5514:
5497:
5493:
5488:
5484:
5479:
5473:
5468:
5465:
5462:
5458:
5448:
5435:
5427:
5423:
5414:
5405:
5397:
5390:
5387:
5375:
5363:
5347:
5339:
5335:
5324:
5308:
5303:
5291:
5287:
5281:
5276:
5273:
5270:
5266:
5255:
5242:
5237:
5233:
5229:
5224:
5220:
5214:
5209:
5206:
5203:
5199:
5195:
5190:
5186:
5182:
5174:
5170:
5159:
5141:
5136:
5133:
5130:
5126:
5122:
5117:
5105:
5100:
5097:
5094:
5090:
5086:
5081:
5071:
5068:
5063:
5051:
5046:
5043:
5040:
5036:
5015:
5007:
5003:
4992:
4976:
4971:
4959:
4954:
4951:
4948:
4944:
4940:
4935:
4931:
4923:
4918:
4914:
4911:
4906:
4902:
4897:
4895:
4890:
4877:
4872:
4868:
4864:
4856:
4852:
4841:
4823:
4818:
4815:
4812:
4808:
4804:
4799:
4789:
4781:
4776:
4772:
4769:
4762:
4758:
4753:
4749:
4744:
4740:
4735:
4731:
4728:
4724:
4717:
4713:
4708:
4704:
4699:
4695:
4690:
4686:
4681:
4671:
4668:
4663:
4659:
4656:
4650:
4637:
4634:
4629:
4619:
4616:
4608:
4596:
4592:
4588:
4583:
4568:
4563:
4560:
4557:
4553:
4532:
4529:
4526:
4521:
4511:
4508:
4503:
4491:
4486:
4483:
4480:
4476:
4466:
4464:
4460:
4455:
4451:
4448:
4444:
4443:
4437:
4424:
4421:
4416:
4406:
4403:
4395:
4383:
4379:
4375:
4370:
4360:
4355:
4340:
4335:
4332:
4329:
4325:
4315:
4311:
4308:
4304:
4300:
4296:
4292:
4288:
4284:
4274:
4270:
4266:
4263:
4259:
4246:
4234:
4222:
4218:
4209:
4206:
4202:
4189:
4181:
4173:
4160:
4156:
4152:
4147:
4145:
4141:
4137:
4131:
4127:
4122:
4121:
4114:
4110:
4103:
4101:
4097:
4093:
4089:
4088:
4087:metric tensor
4083:
4079:
4057:
4045:
4041:
4026:
4014:
4010:
3996:
3993:
3989:
3985:
3980:
3977:
3972:
3969:
3961:
3956:
3951:
3947:
3943:
3927:
3921:
3909:
3905:
3893:
3888:
3881:
3878:
3870:
3866:
3857:
3853:
3844:
3835:
3827:
3820:
3817:
3805:
3793:
3776:
3770:
3767:
3763:
3759:
3755:
3748:
3736:
3732:
3717:
3705:
3701:
3687:
3684:
3675:
3667:
3659:
3655:
3642:
3638:
3632:
3618:
3614:
3611:
3606:
3602:
3594:
3590:
3584:
3580:
3576:
3572:
3571:
3566:
3560:
3557:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3524:
3520:
3504:
3496:
3493:
3480:
3476:
3472:
3468:
3460:
3456:
3450:Newton's laws
3442:
3440:
3436:
3432:
3428:
3424:
3419:
3416:
3412:
3408:
3403:
3399:
3393:
3389:
3385:
3381:
3377:
3372:
3366:
3348:
3344:
3335:
3326:
3322:
3314:
3307:
3304:
3292:
3283:
3276:
3255:
3254:(second kind)
3251:
3247:
3246:
3242:
3241:
3235:
3222:
3216:
3213:
3207:
3196:
3185:
3182:
3179:
3171:
3166:
3162:
3157:
3144:
3138:
3128:
3112:
3107:
3100:
3097:
3085:
3081:
3070:
3052:
3047:
3044:
3041:
3037:
3033:
3028:
3017:
3011:
2999:
2995:
2983:
2978:
2971:
2968:
2956:
2952:
2948:
2943:
2930:
2920:
2917:
2911:
2908:
2895:
2891:
2887:
2881:
2878:
2865:
2861:
2857:
2851:
2848:
2835:
2831:
2820:
2814:
2811:
2798:
2788:
2783:
2768:
2762:
2752:
2745:
2741:
2736:
2732:
2729:
2724:
2720:
2716:
2711:
2706:
2701:
2697:
2694:; no awkward
2690:
2686:
2682:
2675:
2668:
2662:
2658:
2653:
2649:
2645:
2641:
2636:
2633:
2629:
2624:
2620:
2614:
2610:
2603:
2586:
2577:
2570:
2567:
2551:
2543:
2536:
2533:
2517:
2509:
2502:
2499:
2482:
2478:
2470:
2463:
2441:
2437:
2428:
2424:
2411:
2403:
2399:
2386:
2378:
2374:
2360:
2356:
2348:
2323:
2319:
2315:
2309:
2305:
2302:= 1, 2, ...,
2301:
2295:
2282:
2279:
2276:
2268:
2251:
2247:
2235:
2231:
2225:
2220:
2217:
2214:
2210:
2206:
2198:
2191:
2174:
2162:
2146:
2138:
2123:
2109:
2105:
2101:
2099:
2094:
2092:
2083:
2076:
2069:
2062:
2058:
2053:
2047:
2040:
2033:
2026:
2019:
2015:
2010:
2006:
2001:
1999:
1993:
1991:
1987:
1980:
1975:
1971:
1967:
1959:
1955:
1949:
1945:
1941:
1931:
1927:
1920:
1914:
1910:
1903:
1898:
1892:
1888:
1884:
1879:
1874:
1872:
1867:
1863:
1858:
1855:
1848:
1841:
1834:
1827:
1823:
1819:
1809:
1802:
1795:
1788:
1784:
1780:
1775:
1766:
1759:
1755:
1751:
1746:
1742:
1738:
1734:
1730:
1724:
1717:
1713:
1709:
1704:
1699:
1695:
1690:
1686:
1681:
1675:
1673:
1667:
1663:
1658:
1654:
1649:
1645:
1628:
1623:
1619:
1598:
1578:
1570:
1566:
1563:is the total
1548:
1543:
1539:
1533:
1529:
1523:
1518:
1515:
1512:
1508:
1502:
1499:
1494:
1491:
1471:
1468:
1465:
1462:
1459:
1456:
1448:
1444:
1440:
1430:
1428:
1425:second-order
1424:
1420:
1404:
1396:
1392:
1388:
1376:
1372:
1365:
1362:
1354:
1346:
1342:
1338:
1337:Newton's laws
1335:are given by
1334:
1318:
1312:
1309:
1302:
1292:
1286:
1281:
1271:
1268:
1265:
1259:
1256:
1249:
1239:
1233:
1228:
1218:
1212:
1209:
1202:
1192:
1186:
1181:
1166:
1162:
1156:
1152:
1148:
1144:
1139:
1135:
1131:
1122:
1115:
1108:
1101:
1091:
1084:
1077:
1070:
1065:
1060:
1056:
1049:
1042:
1038:
1034:
1024:
1017:
1014:
1010:
1006:
1001:
999:
994:
989:
986:
982:
973:
969:
963:
959:
955:
949:
942:
938:
933:
924:
922:
918:
914:
910:
906:
901:
899:
895:
891:
887:
882:
878:
874:
869:
868:
851:
843:
840:
834:
830:
823:
821:
820:
815:
811:
807:
803:
799:
791:
787:
776:
771:
769:
764:
762:
757:
756:
754:
753:
747:
737:
734:
729:
723:
722:
721:
720:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
593:
586:
585:
578:
574:
570:
566:
563:
562:
558:
555:
553:
550:
548:
545:
543:
540:
536:
533:
532:
531:
528:
526:
523:
521:
518:
516:
513:
512:
508:
503:
502:
494:
491:
489:
486:
482:
479:
477:
474:
473:
472:
469:
467:
464:
462:
459:
457:
452:
449:
445:
442:
441:
437:
434:
431:
427:
423:
422:
418:
415:
413:
410:
408:
405:
403:
398:
396:
393:
391:
388:
386:
383:
382:
375:
374:
363:
360:
358:
355:
353:
350:
348:
345:
343:
340:
338:
335:
334:
332:
331:
326:
323:
322:
317:
316:
310:
309:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
271:
268:
266:
262:
260:
257:
255:
251:
249:
246:
243:
239:
235:
233:
230:
228:
225:
223:
220:
218:
215:
211:
208:
206:
203:
202:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
177:
170:
169:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
133:
131:
128:
126:
123:
122:
115:
114:
108:
105:
103:
100:
98:
95:
94:
92:
91:
87:
68:
65:
55:
49:
35:
34:
31:
28:
27:
23:
22:
19:
29040:Astrophysics
28854:Experimental
28767: /
28763: /
28753:
28629:Optimization
28592:Superalgebra
28451:Field theory
28438:
28424:Mathematical
28402: /
28250:Chaos theory
28233:Gauge theory
28161:Graph theory
28056:Cryptography
27928:
27819:. Retrieved
27815:
27810:David Tong.
27794:
27773:
27765:
27738:
27716:
27694:
27674:
27655:
27635:
27614:
27570:
27566:
27523:
27519:
27468:
27464:
27436:
27416:
27397:
27373:
27352:
27331:
27310:
27290:
27266:
27246:
27237:, R. Feynman
27234:
27216:
27196:
27172:
27145:
27123:
27103:
27082:
27062:
27036:
27010:
27000:
26989:
26952:
26941:
26930:
26919:
26907:
26900:Shabana 2008
26895:
26883:
26871:
26859:
26854:, p. 48
26847:
26835:
26817:10.1142/7689
26807:
26800:
26795:, p. 22
26788:
26783:, p. 14
26776:
26771:, p. 21
26764:
26752:
26747:, p. 21
26740:
26713:
26702:
26654:
26650:
26640:
26635:, p. 51
26628:
26616:
26611:, p. 23
26604:
26583:
26577:
26565:
26560:, p. 17
26553:
26548:, p. 56
26541:
26529:
26524:, p. 53
26517:
26512:, p. 15
26505:
26500:, p. 15
26493:
26481:
26469:
26464:, p. 89
26457:
26445:
26433:
26422:
26417:
26411:Penrose 2007
26406:
26401:, p. 19
26394:
26382:
26372:, retrieved
26350:
26340:
26328:
26320:
26315:
26290:
26286:
26276:
26245:
26222:
26202:. Retrieved
26198:
26171:. Retrieved
26159:
26134:
26122:
26117:, p. 23
26110:
26097:
26078:
26073:coordinates.
