Analytical mechanics

726: 6326: 5842: 4387: 6321:{\displaystyle {\begin{aligned}&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{1}(\mathbf {q} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{2}(\mathbf {q} ,\mathbf {P} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{3}(\mathbf {p} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{4}(\mathbf {p} ,\mathbf {P} ,t)\\\end{aligned}}} 4078: 5399: 739: 4382:{\displaystyle {\begin{aligned}\{A,B\}\equiv \{A,B\}_{\mathbf {q} ,\mathbf {p} }&={\frac {\partial A}{\partial \mathbf {q} }}\cdot {\frac {\partial B}{\partial \mathbf {p} }}-{\frac {\partial A}{\partial \mathbf {p} }}\cdot {\frac {\partial B}{\partial \mathbf {q} }}\\&\equiv \sum _{k}{\frac {\partial A}{\partial q_{k}}}{\frac {\partial B}{\partial p_{k}}}-{\frac {\partial A}{\partial p_{k}}}{\frac {\partial B}{\partial q_{k}}}\,,\end{aligned}}} 1075: 975: 3231: 1016: 5329: 1833: 3003: 8744: 8108: 2251: 867:. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non- 8511: 5071: 8228: 2116: 9002: 8884: 7355: 1258:

as ensembles of particles that interact with each other, rather considering each particle as an isolated unit. In the vectorial approach, forces must be determined individually for each particle, whereas in the analytical approach it is enough to know one single function which contains implicitly all

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during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which

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Development of analytical mechanics has two objectives: (i) increase the range of solvable problems by developing standard techniques with a wide range of applicability, and (ii) understand the mathematical structure of mechanics. In the long run, however, (ii) can help (i) more than a concentration

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do not have to be cyclic, the partition between which coordinates enter the Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian

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Analytical mechanics aims at even more: not at understanding the mathematical structure of a single mechanical problem, but that of a class of problems so wide that they encompass most of mechanics. It concentrates on systems to which Lagrangian or Hamiltonian equations of motion are applicable and

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done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful – since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for

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of the problem. Moreover, an accurate mental or drawn picture can be made for the motion of two bodies, and it can be as real and accurate as the real bodies moving and interacting. In the three-body problem, parameters can also be assigned specific values; however, the solution at these assigned

5048: 1597: 1239:), Newton's approach is difficult to apply. Using a Newtonian approach is possible, under proper precautions, namely isolating each single particle from the others, and determining all the forces acting on it. Such analysis is cumbersome even in relatively simple systems. Newton thought that 960:

The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics.

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relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known as

3988: 7936: 2550: 2131: 1330:, and it is not possible to draw a sharp line between 'simple' and 'not simple' functions. If one speaks merely of 'functions', then every mechanical problem is solved as soon as it has been well stated in differential equations, because given the initial conditions and 3226:{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}=\left({\frac {\partial L}{\partial {\dot {q}}_{1}}},{\frac {\partial L}{\partial {\dot {q}}_{2}}},\cdots {\frac {\partial L}{\partial {\dot {q}}_{N}}}\right)=(p_{1},p_{2}\cdots p_{N})\,,} 3827: 3343: 8378: 8113: 2013: 2941: 5656: 1994: 8892: 8751: 11650: 7844: 929:

Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used in

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coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:

5531: 5324:{\displaystyle T((\lambda {\dot {q}}_{i})^{2},(\lambda {\dot {q}}_{j}\lambda {\dot {q}}_{k}),\mathbf {q} )=\lambda ^{2}T(({\dot {q}}_{i})^{2},{\dot {q}}_{j}{\dot {q}}_{k},\mathbf {q} )\,,\quad L(\mathbf {q} ,\mathbf {\dot {q}} )\,,} 7250: 1366:

values or a collection of such solutions does not reveal the mathematical structure of the problem. As in many other problems, the mathematical structure can be elucidated only by examining the differential equations themselves.

6753: 4887: 852:. A scalar is a quantity, whereas a vector is represented by quantity and direction. The results of these two different approaches are equivalent, but the analytical mechanics approach has many advantages for complex problems. 6345: 3382: 1907: 9558: 9500: 4487: 7599: 7378: 1276:, the principle that this action be stationary under small variation of some other mechanical quantity generates the required set of differential equations. The statement of the principle does not require any special 6696: 7610: 3702: 2784: 3582: 2585: 7141: 3632: 9681: 9639: 9191: 7006: 9390: 9328: 6813: 1828:{\displaystyle {\frac {d\mathbf {q} }{dt}}=\left({\frac {dq_{1}}{dt}},{\frac {dq_{2}}{dt}},\dots ,{\frac {dq_{N}}{dt}}\right)\equiv \mathbf {\dot {q}} =({\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{N}).} 3587:

which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:

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the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as

5847: 4083: 8739:{\displaystyle {\mathcal {H}}(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\mathbf {r} ,t)=\sum _{i=1}^{N}{\dot {\phi }}_{i}(\mathbf {r} ,t)\pi _{i}(\mathbf {r} ,t)-{\mathcal {L}}\,.} 3738: 3242: 11595: 9442: 7052:

Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates. If the Lagrangian of a system has

8103:{\displaystyle {\mathcal {L}}={\mathcal {L}}(\phi _{1},\phi _{2},\dots ,\nabla \phi _{1},\nabla \phi _{2},\dots ,\partial _{t}\phi _{1},\partial _{t}\phi _{2},\ldots ,\mathbf {r} ,t)\,.} 2246:{\displaystyle {\frac {\partial }{\partial \mathbf {q} }}=\left({\frac {\partial }{\partial q_{1}}},{\frac {\partial }{\partial q_{2}}},\dots ,{\frac {\partial }{\partial q_{N}}}\right)} 1232:

and the motion of planets around the Sun. In this approach, Newton's laws describe the motion by a differential equation and then the problem is reduced to the solving of that equation.

2852: 1224:" understood as a single point to which a mass is attached. Newton's method has been successfully applied to a wide range of physical problems, including the motion of a particle in 5548: 898:

on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as

8506:{\displaystyle \pi _{i}(\mathbf {r} ,t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}_{i}}}\,\quad {\dot {\phi }}_{i}\equiv {\frac {\partial \phi _{i}}{\partial t}}} 6554: 1189:

The goal of mechanical theory is to solve mechanical problems, such as arise in physics and engineering. Starting from a physical system—such as a mechanism or a star system—a

11640: 2816: 104: 9597: 9152: 1030: 8538: 8223:{\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}\right)={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}\,,} 6608: 5733: 5701: 11645: 10886: 4816: 2111:{\displaystyle {\boldsymbol {\mathcal {Q}}}={\frac {d}{dt}}\left({\frac {\partial T}{\partial \mathbf {\dot {q}} }}\right)-{\frac {\partial T}{\partial \mathbf {q} }}\,,} 7750: 5379: 1858: 10247:

Kibble, Tom, and Berkshire, Frank H. "Classical Mechanics" (5th Edition). Singapore, World Scientific Publishing Company, 2004.

9266: 8997:{\displaystyle {\frac {\delta }{\delta \phi _{i}}}={\frac {\partial }{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial }{\partial (\partial _{\mu }\phi _{i})}}} 3366: 8879:{\displaystyle {\dot {\phi }}_{i}=+{\frac {\delta {\mathcal {H}}}{\delta \pi _{i}}}\,,\quad {\dot {\pi }}_{i}=-{\frac {\delta {\mathcal {H}}}{\delta \phi _{i}}}\,,} 5435: 10065: 8330:

Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as

9805: 9709: 7350:{\displaystyle {\dot {\mathbf {q} }}=+{\frac {\partial R}{\partial \mathbf {p} }}\,,\quad {\dot {\mathbf {p} }}=-{\frac {\partial R}{\partial \mathbf {q} }}\,,} 1295:

It is not altogether clear what is meant by 'solving' a set of differential equations. A problem is regarded as solved when the particles coordinates at time