26069:
26065:
25937:
25879:
25866:
25730:
25617:
25602:
25590:differential
25585:
25517:
25513:
25433:
25429:
25425:
25418:
25378:
25347:
25327:
25309:
25286:
25273:
25269:
25265:
25261:
25257:
25250:
25246:
25242:
25238:
25234:
25230:
25213:
25081:
25073:
25062:
25059:
25053:
25047:
25041:
25035:
25029:
25023:
25005:
24910:
24900:
24893:
24766:
24762:
24758:
24679:
24488:
24398:as follows:
24393:
24389:
24385:
24381:
24371:
24367:
24363:
24359:
24349:
24345:
24341:
24337:
24327:
24323:
24319:
24315:
24302:
24292:
24273:
24010:
24005:
24001:
23868:
23855:
23851:
23845:
23839:
23825:
23820:
23816:
23812:
23805:
23796:
23792:
23790:
23785:
23779:
23775:
23770:
23763:
23760:
23657:
23652:
23648:
23641:
23538:
23534:
23528:
23526:
23435:
23429:
23423:
23419:
23412:
23409:
23301:
23171:
23166:
23159:
23155:
23151:
23102:
23099:
23090:
23085:
23077:
23072:
23062:
23055:
23049:
23037:
23033:reduced mass
23023:
23016:
23009:
23003:
22999:
22990:
22983:
22979:
22802:
22795:
22788:
22782:
22775:
22769:
22765:
22753:
22746:
22742:
22732:
22720:
22711:
22702:
22693:
22690:
22588:
22331:
22329:
22066:
22064:
21416:
21094:
21090:
21086:
21084:
21066:
21062:
21060:
21054:
21050:
21046:
21042:
21038:
21034:
20924:
20408:
20404:
20400:
20394:
20390:
20386:
20382:
20320:
20316:
19957:
19951:
19748:
19740:
19674:
19672:
19664:
19386:
18797:
18788:
18782:
18778:
18775:
18705:
18701:
18697:
18695:
18641:
18637:
18632:
18628:
18624:
18620:
18616:
18614:
18524:
18520:
18516:
18512:
18510:
18415:
18411:
18407:
18403:
18399:
18394:
18390:
18386:
18382:
18378:
18219:
18215:
18212:
18207:
18203:
18198:
18192:
18176:
18141:
18048:
18042:
18038:
18032:
18030:
17579:
17574:
17569:
17563:
17559:
17555:
17552:
17497:
17494:Conservation
16754:
16579:
16161:
14488:
14484:
14480:
14477:
14320:
14316:
14307:
14305:
13272:
13262:
13257:
13253:
13249:
13244:
13240:
13236:
13231:
13227:
13069:
13065:
13061:
13057:
13053:
13049:
13045:
13041:
12920:
12778:
12774:
12770:
12768:
12692:
12688:
12682:
12542:
11530:
10929:
10463:
10456:
10180:
10174:
10170:
10166:
10162:
10155:
10151:
10147:
10143:
10135:
10131:
10129:
10119:
10110:
10103:
10099:
10089:
10082:
10078:
9639:
9635:
9631:
9627:
9625:
9282:
9278:
9274:
9264:
9260:
9253:
9243:
9239:
9232:
9222:
9215:
9209:
9205:
9200:
9196:
9192:
9187:
9183:
9179:
9177:
8786:
8782:
8778:
8770:
8766:
8764:
8596:
8592:
8591:. Also, the
8588:
8288:
8284:
8279:
8275:
8272:
8270:
7945:
7941:
7937:
7935:
7721:
7717:
7712:
7708:
7702:
7697:
7693:
7688:
7684:
7674:
7670:
7667:
7662:
7658:
7655:
7651:
7646:
7639:
7634:
7630:
7627:
7620:
7615:
7611:
7608:
7602:
7598:
7594:
7592:
7389:
7384:
7380:
7376:
7374:
7358:
7307:
7299:
7259:
7252:
7246:
7173:
7100:
7095:
7091:
7086:
7082:
7078:
7074:
7065:
7060:
7056:
7049:
7044:
7040:
7037:
6406:
6402:
6398:
6393:
6389:
6385:
6378:
6374:
6369:
6365:
6357:
6082:
6076:
6070:
6068:
5956:
5952:
5947:
5943:
5939:
5816:
5811:
5807:
5803:
5799:
5797:
5791:
5787:
5779:
5763:
5761:
5642:
5638:
5633:
5629:
5449:
5256:
4916:
4912:
4909:
4904:
4900:
4898:
4893:
4891:
4774:
4770:
4767:
4760:
4756:
4751:
4747:
4742:
4738:
4733:
4729:
4726:
4715:
4711:
4706:
4702:
4697:
4693:
4688:
4684:
4679:
4677:
4666:
4661:
4657:
4654:
4651:
4467:
4463:Virtual work
4458:
4453:
4449:
4446:
4440:
4438:
4313:
4309:
4306:
4302:
4280:
4272:
4268:
4264:
4261:
4257:
4207:
4203:
4158:
4154:
4150:
4148:
4129:
4125:
4118:
4112:
4108:
4104:
4099:
4095:
4091:
4085:
4074:
3954:
3945:
3941:
3592:
3582:
3578:
3574:
3569:
3568:
3561:
3555:
3550:
3546:
3538:
3534:
3526:
3522:
3518:
3474:
3466:
3464:
3459:Isaac Newton
3434:
3420:
3414:
3410:
3406:
3401:
3391:
3387:
3383:
3379:
3375:
3368:
3256:
3253:
3249:
3244:
3238:
3236:
3169:
3164:
3160:
3158:
2946:
2944:
2757:
2750:
2743:
2739:
2734:
2727:
2722:
2718:
2714:
2709:
2707:
2699:
2688:
2684:
2680:
2673:
2666:
2660:
2656:
2651:
2647:
2637:
2631:
2627:
2622:
2618:
2612:
2321:
2317:
2310:
2303:
2299:
2297:
2110:
2108:(first kind)
2107:
2103:
2097:
2095:
2090:
2081:
2074:
2067:
2060:
2056:
2051:
2045:
2038:
2031:
2024:
2017:
2013:
2008:
2004:
2002:
1994:
1989:
1978:
1973:
1969:
1962:
1957:
1953:
1947:
1943:
1936:
1929:
1925:
1918:
1912:
1908:
1901:
1896:
1890:
1886:
1882:
1875:
1861:
1859:
1853:
1846:
1839:
1832:
1825:
1821:
1817:
1807:
1800:
1793:
1786:
1782:
1778:
1764:
1757:
1753:
1749:
1732:
1731:
1722:
1715:
1711:
1707:
1702:
1697:
1693:
1688:
1684:
1679:
1676:
1665:
1661:
1656:
1652:
1647:
1643:
1446:
1442:
1436:
1422:
1418:
1345:acceleration
1154:
1150:
1146:
1142:
1137:
1120:
1113:
1106:
1099:
1089:
1082:
1075:
1068:
1058:
1054:
1047:
1040:
1029:
1022:
1015:
1008:
1002:
990:
977:
971:
967:
961:
957:
953:
940:
936:
927:Introduction
908:
902:
889:
885:
880:
876:
872:
866:
865:
841:
832:
828:
824:
817:
801:
795:
575: /
571: /
569:displacement
567: /
428: /
390:Displacement
336:
328:
319:
313:Formulations
300:Virtual work
240: /
180:Acceleration
173:Fundamentals
18:
28943:Statistical
28859:Theoretical
28836:Engineering
28769:topics list
28703:Mathematics
28619:Game theory
28520:Topological
28486:Topological
28481:Statistical
28444:Hamiltonian
27055:Landau, Lev
26935:Galley 2013
26912:Gannon 2006
26840:Taylor 2005
26759:, p. 4
26735:, p. 4
26476:, p. 4
23841:Dissipation
23821:generalized
23813:generalized
23529:co-rotating
21089:and length
19960:coordinate
14593:tangent map
13298:Hamiltonian
13239:direction,
10336:and by the
7392:particles,
7245:which is a
4317:, is zero:
4223:(1717—1783)
3587:the law in
3461:(1642–1727)
3433:denoted d/d
3159:Given this
2945:The vector
2710:independent
2696:chain rules
1701:, nor time
1672:dot product
1134:coordinates
792:(1736–1813)
711:von Neumann
378:Core topics
29136:Categories
29060:Geophysics
29050:Biophysics
28894:Analytical
28847:Approaches
28675:Psychology
28639:Statistics
28439:Lagrangian
28066:Statistics
28002:Algorithms
27821:2017-06-08
27155:0201029189
26980:References
26971:Torby 1984
26374:2024-09-23
26293:(6): 123.
26269:Torby 1984
26250:Torby 1984
26225:. London:
26204:2024-09-27
26173:2024-09-27
25613:symmetries
25364:, and the
24589:is called
23106:is simply
19956:, for the
18385:′ =
18224:, so that
18211:′ =
14493:, we have
10521:, we have
10338:chain rule
8587:which are
7346:Maupertuis
7316:solved by
7248:functional
5959:of motion
5819:such that
4299:D'Alembert
3396:gives the
2677:, is just
1443:Lagrangian
1433:Lagrangian
1039:, denoted
867:Lagrangian
646:d'Alembert
626:Maupertuis
589:Scientists
471:Rigid body
145:Kinematics
29010:Molecular
28911:Acoustics
28904:Continuum
28899:Celestial
28889:Newtonian
28876:Classical
28819:Divisions
28680:Sociology
28670:Chemistry
28466:Effective
28461:Conformal
28456:Classical
28328:Geometric
28301:Geometric
27791:Goldstein
27605:119170985
27580:1305.6930
27558:118541484
27533:1402.2610
27478:1210.2745
27037:Mechanics
26679:0002-9505
26307:2190-5444
26049:≠
26036:δ
26012:≠
25999:δ
25975:≠
25962:δ
25906:δ
25903:⋅
25837:˙
25825:∂
25821:∂
25797:−
25779:∂
25775:∂
25770:≡
25752:δ
25748:δ
25722:Footnotes
25547:∫
25516:and time
25480:˙
25477:ϕ
25468:ϕ
25465:∇
25459:ϕ
25397:electrons
25379:In 1948,
25279:momenta.