11264: 6441:{\displaystyle \mathbf {\dot {P}} =-{\frac {\partial K}{\partial \mathbf {Q} }}\,,\quad \mathbf {\dot {Q}} =+{\frac {\partial K}{\partial \mathbf {P} }}\,,} 3478:{\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,} 11633: 11269: 6707: 5043:{\displaystyle {\frac {\partial L}{\partial q_{j}}}=0\,\rightarrow \,{\frac {dp_{j}}{dt}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0} 1266:

Still, deriving the equations of motion of a complicated mechanical system requires a unifying basis from which they follow. This is provided by various

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has not. The two-body problem is solved by formulas involving parameters; their values can be changed to study the class of all solutions, that is, the

7458:{\displaystyle {\frac {d}{dt}}{\frac {\partial R}{\partial {\dot {\boldsymbol {\zeta }}}}}={\frac {\partial R}{\partial {\boldsymbol {\zeta }}}}\,.} 1270:: behind each set of equations there is a principle that expresses the meaning of the entire set. Given a fundamental and universal quantity called 10879: 4410: 2301: 10993: 9018: 8278: 7734:{\displaystyle {\mathcal {Q}}_{r}={\frac {\partial S}{\partial \alpha _{r}}}\,,\quad S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}^{2}\,,} 7507: 5745:

in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies the

1193:

is developed in the form of a differential equation. The model can be solved numerically or analytically to determine the motion of the system.

11628: 10764: 6620: 4710: 10534: 2673:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,} 1516: 770: 7226:{\displaystyle R=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {p} ,{\boldsymbol {\zeta }},{\dot {\boldsymbol {\zeta }}})\,,} 3650: 2732: 11446: 3520: 10872: 10853: 4651:) for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time 3594: 1034: 359: 9645: 9603: 11491: 9158: 6944: 1448: 860: 478: 9334: 9272: 6925:{\displaystyle H=H(\mathbf {q} ,\mathbf {p} ,t)=H\left(\mathbf {q} ,{\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }},t\right)} 6768: 3861:)) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the 1263:. However, the analytical treatment does not require the knowledge of these forces and takes these kinematic conditions for granted. 10998: 8331: 3983:{\displaystyle \{(\mathbf {q} (t),\mathbf {p} (t))\in \mathbb {R} ^{2N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {P}}\,,} 1463:; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of 1139: 451: 4900:

If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are

910:. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the 11407: 2545:{\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {q} ,\mathbf {\dot {q}} ,t)-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)} 1111: 6474:". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem. 4522: 10732: 9854: 1259:

the forces acting on and in the system. Such simplification is often done using certain kinematic conditions which are stated

36: 9811: 11008: 10485: 10458: 10435: 10416: 10395: 10356: 10273: 10224: 10189: 6485:

is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformation

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is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:

1118: 11574: 9749: 9715: 9197: 3822:{\displaystyle {\mathcal {P}}={\mathcal {C}}\times {\mathcal {M}}=\{(\mathbf {q} ,\mathbf {p} )\in \mathbb {R} ^{2N}\}\,,} 3338:{\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)} 11028: 11023: 433: 10211:. Encyclopedia of Physics / Handbuch der Physik. Vol. 2 / 3 / 1. Berlin, Heidelberg: Springer Berlin Heidelberg. 1125: 10527: 10505: 10333: 10314: 10294: 10162: 1176: 1158: 1056: 1002: 763: 9403: 11441: 11125: 3514:). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian: 1197: 828:

Analytical mechanics was developed by many scientists and mathematicians during the 18th century and onward, after

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Set up in this way, although the Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian with

5054:" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates. 725: 11188: 10948: 6762:

is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:

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If the Lagrangian is time-independent the Hamiltonian is also time-independent (i.e. both are constant in time).

2936:{\displaystyle \{\mathbf {q} (t)\in \mathbb {R} ^{N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {C}}\,,} 11115: 10680: 7498: 7493: 2688: 1096: 907: 354: 94: 5651:{\displaystyle \delta {\mathcal {S}}=\delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0\,,} 1989:{\displaystyle {\boldsymbol {\mathcal {Q}}}=({\mathcal {Q}}_{1},{\mathcal {Q}}_{2},\dots ,{\mathcal {Q}}_{N})} 11699: 11120: 11013: 6463: 5776:

The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of

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This formulation identifies the actual path followed by the motion as a selection of the path over which the

1460: 883: 1243:"action equals reaction" would take care of all complications. This is false even for such simple system as 11674: 11018: 10520: 10375:, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3 10104: 8253:

second order partial differential equations in the fields, which in general will be coupled and nonlinear.