25176:−
25161:˙
25151:∂
25143:∂
25128:˙
25101:∑
24975:˙
24952:˙
24941:∂
24933:∂
24904:) is the
24830:∂
24826:∂
24817:∇
24813:⋅
24807:˙
24782:→
24738:˙
24729:−
24726:ϕ
24723:→
24720:ϕ
24709:∇
24697:→
24661:×
24655:˙
24624:¨
24574:ϕ
24568:−
24560:⋅
24554:˙
24527:˙
24467:×
24463:∇
24441:∂
24431:∂
24425:−
24422:ϕ
24418:∇
24414:−
24288:up quarks
24284:electrons
24237:˙
24227:∂
24219:∂
24197:∂
24189:∂
24183:−
24164:˙
24154:∂
24146:∂
24088:˙
24078:∂
24070:∂
24064:−
24048:∂
24040:∂
24034:−
23976:˙
23957:˙
23917:∑
23896:∑
23731:μ
23721:ℓ
23703:˙
23700:θ
23690:μ
23611:μ
23601:ℓ
23568:−
23559:¨
23550:μ
23507:¨
23498:μ
23472:−
23460:˙
23457:θ
23447:μ
23393:ℓ
23384:˙
23381:θ
23365:μ
23353:˙
23350:θ
23344:∂
23329:∂
23318:θ
23272:−
23255:˙
23252:θ
23223:˙
23208:μ
23164:and take
23125:¨
23043:magnitude
22950:⏟
22922:−
22910:˙
22898:μ
22863:⏟
22850:˙
22653:→
22647:¨
22644:θ
22610:→
22604:¨
22569:θ
22566:
22558:ℓ
22547:θ
22544:
22536:ℓ
22531:¨
22515:¨
22512:θ
22480:θ
22477:
22459:˙
22456:θ
22447:˙
22435:ℓ
22415:˙
22412:θ
22400:ℓ
22393:θ
22390:
22384:ℓ
22378:˙
22310:θ
22307:
22292:˙
22289:θ
22282:ℓ
22276:−
22273:θ
22270:
22261:¨
22258:θ
22252:ℓ
22240:¨
22176:θ
22173:
22164:˙
22161:θ
22155:ℓ
22143:˙
22110:˙
22101:∂
22093:∂
22044:θ
22041:
22035:ℓ
22014:˙
22011:θ
21998:ℓ
21978:θ
21975:
21966:˙
21963:θ
21957:ℓ
21951:˙
21927:˙
21878:θ
21875:
21869:ℓ
21841:θ
21838:
21829:˙
21826:θ
21820:ℓ
21797:θ
21794:
21785:˙
21782:θ
21776:ℓ
21767:˙
21722:˙
21686:−
21569:˙
21531:˙
21491:˙
21445:θ
21396:θ
21393:
21384:˙
21381:θ
21375:ℓ
21349:˙
21338:⇒
21332:θ
21329:
21323:ℓ
21320:−
21288:θ
21285:
21276:˙
21273:θ
21267:ℓ
21258:˙
21226:˙
21215:⇒
21209:θ
21206:
21200:ℓ
21145:θ
21033:in which
21015:˙
21012:φ
21006:θ
21003:
20968:˙
20965:φ
20959:∂
20951:∂
20940:φ
20899:˙
20896:φ
20889:θ
20886:
20804:˙
20801:φ
20793:θ
20790:
20784:θ
20781:
20762:−
20753:˙
20750:θ
20678:∂
20670:∂
20649:˙
20646:φ
20638:θ
20635:
20610:˙
20607:θ
20591:−
20585:¨
20541:−
20526:˙
20523:φ
20515:θ
20512:
20477:˙
20474:θ
20445:˙
20371:which is
20355:∇
20351:−
20342:¨
20294:∂
20286:∂
20280:−
20271:¨
20236:¨
20211:˙
20202:∂
20194:∂
20158:˙
20137:˙
20128:∂
20120:∂
20104:∂
20096:∂
20090:−
20078:∂
20070:∂
20038:∂
20030:∂
20011:˙
20002:∂
19994:∂
19911:−
19896:˙
19874:˙
19852:˙
19817:˙
19802:˙
19787:˙
19709:∇
19705:−
19627:˙
19595:¨
19551:˙
19519:¨
19461:˙
19451:∂
19439:˙
19429:∂
19415:∂
19357:˙
19334:∂
19322:˙
19312:∂
19298:∂
19285:∑
19281:−
19272:∂
19264:∂
19258:−
19242:∂
19234:∂
19216:¨
19194:˙
19157:∑
19127:∂
19119:∂
19113:−
19104:∂
19096:∂
19067:˙
19039:˙
19029:∂
19017:˙
19007:∂
18993:∂
18944:˙
18934:∂
18921:∂
18907:∂
18876:∂
18868:∂
18862:−
18847:˙
18837:∂
18829:∂
18581:−
18482:−
18475:α
18468:β
18464:⇒
18454:α
18439:β
18429:α
18347:…
18302:α
18280:α
18274:…
18256:α
18238:α
18090:˙
17982:−
17976:˙
17961:˙
17952:∂
17944:∂
17905:∂
17897:∂
17891:−
17881:˙
17872:−
17863:˙
17854:∂
17846:∂
17837:¨
17820:˙
17799:˙
17790:∂
17782:∂
17744:∂
17736:∂
17730:−
17717:∂
17709:∂
17694:˙
17685:∂
17677:∂
17668:¨
17646:∂
17638:∂
17629:˙
17489:follows.
17448:˙
17437:∂
17429:∂
17420:˙
17400:˙
17389:∂
17381:∂
17372:˙
17341:˙
17310:˙
17269:˙
17259:∂
17251:∂
17217:˙
17192:˙
17182:∂
17174:∂
17151:∑
17101:˙
17040:˙
17029:∂
17021:∂
16999:∗
16985:˙
16974:∂
16966:∂
16928:˙
16885:˙
16825:˙
16785:˙
16726:∂
16718:∂
16706:˙
16686:˙
16675:∂
16667:∂
16642:∂
16634:∂
16613:˙
16553:∂
16545:∂
16533:˙
16506:∗
16492:˙
16481:∂
16473:∂
16445:˙
16414:˙
16403:∂
16395:∂
16373:∗
16357:∂
16349:∂
16316:∂
16308:∂
16296:˙
16276:˙
16265:∂
16257:∂
16232:∂
16224:∂
16199:˙
16096:˙
16057:∂
16044:∂
16023:∂
15999:∑
15973:…
15943:˙
15904:∂
15891:∂
15870:∂
15846:∑
15812:∑
15779:…
15716:∂
15703:∂
15682:∂
15658:∑
15645:˙
15618:∑
15602:˙
15521:∂
15508:∂
15487:∂
15463:∑
15447:˙
15420:∗
15370:˙
15318:˙
15291:∗
15266:˙
15235:˙
15198:˙
15174:∗
15160:˙
15095:˙
15059:˙
14979:∂
14964:∂
14933:∗
14894:∈
14882:˙
14821:∂
14817:∂
14804:⋅
14792:˙
14765:∑
14729:∈
14717:˙
14665:∂
14661:∂
14648:⋅
14636:˙
14609:∑
14566:∗
14517:∗
14450:˙
14440:∂
14429:∂
14394:˙
14354:˙
14273:∂
14263:∂
14257:⋅
14241:∂
14231:∂
14168:−
14159:−
14144:−
14124:˙
14114:∂
14106:∂
14091:˙
14064:∑
14010:˙
13991:˙
13956:∑
13919:˙
13909:∂
13901:∂
13886:˙
13859:∑
13840:˙
13830:∂
13822:∂
13807:˙
13780:∑
13751:˙
13732:˙
13697:∑
13666:˙
13643:∂
13635:→
13626:∂
13615:⋅
13598:˙
13575:∂
13567:→
13558:∂
13535:∑
13459:⋯
13404:−
13382:˙
13372:∂
13364:∂
13349:˙
13322:∑
13205:˙
13202:ϕ
13196:∂
13188:∂
13177:ϕ
13159:˙
13150:∂
13142:∂
13113:˙
13104:∂
13096:∂
13013:˙
13010:ϕ
12998:˙
12995:θ
12983:˙
12968:˙
12953:˙
12941:θ
12883:∂
12875:∂
12854:˙
12844:∂
12836:∂
12799:˙
12740:˙
12730:∂
12722:∂
12644:∂
12635:∂
12628:−
12613:˙
12603:∂
12595:∂
12554:→
12506:∂
12491:∂
12468:˙
12458:∂
12450:∂
12444:−
12425:˙
12415:∂
12407:∂
12371:∑
12335:∂
12320:∂
12283:˙
12273:∂
12264:∂
12257:−
12242:˙
12232:∂
12218:˙
12208:∂
12189:∂
12180:∂
12173:−
12154:˙
12144:∂
12130:˙
12120:∂
12083:˙
12073:∂
12065:∂
12044:˙
12034:∂
12020:˙
12010:∂
11988:˙
11978:∂
11970:∂
11934:∑
11900:∂
11886:˙
11876:∂
11858:˙
11848:∂
11839:∂
11832:−
11814:∂
11799:∂
11780:∂
11771:∂
11764:−
11744:˙
11734:∂
11720:˙
11710:∂
11692:˙
11682:∂
11674:∂
11637:∑
11610:∂
11601:∂
11594:−
11579:˙
11569:∂
11561:∂
11503:∂
11489:˙
11479:∂
11453:∂
11438:∂
11366:∂
11360:∂
11339:∂
11320:˙
11279:∂
11266:∂
11245:∂
11232:∑
11208:∂
11191:∂
11126:∂
11113:∂
11092:∂
11073:˙
11032:∂
11019:∂
10998:∂
10985:∑
10965:∂
10957:˙
10941:∂
10902:∂
10887:∂
10866:˙
10856:∂
10848:˙
10832:∂
10786:∂
10771:∂
10753:˙
10712:∂
10697:∂
10685:∑
10675:˙
10627:∂
10612:∂
10575:∂
10560:∂
10548:∑
10427:∂
10414:∂
10393:˙
10383:∂
10370:∂
10300:˙
10275:˙
10218:˙
10025:−
9881:∫
9846:˙
9787:∫
9745:˙
9681:∫
9600:˙
9577:∂
9552:∂
9540:∑
9527:∂
9502:∂
9362:˙
9322:˙
9135:∂
9120:∂
9108:λ
9087:∑
9027:∂
9012:∂
9000:λ
8979:∑
8955:⏟
8936:∂
8928:∂
8922:−
8898:−
8892:⏟
8876:˙
8864:∂
8856:∂
8831:−
8813:∂
8805:∂
8710:⇒
8688:˙
8685:λ
8678:∂
8665:∂
8640:−
8628:λ
8624:∂
8611:∂
8551:∂
8536:∂
8524:λ
8503:∑
8484:˙
8472:∂
8464:∂
8439:−
8421:∂
8413:∂
8406:⇒
8384:˙
8372:∂
8359:∂
8334:−
8316:∂
8303:∂
8230:δ
8227:⋅
8204:∂
8189:∂
8177:λ
8156:∑
8137:˙
8125:∂
8117:∂
8092:−
8074:∂
8066:∂
8038:∑
8009:∫
7986:δ
7958:∫
7870:λ
7849:∑
7833:…
7818:˙
7794:˙
7779:…
7552:δ
7549:⋅
7529:˙
7517:∂
7509:∂
7484:−
7466:∂
7458:∂
7430:∑
7401:∫
7330:L'Hôpital
7277:δ
7193:∫
7142:δ
7114:∫
6998:δ
6978:˙
6968:∂
6960:∂
6935:−
6919:∂
6911:∂
6883:∑
6854:∫
6807:δ
6792:˙
6782:∂
6774:∂
6745:∑
6707:δ
6692:˙
6682:∂
6674:∂
6649:−
6631:δ
6616:˙
6606:∂
6598:∂
6555:δ
6539:∂
6531:∂
6503:∑
6474:∫
6451:δ
6423:∫
6311:δ
6297:≡
6263:δ
6260:≡
6248:˙
6238:δ
6217:˙
6207:δ
6192:˙
6182:∂
6174:∂
6155:δ
6139:∂
6131:∂
6103:∑
6093:δ
6083:variation
6037:˙
6027:∂
6019:∂
5994:−
5978:∂
5970:∂
5909:∂
5901:∂
5895:−
5880:˙
5870:∂
5862:∂
5735:∂
5727:∂
5721:−
5706:˙
5696:∂
5688:∂
5597:δ
5571:∂
5563:∂
5557:−
5542:˙
5532:∂
5524:∂
5494:−
5459:∑
5420:∂
5412:∂
5406:−
5391:˙
5381:∂
5373:∂
5332:∂
5315:∂
5309:⋅
5267:∑
5230:δ
5200:∑
5183:δ
5167:∂
5150:∂
5127:∑
5123:⋅
5091:∑
5072:δ
5069:⋅
5037:∑
5000:∂
4983:∂
4977:⋅
4945:∑
4896:of time.
4865:δ
4849:∂
4832:∂
4809:∑
4790:δ
4620:δ
4617:⋅
4589:−
4554:∑
4512:δ
4509:⋅
4477:∑
4407:δ
4404:⋅
4376:−
4326:∑
4120:geodesics
4042:ξ
4011:ξ
3906:ξ
3894:≡
3882:˙
3879:ξ
3854:ξ
3850:∂
3842:∂
3836:−
3821:˙
3818:ξ
3811:∂
3803:∂
3733:ξ
3702:ξ
3672:Γ
3639:ξ
3341:∂
3333:∂
3308:˙
3298:∂
3290:∂
3208:˙
3136:∂
3119:∂
3101:˙
3078:∂
3061:∂
3038:∑
2972:˙
2767:functions
2571:˙
2561:∂
2557:∂
2537:˙
2527:∂
2523:∂
2503:˙
2493:∂
2489:∂
2479:≡
2464:˙
2452:∂
2448:∂
2421:∂
2417:∂
2396:∂
2392:∂
2371:∂
2367:∂
2357:≡
2339:∂
2335:∂
2259:∂
2244:∂
2232:λ
2211:∑
2192:˙
2180:∂
2172:∂
2147:−
2129:∂
2121:∂
1611:has mass
1509:∑
1466:−
1355:∑
1269:…
898:potential
691:Liouville
573:frequency
493:Vibration
210:potential
135:Continuum
130:Celestial
107:Textbooks
28755:Category
28404:analysis
28323:Exterior
28296:Exterior
28176:Analysis
28138:Discrete
28012:analysis
27843:, page 4
27511:14591873
27503:23679733
27144:(1980).
27061:(1975).
26438:Kay 1988
25631:See also
25588:is a 3D
25260:′(
25066:is not.
23861:Rayleigh
23169:= |
22633:, while
19683:gradient
19661:Examples
18564:′
18541:′
17570:extremal
13487:. From:
10421:′
10377:′
10195:′
9713:′
9658:′
9299:′
8672:′
8618:′
8366:′
8310:′
7993:′
7736:′
7650:for all
7354:Hamilton
6373:to the ∂
5951:obtains
5028:so that
4545:so that
4140:Einstein
3958:are the
1871:friction
1439:energies
1161:velocity
892:are the
884:, where
746:Category
671:Hamilton
656:Lagrange
651:Clairaut
616:Horrocks
577:velocity
547:Pendulum
535:reactive
507:Rotation
476:dynamics
426:Inertial
412:Friction
295:Velocity
270:Momentum
150:Kinetics
140:Dynamics
118:Branches
102:Timeline
29094:Related
28978:General
28973:Special
28831:Applied
28765:outline
28696:Related
28665:Biology
28515:Bosonic
28476:Quantum
28426:physics
28392: (
28124:Solvers
27585:Bibcode
27538:Bibcode
27483:Bibcode
26659:Bibcode
26227:Quercus
26160:Physics
25584:where d
25401:photons
25381:Feynman
23848:viscous
23031:is the
19685:∇ of a
18047:, then
17356:yields
14591:is the
7322:Leibniz
5955:or the
5641:or the
4894:instant
4084:of the
3944:is the
3591:is the
2737:-tuple
1960:, then
1852:, ...,
1838:, ...,
1813:, ...).
1799:, ...,
1770:, ...).
1728:, ...).
1053:, ...,
1028:, ...,
917:minimum
913:maximum
894:kinetic
798:physics
706:Koopman
666:Poisson
661:Laplace
606:Huygens
601:Galileo
446: (
385:Damping
238:Inertia
232:Impulse
205:kinetic
155:Statics
125:Applied
97:History
29005:Atomic
28960:Modern
28810:Major
28338:Vector
28333:Tensor
28311:Vector
28306:Tensor
28007:design
27780:
27746:
27725:
27702:
27681:
27662:
27643:
27622:
27603:
27556:
27509:
27501:
27452:
27423:
27404:
27381:
27360:
27339:
27318:
27297:
27274:
27253:
27224:
27203:
27180:
27152:
27130:
27111:
27090:
27069:
27043:
27017:
26823:
26677:
26592:
26365:
26305:
26233:
25409:optics
25354:action
25300:Optics
24756:where
24680:Under
24280:charge
24000:where
23817:always
23175:|
23084:and a
23069:|
23047:|
23035:, and
22977:where
22065:Since
20254:hence
17568:be an
17500:closed
15336:where
14872:
14862:
14707:
14697:
14553:where
14204:where
13269:Energy
13068:, and
13040:where
9491:means
9433:where
8790:gives
7342:Fermat
7334:Newton
7176:action
6081:. The
3940:where
3517:where
3252:
2756:, ...
2325:, and
2298:where
2106:
2087:, ...)
2073:, ...
2044:, ...
2030:, ...
1981:) = 0.
1950:) = 0,
1932:) = 0,
1915:) = 0,
1893:) = 0.
1743:(e.g.
1735:, the
1484:where
1159:. The
1013:masses
921:saddle
744:
696:Appell
681:Cauchy
676:Jacobi
621:Halley
611:Newton
596:Kepler
448:linear
444:Motion
290:Torque
265:Moment
200:Energy
190:Couple
28471:Gauge
27816:DAMTP
27601:S2CID
27575:arXiv
27554:S2CID
27528:arXiv
27507:S2CID
27473:arXiv
26195:(PDF)
26090:Notes
25358:phase
25207:(see
24297:with
23819:with
23660:(see
23088:term
23075:term
14475:Proof
12773:does
10461:Proof
9228:]
9214:[
7705:) = 0
7350:Euler
7071:) = 0
3553:when
3243:, or
1934:...,
1705:, so
1341:force
981:torus
964:) = 0
919:, or
701:Gibbs
686:Routh
641:Euler
280:Speed
275:Space
217:Force
28931:Wave
28826:Pure
27778:ISBN
27744:ISBN
27723:ISBN
27700:ISBN
27679:ISBN
27660:ISBN
27641:ISBN
27620:ISBN
27499:PMID
27450:ISBN
27421:ISBN
27402:ISBN
27379:ISBN
27358:ISBN
27337:ISBN
27316:ISBN
27295:ISBN
27272:ISBN
27251:ISBN
27222:ISBN
27201:ISBN
27178:ISBN
27150:ISBN
27128:ISBN
27109:ISBN
27088:ISBN
27067:ISBN
27041:ISBN
27015:ISBN
26821:ISBN
26675:ISSN
26590:ISBN
26363:ISBN
26303:ISSN
26231:ISBN
25399:and
25395:for
25332:and
25045:and
25014:and
24605:law
24376:and
24332:and
24286:and
24278:and
24276:mass
24112:and
22718:and
22700:and
22685:and
21437:and
21069:/2.
21041:and
20923:The
20319:and
18636:and
18619:and
18410:by (
17119:and
16955:and
15080:and
14919:and
13296:the
13056:and
13044:and
10097:and
9638:and
9630:and
9251:and
9195:′ =
7716:for
7645:) =
7626:) =
7258:and
7055:) =
6401:for
6377:/∂(d
4439:The
4080:the
3948:-th
3558:= 0.
3549:and
2100:are
1641:and
896:and
888:and
285:Time
248:Mass
28926:Ray
27593:doi
27546:doi
27491:doi
27469:110
27442:doi
26813:doi
26667:doi
26355:doi
26295:doi
26291:130
26164:doi
25623:or
25407:in
25391:to
25372:of
25348:In
25264:, d
25241:, d
25211:).
23664:):
23439:is
23337:rel
23299:so
23190:rel
23094:rel
22961:rel
22794:)/(
22768:= (
22563:sin
22541:cos
22474:sin
22387:cos
22334:is
22304:sin
22267:cos
22211:is
22170:cos
22038:cos
21972:cos
21872:cos
21835:sin
21791:cos
21390:sin
21326:cos
21282:cos
21203:sin
20994:sin
20877:sin
20787:cos
20778:sin
20626:sin
20503:sin
18045:= 0
15401:def
12775:not
10093:fin
10086:fin
10017:fin
10004:fin
9902:fin
9808:fin
9702:fin
9268:fin
9257:fin
9226:fin
6397:)/d
6356:of
5800:may
4737:= (
4692:= (
4285:is
4142:'s
3573:= (
3545:of
3477:is
3409:= 3
3404:to
3382:, d
2742:= (
2717:= 3
2705:).