6572: 2576: 2416: 2000:(script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and 1843: 756: 743: 504: 427: 349: 192: 11060: 10817: 10129: 9948: 8346:. It is a question of determining the correct Lagrangian density to generate the correct field equation. 423: 224: 10208:

Principles of Classical Mechanics and Field Theory / Prinzipien der Klassischen Mechanik und Feldtheorie

202: 11704: 11678: 11142: 10648: 10621: 9938:, i.e. the action (and hence the Lagrangian) does not change under a transformation parameterized by a 9935: 8335: 6507: 899: 643: 532: 458: 318: 251: 8513:

where in this context the overdot denotes a partial time derivative, not a total time derivative. The

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and of parameters defining the initial positions and velocities. However, 'simple function' is not a

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is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.

11395: 11390: 11299: 11259: 10975: 10754: 10475: 10124: 8350: 7839:{\displaystyle \mathbf {a} _{k}={\ddot {\mathbf {r} }}_{k}={\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}} 7035: 5426: 1440:

Generalized coordinates incorporate constraints on the system. There is one generalized coordinate

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which provide arithmetical solutions to mechanical problems to any desired degree of accuracy, the

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may contain an excessive amount of intricate detail that may interest only a particular audience

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involve generalized accelerations, the second time derivatives of the generalized coordinates:

4513: 2572: 1362: 1308: 931: 822: 668: 485: 9235: 5526:{\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt\,.} 11506: 11486: 11424: 11353: 11132: 11110: 11085: 10985: 10843: 10812: 10675: 10596: 10094: 10089: 9396: 3373: 2993: 2966: 2266: 1398: 1343: 1267: 1240: 887: 678: 653: 339: 157: 10179: 10152: 9012:

first order partial differential equations, which in general will be coupled and nonlinear.

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The study of the solutions of the Hamilton–Jacobi equations leads naturally to the study of

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When a mechanical system contains many particles, however (such as a complex mechanism or a

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as well as generalized forces mentioned above in D'Alembert's principle. The equations are

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and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):

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The configuration space can be defined more generally, and indeed more deeply, in terms of

2827: 2412: 1251:. In more complicated systems, the vectorial approach cannot give an adequate description. 943: 915: 879: 788: 698: 658: 566: 562: 554: 544: 334: 83: 10289:. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. 9083:

Each transformation can be described by an operator (i.e. function acting on the position

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variables to change them). The following are the cases when the operator does not change

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Following are overlapping properties between the Lagrangian and Hamiltonian functions.

4505: 4497: 4400:, and substituting Hamilton's equations into the result leads to the time evolution of 2843: 1997: 1358: 1190: 935: 923: 903: 868: 603: 344: 219: 187: 147: 11496: 11232: 11210: 11137: 11098: 11075: 10864: 10848: 10797: 10742: 10722: 10608: 10501: 10481: 10454: 10431: 10412: 10391: 10352: 10329: 10310: 10290: 10269: 10230: 10220: 10206: 10185: 10158: 6748:{\displaystyle \mathbf {p} ={\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }}} 5750: 4501: 3833: 2723: 2421:

The introduction of generalized coordinates and the fundamental Lagrangian function:

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are derived from the scalar quantity by some underlying principle about the scalar's