2615:to
2605:∂/∂
2089:is
2050:is
2007:or
2003:If
1569:sum
1347:",
1145:= (
1105:= (
1074:= (
796:In
29138::
27814:.
27599:.
27591:.
27583:.
27571:88
27569:.
27552:.
27544:.
27536:.
27524:29
27522:.
27505:.
27497:.
27489:.
27481:.
27467:.
27448:.
27396:.
27245:.
27195:.
27166:;
27057:;
27031:;
26963:^
26819:.
26725:^
26687:^
26673:.
26665:.
26655:73
26653:.
26649:.
26425:.
26361:,
26349:,
26301:.
26289:.
26285:.
26257:^
26213:^
26197:.
26182:^
26158:.
26146:^
25627:.
25432:,
25411:.
25376:.
25352:,
25272:,
25268:/d
25249:,
25245:/d
24684::
24392:,
24384:=
24370:,
24362:=
24348:,
24340:=
24326:,
24318:=
24255:0.
24006:jk
23867:,
23859:,
23778:,
23651:/d
23647:−d
23537:/d
23422:/d
23177:,
23158:,
23097:.
23081:cm
23061:−
23022:+
23015:/(
23002:=
22989:+
22982:=
22874:cm
22801:+
22781:+
22752:−
22745:=
22673:.
22575:0.
22316:0.
21047:dt
21043:dφ
21037:,
20911:0.
20393:,
20389:,
19689:,
18783:AB
18706:AB
18387:βt
18381:,
18201:,
18041:/∂
17558:=
17550:.
14483:=
14319:→
13064:,
12663:0.
10927:.
10173:,
10165:=
10154:,
10146:=
10122:.
10114:st
10109:,
10107:st
10088:,
10052:st
10039:st
9890:st
9796:st
9690:st
9287::
9281:,
9259:=
9247:st
9238:=
9236:st
9221:,
9219:st
9199:+
9197:aL
8785:−
8781:=
8751:0.
8257:0.
7701:,
7579:0.
7352:,
7348:,
7344:,
7328:,
7324:,
7305:.
7286:0.
7160:0.
7105::
7092:δq
7083:δq
7079:δL
7057:δq
7041:δq
6403:δq
6390:δq
6384:/d
6366:δq
6071:if
6055:0.
5946:−
5942:=
5794:).
5645:,
5630:δq
4782:,
4757:δz
4755:,
4748:δy
4746:,
4739:δx
4710:,
4701:,
4638:0.
4445:,
4425:0.
4161:,
4128:≠
4111:=
4098:,
4094:,
4077:bc
3955:bc
3585:),
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3577:,
3527:on
3413:−
3390:,
3386:/d
2749:,
2721:−
2691:,2
2683:/∂
2674:dt
2667:dz
2665:=
2663:,2
2621:+
2080:,
2066:,
2037:,
2023:,
1977:,
1946:,
1928:,
1911:,
1889:,
1856:).
1845:,
1831:,
1820:=
1806:,
1792:,
1781:=
1763:,
1752:=
1721:,
1710:=
1660:·
1651:=
1153:,
1149:,
1119:,
1112:,
1097:,
1088:,
1081:,
1062:.
1046:,
1021:,
960:,
915:,
879:−
875:=
831:,
822:.
800:,
28803:e
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28789:v
28688:"
28684:"
28396:)
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27971:t
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27824:.
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27731:.
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26669::
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26239:.
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26176:.
26166::
26070:k
26066:r
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26031:z
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26023:C
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26015:0
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26003:y
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25990:k
25986:C
25981:,
25978:0
25970:k
25966:x
25957:x
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25938:k
25924:0
25921:=
25916:k
25911:r
25898:k
25893:C
25874:.
25844:k
25833:r
25813:t
25809:d
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25789:k
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25535:(
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25490:r
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25456:(
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25436:)
25434:t
25430:r
25428:(
25426:ϕ
25276:)
25274:t
25270:t
25266:p
25262:p
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25251:t
25247:t
25243:q
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25237:(
25235:L
25179:L
25168:i
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25146:L
25135:i
25125:q
25116:n
25111:1
25108:=
25105:i
25097:=
25094:H
25063:v
25060:m
25054:p
25048:A
25042:ϕ
25036:r
25030:r
25024:p
24992:.
24988:A
24984:q
24981:+
24971:r
24964:m
24961:=
24948:r
24936:L
24927:=
24923:p
24911:A
24901:r
24879:,
24873:t
24870:d
24865:f
24862:d
24856:q
24853:+
24850:L
24847:=
24844:f
24840:)
24833:t
24821:+
24803:r
24795:(
24791:q
24788:+
24785:L
24779:L
24769:)
24767:t
24765:,
24763:r
24761:(
24759:f
24744:,
24735:f
24716:,
24713:f
24705:+
24701:A
24693:A
24665:B
24651:r
24644:q
24641:+
24637:E
24633:q
24630:=
24620:r
24613:m
24577:,
24571:q
24564:A
24550:r
24542:q
24539:+
24534:2
24523:r
24515:m
24509:2
24506:1
24500:=
24497:L
24475:.
24471:A
24459:=
24455:B
24450:,
24444:t
24435:A
24411:=
24407:E
24396:)
24394:t
24390:r
24388:(
24386:B
24382:B
24374:)
24372:t
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24366:(
24364:E
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24352:)
24350:t
24346:r
24344:(
24342:A
24338:A
24330:)
24328:t
24324:r
24322:(
24320:ϕ
24316:ϕ
24303:q
24252:=
24244:j
24234:q
24222:D
24213:+
24205:j
24201:q
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24179:)
24171:j
24161:q
24149:L
24140:(
24133:t
24129:d
24123:d
24095:j
24085:q
24073:D
24056:j
24052:q
24043:V
24031:=
24026:j
24022:Q
24011:D
24002:C
23988:,
23983:k
23973:q
23964:j
23954:q
23945:k
23942:j
23938:C
23932:m
23927:1
23924:=
23921:k
23911:m
23906:1
23903:=
23900:j
23890:2
23887:1
23882:=
23879:D
23869:D
23856:i
23852:F
23782:)
23780:θ
23776:r
23774:(
23764:ℓ
23747:.
23739:3
23735:r
23725:2
23715:=
23710:2
23693:r
23687:=
23681:f
23678:c
23673:F
23653:r
23649:V
23642:μ
23627:.
23619:3
23615:r
23605:2
23595:+
23589:r
23585:d
23579:V
23575:d
23565:=
23556:r
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23535:θ
23533:d
23513:.
23504:r
23495:=
23489:r
23486:d
23481:V
23478:d
23467:2
23450:r
23436:r
23430:ℓ
23424:t
23420:θ
23418:d
23413:r
23396:.
23390:=
23373:2
23369:r
23362:=
23333:L
23323:=
23314:p
23302:θ
23287:,
23284:)
23281:r
23278:(
23275:V
23268:)
23262:2
23243:2
23239:r
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23230:2
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23212:(
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23200:1
23195:=
23186:L
23172:r
23167:r
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23160:θ
23156:r
23154:(
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23134:0
23131:=
23121:R
23114:M
23103:R
23091:L
23078:L
23066:1
23063:r
23059:2
23056:r
23050:r
23038:V
23029:)
23027:2
23024:m
23020:1
23017:m
23013:2
23010:m
23007:1
23004:m
23000:μ
22994:2
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22942:|
22937:r
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22928:(
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22906:r
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22890:1
22881:+
22870:L
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22838:M
22833:2
22830:1
22821:=
22818:L
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22806:2
22803:m
22799:1
22796:m
22792:2
22789:r
22786:2
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22779:1
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22721:r
22715:1
22712:r
22706:2
22703:m
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22656:0
22613:0
22601:x
22572:=
22555:g
22550:+
22528:x
22521:+
22489:;
22486:0
22483:=
22471:)
22468:g
22465:+
22444:x
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22432:m
22429:+
22425:]
22421:)
22404:2
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22087:=
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21912:m
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21902:(
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21866:g
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21860:+
21856:]
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21816:(
21811:+
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21764:x
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21751:[
21747:m
21742:2
21739:1
21734:+
21729:2
21719:x
21712:M
21707:2
21704:1
21697:=
21689:V
21683:T
21678:=
21673:L
21646:d
21643:n
21640:e
21637:p
21632:y
21628:g
21625:m
21622:=
21619:V
21598:)
21592:2
21586:d
21583:n
21580:e
21577:p
21566:y
21559:+
21554:2
21548:d
21545:n
21542:e
21539:p
21528:x
21520:(
21516:m
21511:2
21508:1
21503:+
21498:2
21488:x
21481:M
21476:2
21473:1
21468:=
21465:T
21425:x
21399:.
21372:=
21366:d
21363:n
21360:e
21357:p
21346:y
21317:=
21311:d
21308:n
21305:e
21302:p
21297:y
21264:+
21255:x
21249:=
21243:d
21240:n
21237:e
21234:p
21223:x
21197:+
21194:x
21191:=
21185:d
21182:n
21179:e
21176:p
21171:x
21125:x
21105:x
21095:M
21091:ℓ
21087:m
21067:π
21063:θ
21055:φ
21051:p
21045:/
21039:θ
21035:r
21021:,
20998:2
20988:2
20984:r
20980:m
20977:=
20954:L
20945:=
20936:p
20925:φ
20908:=
20905:)
20881:2
20871:2
20867:r
20863:m
20860:(
20854:t
20850:d
20844:d
20822:,
20819:0
20816:=
20811:2
20773:2
20769:r
20765:m
20759:)
20742:2
20738:r
20734:m
20731:(
20725:t
20721:d
20715:d
20693:,
20690:0
20687:=
20681:r
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20664:+
20661:)
20656:2
20630:2
20622:+
20617:2
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20597:r
20594:m
20582:r
20576:m
20556:.
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20544:V
20538:)
20533:2
20507:2
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20465:2
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20435:(
20430:2
20427:m
20422:=
20419:L
20409:φ
20405:θ
20401:r
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20391:θ
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20385:(
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20348:=
20338:r
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20317:y
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20242:,
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20208:x
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20146:=
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20087:=
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19938:.
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19914:V
19908:)
19903:2
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19886:+
19881:2
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19839:m
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19826:=
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19757:L
19727:.
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19702:=
19698:F
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19592:q
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18895:0
18892:=
18884:i
18880:q
18871:L
18854:i
18844:q
18832:L
18820:t
18817:d
18813:d
18779:L
18762:.
18757:B
18754:A
18750:L
18746:+
18741:B
18737:L
18733:+
18728:A
18724:L
18720:=
18717:L
18702:L
18698:L
18682:.
18677:B
18673:L
18669:+
18664:A
18660:L
18656:=
18653:L
18642:B
18638:L
18633:A
18629:L
18625:L
18621:B
18617:A
18596:.
18589:2
18586:N
18578:1
18573:)
18568:l
18561:l
18555:(
18550:=
18545:t
18538:t
18525:t
18521:l
18517:t
18513:l
18497:.
18490:2
18487:N
18479:1
18471:=
18458:N
18450:=
18443:2
18433:2
18416:β
18414:/
18412:α
18408:T
18404:β
18402:/
18400:α
18395:k
18391:v
18383:t
18379:β
18365:)
18360:N
18355:r
18350:,
18344:,
18339:2
18334:r
18329:,
18324:1
18319:r
18314:(
18311:V
18306:N
18298:=
18295:)
18290:N
18285:r
18277:,
18271:,
18266:2
18261:r
18253:,
18248:1
18243:r
18235:(
18232:V
18220:k
18216:r
18213:α
18208:k
18204:r
18199:α
18177:E
18162:V
18159:+
18156:T
18153:=
18150:E
18122:.
18114:=
18111:)
18108:t
18105:,
18102:)
18099:t
18096:(
18086:q
18079:,
18076:)
18073:t
18070:(
18066:q
18062:(
18059:H
18049:H
18043:t
18039:L
18037:∂
18033:L
18010:t
18007:d
18002:H
17999:d
17993:=
17989:)
17985:L
17973:q
17958:q
17947:L
17937:(
17930:t
17927:d
17923:d
17918:=
17908:t
17900:L
17878:L
17860:q
17849:L
17833:q
17826:+
17816:q
17808:)
17796:q
17785:L
17776:(
17769:t
17766:d
17762:d
17757:=
17747:t
17739:L
17720:t
17712:L
17703:+
17691:q
17680:L
17664:q
17657:+
17650:q
17641:L
17625:q
17618:=
17608:t
17605:d
17600:L
17597:d
17580:L
17575:q
17566:)
17564:t
17562:(
17560:q
17556:q
17534:E
17510:L
17477:E
17457:.
17444:q
17432:L
17416:q
17409:=
17396:Q
17384:L
17368:Q
17337:Q
17306:q
17276:i
17266:q
17254:L
17224:i
17214:q
17207:d
17199:i
17189:q
17177:L
17166:n
17161:1
17158:=
17155:i
17130:,
17127:t
17107:,
17097:q
17074:,
17070:q
17049:.
17036:q
17024:L
17015:=
17012:)
17008:q
17004:(
16995:F
16981:Q
16969:L
16943:,
16940:)
16937:t
16934:,
16924:q
16917:,
16913:q
16909:(
16906:L
16903:d
16900:=
16897:)
16894:t
16891:,
16881:Q
16874:,
16870:Q
16866:(
16863:L
16860:d
16840:.
16837:)
16834:t
16831:,
16821:q
16814:,
16810:q
16806:(
16803:L
16800:=
16797:)
16794:t
16791:,
16781:Q
16774:,
16770:Q
16766:(
16763:L
16741:.
16738:t
16735:d
16729:t
16721:L
16712:+
16702:q
16695:d
16682:q
16670:L
16661:+
16657:q
16653:d
16646:q
16637:L
16628:=
16625:)
16622:t
16619:,
16609:q
16602:,
16598:q
16594:(
16591:L
16588:d
16562:.
16556:t
16548:L
16539:+
16529:q
16522:d
16519:)
16515:q
16511:(
16502:F
16488:Q
16476:L
16467:+
16463:q
16459:d
16455:)
16451:)
16441:q
16434:,
16430:q
16426:(
16423:G
16410:Q
16398:L
16389:+
16386:)
16382:q
16378:(
16369:F
16361:Q
16352:L
16342:(
16338:=
16328:t
16325:d
16319:t
16311:L
16302:+
16292:Q
16285:d
16272:Q
16260:L
16251:+
16247:Q
16243:d
16236:Q
16227:L
16218:=
16211:)
16208:t
16205:,
16195:Q
16188:,
16184:Q
16180:(
16177:L
16174:d
16144:.
16140:q
16136:d
16131:n
16126:1
16123:=
16120:k
16117:,
16114:i
16109:)
16103:j
16093:q
16083:q
16076:|
16065:k
16061:q
16052:j
16048:q
16037:i
16033:F
16027:2
16014:n
16009:1
16006:=
16003:j
15994:(
15989:=
15984:T
15979:n
15976:,
15970:,
15967:1
15964:=
15961:i
15956:)
15950:j
15940:q
15930:q
15923:|
15912:k
15908:q
15899:j
15895:q
15884:i
15880:F
15874:2
15861:n
15856:1
15853:=
15850:j
15840:k
15836:q
15832:d
15827:n
15822:1
15819:=
15816:k
15807:(
15802:=
15790:T
15785:n
15782:,
15776:,
15773:1
15770:=
15767:i
15762:)
15756:k
15752:q
15748:d
15742:q
15735:|
15724:k
15720:q
15711:j
15707:q
15696:i
15692:F
15686:2
15673:n
15668:1
15665:=
15662:k
15652:j
15642:q
15633:n
15628:1
15625:=
15622:j
15613:(
15608:=
15598:q
15589:n
15584:1
15581:=
15578:j
15575:,
15572:i
15567:)
15561:k
15557:q
15553:d
15547:q
15540:|
15529:k
15525:q
15516:j
15512:q
15501:i
15497:F
15491:2
15478:n
15473:1
15470:=
15467:k
15458:(
15453:=
15443:q
15436:)
15433:)
15429:q
15425:(
15416:F
15412:(
15409:d
15396:=
15383:q
15379:d
15376:)
15366:q
15359:,
15355:q
15351:(
15348:G
15324:,
15314:q
15307:d
15304:)
15300:q
15296:(
15287:F
15283:+
15279:q
15275:d
15272:)
15262:q
15255:,
15251:q
15247:(
15244:G
15241:=
15231:Q
15224:d
15204:.
15194:q
15187:)
15183:q
15179:(
15170:F
15166:=
15156:Q
15131:Q
15127:d
15107:,
15102:i
15092:Q
15066:i
15056:q
15029:n
15024:1
15021:=
15018:j
15015:,
15012:i
15007:)
15001:q
14995:|
14987:j
14983:q
14972:i
14968:F
14954:(
14949:=
14946:)
14942:q
14938:(
14929:F
14907:,
14903:}
14898:R
14889:i
14879:Q
14867:|
14858:)
14853:)
14849:q
14845:(
14842:F
14837:|
14829:i
14825:Q
14808:(
14799:i
14789:Q
14780:n
14775:1
14772:=
14769:i
14760:{
14738:}
14733:R
14724:i
14714:q
14702:|
14693:)
14687:q
14681:|
14673:i
14669:q
14652:(
14643:i
14633:q
14624:n
14619:1
14616:=
14613:i
14604:{
14579:)
14575:q
14571:(
14562:F
14541:,
14537:q
14533:d
14530:)
14526:q
14522:(
14513:F
14509:=
14505:Q
14501:d
14491:)
14489:q
14487:(
14485:F
14481:Q
14457:i
14447:q
14436:/
14432:L
14409:.
14406:)
14403:t
14400:,
14390:Q
14383:,
14379:Q
14375:(
14372:E
14369:=
14366:)
14363:t
14360:,
14350:q
14343:,
14339:q
14335:(
14332:E
14321:Q
14317:q
14308:t
14281:j
14277:q
14267:r
14249:i
14245:q
14235:r
14225:=
14220:j
14217:i
14213:a
14192:E
14189:=
14186:V
14183:+
14180:T
14177:=
14174:)
14171:V
14165:T
14162:(
14156:T
14153:2
14150:=
14147:L
14140:)
14131:i
14121:q
14109:L
14098:i
14088:q
14079:n
14074:1
14071:=
14068:i
14059:(
14055:=
14052:H
14033:T
14030:2
14027:=
14023:)
14017:j
14007:q
13998:i
13988:q
13979:j
13976:i
13972:a
13966:j
13963:,
13960:i
13952:2
13948:(
13942:2
13939:m
13934:=
13926:k
13916:q
13904:T
13893:k
13883:q
13874:n
13869:1
13866:=
13863:k
13855:=
13847:k
13837:q
13825:L
13814:k
13804:q
13795:n
13790:1
13787:=
13784:k
13758:j
13748:q
13739:i
13729:q
13720:j
13717:i
13713:a
13707:j
13704:,
13701:i
13691:2
13688:m
13683:=
13679:)
13673:j
13663:q
13651:j
13647:q
13632:r
13619:(
13611:)
13605:i
13595:q
13583:i
13579:q
13564:r
13551:(
13545:j
13542:,
13539:i
13529:2
13526:m
13521:=
13516:2
13512:v
13506:2
13503:m
13498:=
13495:T
13475:)
13470:n
13466:q
13462:,
13456:,
13451:1
13447:q
13443:(
13439:r
13435:=
13431:r
13410:.
13407:L
13399:)
13389:i
13379:q
13367:L
13356:i
13346:q
13337:n
13332:1
13329:=
13326:i
13316:(
13311:=
13308:H
13284:,
13281:L
13263:φ
13258:φ
13254:p
13250:s
13245:s
13241:p
13237:z
13232:z
13228:p
13214:,
13191:L
13182:=
13173:p
13168:,
13156:s
13145:L
13136:=
13131:s
13127:p
13122:,
13110:z
13099:L
13090:=
13085:z
13081:p
13070:φ
13066:s
13062:z
13058:φ
13054:θ
13050:s
13046:z
13042:r
13028:,
13025:)
13022:t
13019:,
13004:,
12989:,
12980:r
12974:,
12965:z
12959:,
12950:s
12944:,
12938:,
12935:r
12932:(
12929:L
12902:0
12899:=
12891:i
12887:q
12878:L
12869:=
12861:i
12851:q
12839:L
12827:t
12823:d
12817:d
12811:=
12806:i
12796:p
12781:i
12779:q
12771:L
12755:.