613: 570: 527: 522: 463: 239: 229: 122: 5410:(only some are shown). The path taken by the system (red) has a stationary action (δ 4882:{\displaystyle K=H+{\frac {\partial }{\partial t}}G(\mathbf {q} ,\mathbf {p} ,t)\,,} 2846:. The general solutions form a set of possible configurations as functions of time: 2288:, and if the position vector can be written in terms of the generalized coordinates 11414: 11205: 11193: 11176: 11154: 11103: 11055: 10916: 10787: 10727: 10653: 10212: 9687: 9228: 5422: 4692: 4393: 2568: 2007: 1902:{\displaystyle \delta W={\boldsymbol {\mathcal {Q}}}\cdot \delta \mathbf {q} =0\,,} 1354: 1271: 919: 833: 814: 805: 708: 688: 633: 628: 574: 549: 404: 262: 207: 182: 9553:{\displaystyle \mathbf {p} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {p} } 9495:{\displaystyle \mathbf {r} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {r} } 11520: 11476: 11303: 11247: 11242: 11181: 11003: 10928: 10807: 10717: 10663: 10563: 10448: 9910: 8272: 4696: 3999: 2280: 1587: 703: 648: 598: 593: 512: 10500:

Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973,

10071:, which can be length, angle of rotation, or time. The corresponding momenta to 1471:

of the position space in question (usually 3 for 3d space), while the number of

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particle, the second time derivative of its position vector. Each acceleration

7031: 5676: 3863: 3841: 3640: 2954: 2710: 2560: 2125: 810: 730: 638: 539: 256: 10216: 10157:(4th ed.). New York: Dover Publications Inc. Introduction, pp. xxi–xxix. 7030:. In this formulation, the solutions of the Hamilton–Jacobi equations are the 1428: 11693: 11481: 11436: 11419: 11065: 10938: 10747: 10234: 2706: 618: 445: 10512: 4482:{\displaystyle {\frac {dA}{dt}}=\{A,H\}+{\frac {\partial A}{\partial t}}\,.} 1037:

any relevant information, and removing excessive detail that may be against

11501: 11380: 11159: 11070: 10965: 10772: 8265: 8261: 8257: 7902: 7594:{\displaystyle \alpha _{r}={\ddot {q}}_{r}={\frac {d^{2}q_{r}}{dt^{2}}}\,,} 2393: 2270: 1850: 1304: 1209: 1196:

Newton's vectorial approach to mechanics describes motion with the help of

954: 882:(using generalized coordinates and corresponding generalized velocities in 837: 683: 608: 297: 177: 1326:) as in the time of Newton but most generally as a quantity determined by 1284:. This means that the analytical equations of motion do not change upon a 11612: 11528: 9914: 9015:

Again, the volume integral of the Hamiltonian density is the Hamiltonian

6691:{\displaystyle G_{2}(\mathbf {q} ,t)={\mathcal {S}}(\mathbf {q} ,t)+C\,,} 5398: 3728: 3712: 3369: 2823: 1505:

degrees of freedom, the generalized coordinates can be collected into an

891: 11548: 10792: 10782: 10430:, R.J.A. Lambourne, Open University, Cambridge University Press, 2010, 10309:

McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994,

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of the Lagrangian replaces the generalized coordinates and velocities (

2830:). The particular solution to the Euler–Lagrange equations is called a 2399: 2274: 2006:

are the generalized coordinates. This leads to the generalized form of

1248: 1099: in this section. Unsourced material may be challenged and removed. 468: 142: 5735:. The path for which action is least is the path taken by the system. 3697:{\displaystyle {\mathcal {M}}=\{\mathbf {p} \in \mathbb {R} ^{N}\}\,.} 2779:{\displaystyle {\mathcal {C}}=\{\mathbf {q} \in \mathbb {R} ^{N}\}\,,} 11596:

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

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A general way to find the equations of motion from the action is the

3577:{\displaystyle {\frac {dH}{dt}}=-{\frac {\partial L}{\partial t}}\,,} 1468: 490: 4893:

is a frequently used letter in this case). This property is used in

4508:(indicated by hats (^)), and the Poisson bracket is replaced by the 3637:

Analogous to the configuration space, the set of all momenta is the

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properties of motion representing the system as a whole—usually its

5754: 5339:, equal to the total kinetic and potential energies of the system: 4609: 3627:{\displaystyle \mathbf {\dot {p}} ={\boldsymbol {\mathcal {Q}}}\,.} 2950: 2947: 2391:. For time-independent situations, the constraints are also called 1374:

on specific problems for which methods have already been designed.