12747:i
12737:q
12725:L
12716:=
12711:i
12707:p
12695:i
12693:q
12660:=
12652:i
12648:Q
12639:L
12620:i
12610:Q
12598:L
12586:t
12583:d
12579:d
12557:Q
12551:q
12525:0
12522:=
12514:i
12510:q
12499:k
12495:Q
12484:)
12475:k
12465:Q
12453:L
12440:)
12432:k
12422:Q
12410:L
12401:(
12394:t
12391:d
12387:d
12381:(
12375:k
12363:0
12360:=
12356:)
12351:)
12343:i
12339:q
12328:k
12324:Q
12314:(
12307:t
12304:d
12300:d
12290:k
12280:Q
12268:L
12249:i
12239:q
12225:k
12215:Q
12197:k
12193:Q
12184:L
12169:)
12161:i
12151:q
12137:k
12127:Q
12114:(
12107:t
12104:d
12100:d
12090:k
12080:Q
12068:L
12059:+
12051:i
12041:q
12027:k
12017:Q
12003:)
11995:k
11985:Q
11973:L
11964:(
11957:t
11954:d
11950:d
11944:(
11938:k
11926:0
11923:=
11919:)
11910:i
11905:q
11893:k
11883:Q
11865:k
11855:Q
11843:L
11824:i
11819:q
11807:k
11803:Q
11788:k
11784:Q
11775:L
11760:)
11751:i
11741:q
11727:k
11717:Q
11699:k
11689:Q
11677:L
11667:(
11660:t
11657:d
11653:d
11647:(
11641:k
11629:0
11626:=
11618:i
11614:q
11605:L
11586:i
11576:q
11564:L
11552:t
11549:d
11545:d
11511:k
11507:q
11496:i
11486:Q
11473:=
11469:)
11461:k
11457:q
11446:i
11442:Q
11432:(
11425:t
11422:d
11418:d
11396:)
11393:t
11390:,
11386:q
11382:(
11374:j
11370:q
11363:t
11353:k
11349:Q
11343:2
11332:+
11327:k
11317:q
11309:)
11306:t
11303:,
11299:q
11295:(
11287:j
11283:q
11274:k
11270:q
11259:i
11255:Q
11249:2
11236:k
11228:=
11224:)
11216:j
11212:q
11200:i
11196:Q
11185:(
11178:t
11175:d
11171:d
11149:)
11146:t
11143:,
11139:q
11135:(
11129:t
11121:j
11117:q
11106:k
11102:Q
11096:2
11085:+
11080:k
11070:q
11062:)
11059:t
11056:,
11052:q
11048:(
11040:k
11036:q
11027:j
11023:q
11012:i
11008:Q
11002:2
10989:k
10981:=
10973:j
10969:q
10952:i
10948:Q
10910:k
10906:q
10895:i
10891:Q
10881:=
10873:k
10863:q
10843:i
10839:Q
10809:)
10806:t
10803:,
10799:q
10795:(
10789:t
10779:k
10775:Q
10765:+
10760:k
10750:q
10742:)
10739:t
10736:,
10732:q
10728:(
10720:k
10716:q
10705:i
10701:Q
10689:k
10681:=
10670:i
10666:Q
10642:,
10639:t
10636:d
10630:t
10620:k
10616:Q
10606:+
10600:k
10596:q
10591:d
10583:k
10579:q
10568:i
10564:Q
10552:k
10544:=
10538:i
10534:Q
10529:d
10509:)
10506:t
10503:,
10499:q
10495:(
10490:i
10486:Q
10482:=
10477:i
10473:Q
10443:.
10435:i
10431:Q
10418:L
10408:=
10400:i
10390:Q
10374:L
10361:t
10357:d
10351:d
10324:,
10321:)
10318:t
10315:,
10312:)
10309:t
10306:,
10296:Q
10289:,
10285:Q
10281:(
10271:q
10264:,
10261:)
10258:t
10255:,
10251:Q
10247:(
10243:q
10239:(
10236:L
10233:=
10230:)
10227:t
10224:,
10214:Q
10207:,
10203:Q
10199:(
10192:L
10181:L
10177:)
10175:t
10171:Q
10169:(
10167:q
10163:q
10158:)
10156:t
10152:q
10150:(
10148:Q
10144:Q
10136:Q
10132:q
10120:q
10116:)
10111:t
10104:P
10102:(
10100:f
10095:)
10090:t
10083:P
10081:(
10079:f
10060:,
10057:)
10048:t
10044:,
10035:P
10031:(
10028:f
10022:)
10013:t
10009:,
10000:P
9996:(
9993:f
9990:+
9987:]
9983:q
9979:[
9976:S
9973:=
9963:t
9960:d
9953:t
9949:d
9943:)
9940:t
9937:,
9934:)
9931:t
9928:(
9924:q
9920:(
9917:f
9913:d
9898:t
9886:t
9877:+
9874:t
9871:d
9867:)
9864:t
9861:,
9858:)
9855:t
9852:(
9842:q
9835:,
9832:)
9829:t
9826:(
9822:q
9818:(
9815:L
9804:t
9792:t
9783:=
9773:t
9770:d
9766:)
9763:t
9760:,
9757:)
9754:t
9751:(
9741:q
9734:,
9731:)
9728:t
9725:(
9721:q
9717:(
9710:L
9698:t
9686:t
9677:=
9670:]
9666:q
9662:[
9655:S
9640:S
9636:S
9632:L
9628:L
9612:.
9607:i
9597:q
9585:i
9581:q
9572:)
9569:t
9566:,
9562:q
9558:(
9555:f
9544:i
9536:+
9530:t
9522:)
9519:t
9516:,
9512:q
9508:(
9505:f
9476:t
9472:d
9466:)
9463:t
9460:,
9456:q
9452:(
9449:f
9445:d
9421:,
9415:t
9411:d
9405:)
9402:t
9399:,
9395:q
9391:(
9388:f
9384:d
9377:+
9374:)
9371:t
9368:,
9358:q
9351:,
9347:q
9343:(
9340:L
9337:=
9334:)
9331:t
9328:,
9318:q
9311:,
9307:q
9303:(
9296:L
9285:)
9283:t
9279:q
9277:(
9275:f
9270:)
9265:t
9263:(
9261:q
9254:P
9249:)
9244:t
9242:(
9240:q
9233:P
9223:t
9216:t
9210:q
9206:L
9201:b
9193:L
9188:b
9184:a
9180:L
9153:,
9145:k
9140:r
9128:i
9124:f
9112:i
9102:C
9097:1
9094:=
9091:i
9083:=
9078:k
9073:C
9051:,
9048:0
9045:=
9037:k
9032:r
9020:i
9016:f
9004:i
8994:C
8989:1
8986:=
8983:i
8975:+
8968:k
8963:N
8946:k
8941:r
8931:V
8915:+
8908:k
8903:F
8883:k
8872:r
8859:T
8847:t
8843:d
8837:d
8823:k
8818:r
8808:T
8787:V
8783:T
8779:L
8774:k
8771:r
8767:V
8748:=
8745:)
8742:t
8739:,
8734:k
8729:r
8724:(
8719:i
8715:f
8706:0
8703:=
8695:i
8669:L
8656:t
8652:d
8646:d
8632:i
8615:L
8597:i
8593:λ
8575:,
8572:0
8569:=
8561:k
8556:r
8544:i
8540:f
8528:i
8518:C
8513:1
8510:=
8507:i
8499:+
8491:k
8480:r
8467:L
8455:t
8451:d
8445:d
8431:k
8426:r
8416:L
8402:0
8399:=
8391:k
8380:r
8363:L
8350:t
8346:d
8340:d
8326:k
8321:r
8307:L
8289:k
8285:r
8280:k
8276:r
8273:δ
8254:=
8251:t
8247:d
8240:k
8235:r
8223:)
8214:k
8209:r
8197:i
8193:f
8181:i
8171:C
8166:1
8163:=
8160:i
8152:+
8144:k
8133:r
8120:L
8108:t
8104:d
8098:d
8084:k
8079:r
8069:L
8059:(
8053:N
8048:1
8045:=
8042:k
8030:2
8026:t
8018:1
8014:t
8005:=
8002:t
7998:d
7990:L
7979:2
7975:t
7967:1
7963:t
7946:k
7942:r
7938:t
7922:.
7919:)
7916:t
7913:,
7908:k
7903:r
7898:(
7893:i
7889:f
7885:)
7882:t
7879:(
7874:i
7864:C
7859:1
7856:=
7853:i
7845:+
7842:)
7839:t
7836:,
7830:,
7825:2
7814:r
7806:,
7801:1
7790:r
7782:,
7776:,
7771:2
7766:r
7761:,
7756:1
7751:r
7746:(
7743:L
7740:=
7733:L
7722:C
7718:i
7713:i
7709:λ
7703:t
7698:k
7694:r
7692:(
7689:i
7685:f
7675:k
7671:r
7668:δ
7663:k
7659:r
7656:δ
7652:k
7647:0
7643:2
7640:t
7638:(
7635:k
7631:r
7628:δ
7624:1
7621:t
7619:(
7616:k
7612:r
7609:δ
7603:k
7599:r
7595:L
7576:=
7573:t
7569:d
7562:k
7557:r
7545:)
7536:k
7525:r
7512:L
7500:t
7496:d
7490:d
7476:k
7471:r
7461:L
7451:(
7445:N
7440:1
7437:=
7434:k
7422:2
7418:t
7410:1
7406:t
7390:N
7385:k
7381:r
7377:L
7283:=
7280:S
7263:2
7260:t
7256:1
7253:t
7233:,
7230:t
7226:d
7221:L
7214:2
7210:t
7202:1
7198:t
7189:=
7186:S
7157:=
7154:t
7150:d
7145:L
7135:2
7131:t
7123:1
7119:t
7096:j
7087:j
7075:j
7069:2
7066:t
7064:(
7061:j
7053:1
7050:t
7048:(
7045:j
7020:.