1244: 1221: 1205: 872: 841: 409: 292: 267: 10390:, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, 10328:, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, 9676:{\displaystyle \mathbf {p} \rightarrow \mathbf {p} +m\mathbf {v} } 9634:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {v} t} 6567:. If this does not hold then the transformation is not canonical. 2380:

in cases when the constraints vary with time, not just because of

1353:

Still, though lacking precise definitions, it is obvious that the

1338:. This is a fact especially at present with the modern methods of 9186:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {a} } 7001:{\displaystyle W(\mathbf {q} )={\mathcal {S}}(\mathbf {q} ,t)+Et} 949:

Analytical mechanics is used widely, from fundamental physics to

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Synge, J. L. (1960). "Classical dynamics". In Flügge, S. (ed.).

10184:(4th ed.). New York: Dover Publications Inc. pp. 3–6. 9385:{\displaystyle \mathbf {p} (t)\rightarrow \mathbf {p} (t+t_{0})} 9323:{\displaystyle \mathbf {r} (t)\rightarrow \mathbf {r} (t+t_{0})} 9070: 6808:{\displaystyle H=-{\frac {\partial {\mathcal {S}}}{\partial t}}} 4795:

Analogously, the Hamiltonian is invariant under addition of the

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If the curvilinear coordinate system is defined by the standard

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European Community on Computational Methods in Applied Sciences

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times are fixed. The term "path" or "trajectory" refers to the

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property that is lacking in the vectorial equations of motion.

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The corresponding "momentum" field densities conjugate to the

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and the Euler–Lagrange equations have an analogue for fields:

5414:= 0) under small changes in the configuration of the system (δ 8249:

scalar fields, these Lagrangian field equations are a set of

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International Council for Industrial and Applied Mathematics

2347:{\displaystyle \mathbf {r} =\mathbf {r} (\mathbf {q} (t),t)} 10411:

Quantum Field Theory, D. McMahon, Mc Graw Hill (US), 2008,

10268:, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, 9061:{\displaystyle H=\int _{\mathcal {V}}{\mathcal {H}}\,dV\,.} 8321:{\displaystyle L=\int _{\mathcal {V}}{\mathcal {L}}\,dV\,.} 4792:. In other words, the Lagrangian of a system is not unique. 3836:

of the configuration space and generalized momentum space.

3720: 848:, of the constituents of the system; it can also be called 282: 245: 10480:. Cambridge University Press. p. §12.3, pp. 432–439. 7106:, with the rest of the coordinates non-cyclic and denoted 4599:{\displaystyle \{A,B\}\rightarrow {\frac {1}{i\hbar }}\,.} 3839:

A particular solution to Hamilton's equations is called a

3492:

first-order ordinary differential equations, one for each

1382: 9889:{\displaystyle \mathbf {p} \rightarrow -\mathbf {p} (-t)} 7898:

Generalized coordinates apply to discrete particles. For

5425:

is another quantity in analytical mechanics defined as a

1837: 75:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}} 16:

Overview of mechanics based on the least action principle

10067:

the Lagrangian describes the same motion independent of

9843:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} (-t)} 9008:

fields, these Hamiltonian field equations are a set of 2

9004:

must be used instead of merely partial derivatives. For

4773:{\displaystyle L'=L+{\frac {d}{dt}}F(\mathbf {q} ,t)\,,} 1254:

The analytical approach simplifies problems by treating

9934:

symmetry transformation of the action corresponds to a

7862:

is expressed in terms of the generalized accelerations

4072:

is defined by the generalized coordinates and momenta:

1579:{\displaystyle \mathbf {q} =(q_{1},q_{2},\dots ,q_{N})} 875:, in which case one may revert to Newtonian mechanics. 10894: 4908:, this immediately follows from Lagrange's equations: 964: 11265:

Numerical methods for ordinary differential equations

9951: 9857: 9814: 9793: 9772:{\displaystyle \mathbf {p} \rightarrow -\mathbf {p} } 9752: 9738:{\displaystyle \mathbf {r} \rightarrow -\mathbf {r} } 9718: 9697: 9648: 9606: 9574: 9509: 9451: 9406: 9337: 9275: 9238: 9200: 9161: 9129: 9021: 8895: 8754: 8546: 8522: 8381: 8281: 8116: 7939: 7753: 7613: 7510: 7381: 7253: 7144: 6947: 6831: 6771: 6710: 6623: 6592: 6510: 6348: 5845: 5717: 5685: 5551: 5438: 5345: 5074: 5068:

the Lagrangian is explicitly time-independent, then:

4914: 4819: 4713: 4525: 4413: 4081: 4004:

All dynamical variables can be derived from position

3876: 3741: 3653: 3597: 3523: 3385: 3354: 3245: 3006: 2855: 2795: 2735: 2588: 2430: 2304: 2134: 2016: 1915: 1861: 1600: 1519: 39: 11641:

Société de Mathématiques Appliquées et Industrielles

11634:

Japan Society for Industrial and Applied Mathematics

11270:

Numerical methods for partial differential equations

10984: 10383: 10381: 9217:{\displaystyle \mathbf {p} \rightarrow \mathbf {p} } 5679:

of the system as a path through configuration space

4496:

is closely related to the equation of motion in the

1590:(here denoted by an overdot) of this tuple give the 11457: 7876:are expressed in terms the generalized coordinates 7368:Lagrangian equations in the non cyclic coordinates 6578:By setting the canonically transformed Hamiltonian 5788:) allows the Hamiltonian in one set of coordinates 1370:that include a very wide range of problems indeed. 855:Analytical mechanics takes advantage of a system's 799:is a collection of closely related formulations of 10059: 9888: 9842: 9799: 9771: 9737: 9703: 9675: 9633: 9591: 9552: 9494: 9436: 9384: 9322: 9260: 9216: 9185: 9146: 9060: 8996: 8878: 8738: 8532: 8505: 8320: 8222: 8102: 7838: 7733: 7593: 7457: 7349: 7225: 7000: 6924: 6807: 6747: 6690: 6602: 6548: 6440: 6320: 5727: 5695: 5650: 5525: 5373: 5323: 5042: 4881: 4772: 4690:The Lagrangian is invariant under addition of the 4598: 4481: 4381: 3982: 3821: 3696: 3626: 3576: 3477: 3360: 3337: 3225: 2935: 2810: 2778: 2672: 2544: 2346: 2245: 2110: 1988: 1901: 1827: 1578: 878:Two dominant branches of analytical mechanics are 74: 10407: 10405: 10403: 10378: 9530: 9472: 9419: 8256:This scalar field formulation can be extended to 7240:Hamiltonian equations for the cyclic coordinates 6582:= 0, and the type-2 generating function equal to 6501:to be canonical is the Poisson bracket be unity, 6339:such that the transformed Hamiltonian system is: 4068:) are two scalar valued dynamical variables, the 1848:D'Alembert's principle states that infinitesimal 11691: 10261: 10259: 10257: 10255: 10253: 5335:is a constant, then the Hamiltonian will be the 4610:Properties of the Lagrangian and the Hamiltonian 4504:, in which classical dynamical variables become 2571:of the entire system, then either following the 890:(using coordinates and corresponding momenta in 5764: 4687:), which would mean explicit time-independence. 11629:Society for Industrial and Applied Mathematics 10400: 9437:{\displaystyle R(\mathbf {\hat {n}} ,\theta )} 3726:The set of all positions and momenta form the 10880: 10542: 10528: 10250: 9071:Symmetry, conservation, and Noether's theorem 3719:) as used in quantum mechanics and theory of 859:to solve problems. The constraints limit the 764: 11447:Supersymmetric theory of stochastic dynamics 10473: 6537: 6511: 5393: 4538: 4526: 4449: 4437: 4117: 4104: 4098: 4086: 3963: 3877: 3812: 3772: 3687: 3664: 2916: 2856: 2769: 2746: 1318:) is not regarded as a formal expression in 10450:Mathematical methods of classical mechanics 8349: 5064:of degree 2 of the generalized velocities, 4618:All the individual generalized coordinates 2397:, for time-dependent cases they are called 1459:), i.e. each way the system can change its 1003:Learn how and when to remove these messages 10887: 10873: 10535: 10521: 7891: 