7017:t
7013:d
7006:j
7002:q
6994:)
6985:j
6975:q
6963:L
6951:t
6947:d
6941:d
6927:j
6923:q
6914:L
6904:(
6898:n
6893:1
6890:=
6887:j
6875:2
6871:t
6863:1
6859:t
6850:+
6843:2
6839:t
6831:1
6827:t
6821:]
6815:j
6811:q
6799:j
6789:q
6777:L
6767:[
6760:n
6755:1
6752:=
6749:j
6741:=
6731:t
6727:d
6721:)
6715:j
6711:q
6699:j
6689:q
6677:L
6665:t
6661:d
6655:d
6645:)
6639:j
6635:q
6623:j
6613:q
6601:L
6591:(
6584:t
6580:d
6574:d
6568:+
6563:j
6559:q
6547:j
6543:q
6534:L
6524:(
6518:n
6513:1
6510:=
6507:j
6495:2
6491:t
6483:1
6479:t
6470:=
6463:t
6459:d
6454:L
6444:2
6440:t
6432:1
6428:t
6407:j
6399:t
6394:j
6386:t
6381:j
6379:q
6375:L
6370:j
6358:L
6340:,
6334:t
6330:d
6324:)
6319:j
6315:q
6308:(
6304:d
6291:t
6287:d
6279:j
6275:q
6270:d
6255:j
6245:q
6234:,
6230:)
6224:j
6214:q
6199:j
6189:q
6177:L
6168:+
6163:j
6159:q
6147:j
6143:q
6134:L
6124:(
6118:n
6113:1
6110:=
6107:j
6099:=
6096:L
6052:=
6044:j
6034:q
6022:L
6010:t
6006:d
6000:d
5986:j
5982:q
5973:L
5948:V
5944:T
5940:L
5925:,
5917:j
5913:q
5904:V
5887:j
5877:q
5865:V
5853:t
5849:d
5843:d
5837:=
5832:j
5828:Q
5817:V
5812:i
5808:Q
5804:V
5792:q
5788:S
5780:q
5743:j
5739:q
5730:T
5713:j
5703:q
5691:T
5679:t
5675:d
5669:d
5663:=
5658:j
5654:Q
5634:j
5616:,
5613:0
5610:=
5605:j
5601:q
5593:]
5588:)
5579:j
5575:q
5566:T
5549:j
5539:q
5527:T
5515:t
5511:d
5505:d
5498:(
5489:j
5485:Q
5480:[
5474:n
5469:1
5466:=
5463:j
5436:.
5428:j
5424:q
5415:T
5398:j
5388:q
5376:T
5364:t
5360:d
5354:d
5348:=
5340:j
5336:q
5325:k
5320:r
5304:k
5299:a
5292:k
5288:m
5282:N
5277:1
5274:=
5271:k
5243:.
5238:j
5234:q
5225:j
5221:Q
5215:n
5210:1
5207:=
5204:j
5196:=
5191:j
5187:q
5175:j
5171:q
5160:k
5155:r
5142:n
5137:1
5134:=
5131:j
5118:k
5113:N
5106:N
5101:1
5098:=
5095:k
5087:=
5082:k
5077:r
5064:k
5059:N
5052:N
5047:1
5044:=
5041:k
5016:,
5008:j
5004:q
4993:k
4988:r
4972:k
4967:N
4960:N
4955:1
4952:=
4949:k
4941:=
4936:j
4932:Q
4917:k
4913:r
4910:δ
4905:k
4901:N
4878:.
4873:j
4869:q
4857:j
4853:q
4842:k
4837:r
4824:n
4819:1
4816:=
4813:j
4805:=
4800:k
4795:r
4775:k
4771:r
4768:δ
4764:)
4761:k
4752:k
4743:k
4734:k
4730:r
4727:δ
4719:)
4716:k
4712:z
4707:k
4703:y
4698:k
4694:x
4689:k
4685:r
4680:k
4667:N
4662:k
4658:r
4655:δ
4635:=
4630:k
4625:r
4614:)
4609:k
4604:a
4597:k
4593:m
4584:k
4579:N
4574:(
4569:N
4564:1
4561:=
4558:k
4533:,
4530:0
4527:=
4522:k
4517:r
4504:k
4499:C
4492:N
4487:1
4484:=
4481:k
4454:k
4450:r
4447:δ
4422:=
4417:k
4412:r
4401:)
4396:k
4391:a
4384:k
4380:m
4371:k
4366:C
4361:+
4356:k
4351:N
4346:(
4341:N
4336:1
4333:=
4330:k
4314:k
4310:r
4307:δ
4303:N
4275:.
4273:N
4269:m
4265:r
4262:δ
4258:C
4190:.
4186:N
4182:+
4178:C
4174:=
4170:F
4159:C
4155:N
4151:F
4132:,
4130:0
4126:F
4115:,
4113:0
4109:F
4100:c
4096:b
4092:a
4075:g
4058:t
4054:d
4046:c
4037:d
4027:t
4023:d
4015:b
4006:d
3997:c
3994:b
3990:g
3986:m
3981:2
3978:1
3973:=
3970:T
3946:a
3942:F
3928:,
3922:t
3918:d
3910:a
3901:d
3889:a
3871:,
3867:)
3858:k
3845:T
3828:k
3806:T
3794:t
3790:d
3784:d
3777:(
3771:k
3768:a
3764:g
3760:=
3756:)
3749:t
3745:d
3737:c
3728:d
3718:t
3714:d
3706:b
3697:d
3688:c
3685:b
3676:a
3668:+
3660:2
3656:t
3651:d
3643:a
3633:2
3628:d
3619:(
3615:m
3612:=
3607:a
3603:F
3583:ξ
3579:ξ
3575:ξ
3570:ξ
3556:t
3551:v
3547:r
3539:t
3535:r
3523:F
3519:a
3505:,
3501:a
3497:m
3494:=
3490:F
3475:m
3467:N
3435:t
3415:C
3411:N
3407:n
3402:N
3394:)
3392:t
3388:t
3384:q
3380:q
3378:(
3376:L
3349:j
3345:q
3336:L
3327:=
3323:)
3315:j
3305:q
3293:L
3284:(
3277:t
3273:d
3267:d
3223:.
3220:)
3217:t
3214:,
3204:q
3197:,
3193:q
3189:(
3186:T
3183:=
3180:T
3165:k
3161:v
3145:.
3139:t
3129:k
3124:r
3113:+
3108:j
3098:q
3086:j
3082:q
3071:k
3066:r
3053:n
3048:1
3045:=
3042:j
3034:=
3029:k
3024:v
3018:,
3012:t
3008:d
3000:j
2996:q
2991:d
2984:=
2979:j
2969:q
2947:q
2931:.
2926:)
2921:t
2918:,
2915:)
2912:t
2909:,
2905:q
2901:(
2896:k
2892:z
2888:,
2885:)
2882:t
2879:,
2875:q
2871:(
2866:k
2862:y
2858:,
2855:)
2852:t
2849:,
2845:q
2841:(
2836:k
2832:x
2826:(
2821:=
2818:)
2815:t
2812:,
2808:q
2804:(
2799:k
2794:r
2789:=
2784:k
2779:r
2763:)
2760:n
2758:q
2754:2
2751:q
2747:1
2744:q
2740:q
2735:n
2728:n
2723:C
2719:N
2715:n
2703:2
2700:z
2689:z
2685:v
2681:L
2679:∂
2672:/
2670:2
2661:z
2657:v
2652:z
2648:L
2632:C
2628:N
2623:C
2619:N
2617:3
2613:N
2587:)
2578:k
2568:z
2552:,
2544:k
2534:y
2518:,
2510:k
2500:x
2483:(
2471:k
2460:r
2442:,
2438:)
2429:k
2425:z
2412:,
2404:k
2400:y
2387:,
2379:k
2375:x
2361:(
2349:k
2344:r
2322:i
2318:f
2313:i
2311:λ
2304:N
2300:k
2283:,
2280:0
2277:=
2269:k
2264:r
2252:i
2248:f
2236:i
2226:C
2221:1
2218:=
2215:i
2207:+
2199:k
2188:r
2175:L
2163:t
2159:d
2153:d
2139:k
2134:r
2124:L
2085:2
2082:v
2078:1
2075:v
2071:2
2068:r
2064:1
2061:r
2059:(
2057:L
2048:)
2046:t
2042:2
2039:v
2035:1
2032:v
2028:2
2025:r
2021:1
2018:r
2016:(
2014:L
2009:V
2005:T
1979:t
1974:k
1970:r
1968:(
1965:i
1963:f
1958:i
1954:k
1948:t
1944:r
1942:(
1939:C
1937:f
1930:t
1926:r
1924:(
1922:2
1919:f
1913:t
1909:r
1907:(
1905:1
1902:f
1897:C
1891:t
1887:r
1885:(
1883:f
1862:L
1854:t
1850:2
1847:v
1843:1
1840:v
1836:2
1833:r
1829:1
1826:r
1824:(
1822:V
1818:V
1811:2
1808:v
1804:1
1801:v
1797:2
1794:r
1790:1
1787:r
1785:(
1783:V
1779:V
1768:2
1765:r
1761:1
1758:r
1756:(
1754:V
1750:V
1733:V
1726:2
1723:v
1719:1
1716:v
1714:(
1712:T
1708:T
1703:t
1698:k
1694:r
1689:k
1685:v
1680:T
1666:k
1662:v
1657:k
1653:v
1648:k
1644:v
1629:,
1624:k
1620:m
1599:k
1579:N
1549:2
1544:k
1540:v
1534:k
1530:m
1524:N
1519:1
1516:=
1513:k
1503:2
1500:1
1495:=
1492:T
1472:,
1469:V
1463:T
1460:=
1457:L
1423:N
1419:N
1405:,
1397:2
1393:t
1389:d
1383:r
1377:2
1373:d
1366:m
1363:=
1359:F
1319:.
1313:t
1310:d
1303:N
1298:r
1293:d
1287:=
1282:N
1277:v
1272:,
1266:,
1260:t
1257:d
1250:2
1245:r
1240:d
1234:=
1229:2
1224:v
1219:,
1213:t
1210:d
1203:1
1198:r
1193:d
1187:=
1182:1
1177:v
1157:)
1155:z
1151:y
1147:x
1143:r
1138:N
1126:)
1124:2
1121:z
1117:2
1114:y
1110:2
1107:x
1103:2
1100:r
1095:)
1093:1
1090:z
1086:1
1083:y
1079:1
1076:x
1072:1
1069:r
1059:N
1055:r
1051:2
1048:r
1044:1
1041:r
1032:N
1030:m
1026:2
1023:m
1019:1
1016:m
1009:N
972:N
968:C
962:y
958:x
956:(
954:f
941:N
937:C
909:L
890:V
886:T
881:V
877:T
873:L
852:L
842:M
835:)
833:L
829:M
827:(
774:e
767:t
760:v
450:)
69:t
66:d
60:p
56:d
50:=
45:F
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