2997:conjugate to the generalized coordinates: 771: 757: 10146: 10144: 9054: 9047: 8872: 8811: 8732: 8452: 8314: 8307: 8216: 8096: 7886: 7727: 7658: 7587: 7451: 7343: 7296: 7219: 7131:, they can be removed by introducing the 6684: 6434: 6389: 5644: 5519: 5367: 5317: 5283: 4952: 4948: 4875: 4766: 4592: 4475: 4371: 3976: 3959: 3939: 3935: 3922: 3815: 3799: 3690: 3677: 3620: 3570: 3471: 3426: 3219: 2929: 2912: 2892: 2888: 2878: 2798: 2772: 2759: 2726:of all possible generalized coordinates: 2666: 2575:or using the above formula – lead to the 2104: 2010:in the language of analytical mechanics: 1895: 1177:Learn how and when to remove this message 1159:Learn how and when to remove this message 1057:Learn how and when to remove this message 894:). Both formulations are equivalent by a 10371:Encyclopaedia of Physics (2nd Edition), 10278: 5397: 2960: 10446: 10344: 10285:Torby, Bruce (1984). "Energy Methods". 10181:The variational principles of mechanics 10177: 10171: 10154:The variational principles of mechanics 10150: 10135: 7444: 7414: 7207: 7197: 3615: 2991:); the generalized coordinates and the 2406: 2019: 1918: 1873: 1383:Generalized coordinates and constraints 11692: 10733:Atomic, molecular, and optical physics 10440: 10428:Relativity, Gravitation, and Cosmology 10141: 8540:is defined by analogy with mechanics: 7487: 4016:, and written as a function of these: 3993: 2354:and this relation holds for all times 1838:D'Alembert's principle of virtual work 1451:(for convenience labelled by an index 10868: 10516: 10467: 10453:(2nd ed.). Springer. Chapter 8. 10284: 10204: 7041: 6451:the above transformations are called 4671:), not simply as a parameter through 1299:are expressed as simple functions of 453:Newton's law of universal gravitation 2716:The Lagrangian formulation uses the 1212:. These quantities characterise the 1097:adding citations to reliable sources 1068: 1009: 968: 7015:for a time-independent Hamiltonian 4801:time derivative of any function of 1377: 1357:has a simple solution, whereas the 1280:, and all results are expressed in 965:Motivation for analytical mechanics 836:quantities of motion, particularly 434:Mechanics of planar particle motion 42: 13: 10896:Industrial and applied mathematics 9042: 9034: 8969: 8962: 8958: 8947: 8927: 8923: 8849: 8788: 8727: 8549: 8525: 8494: 8479: 8427: 8420: 8415: 8302: 8294: 8200: 8193: 8188: 8153: 8146: 8139: 8134: 8118: 8054: 8031: 8008: 7992: 7952: 7942: 7642: 7634: 7617: 7440: 7432: 7408: 7400: 7332: 7324: 7285: 7277: 6967: 6937:Hamilton's characteristic function 6900: 6893: 6888: 6796: 6789: 6784: 6734: 6727: 6722: 6656: 6595: 6423: 6415: 6378: 6370: 6270: 6266: 6153: 6149: 6036: 6032: 5919: 5915: 5720: 5688: 5557: 5441: 5009: 5001: 4926: 4918: 4838: 4834: 4466: 4458: 4355: 4347: 4328: 4320: 4298: 4290: 4271: 4263: 4229: 4221: 4204: 4196: 4179: 4171: 4154: 4146: 3971: 3764: 3754: 3744: 3707:("Momentum space" also refers to " 3656: 3561: 3553: 3460: 3452: 3415: 3407: 3144: 3136: 3102: 3094: 3063: 3055: 3026: 3018: 2924: 2832:(configuration) path or trajectory 2738: 2655: 2647: 2619: 2611: 2222: 2218: 2191: 2187: 2166: 2162: 2141: 2137: 2093: 2085: 2057: 2049: 1972: 1949: 1932: 14: 11716: 11126:Stochastic differential equations 6549:{\displaystyle \{Q_{i},P_{i}\}=1} 5796:to be transformed into a new set 4655:as a variable in addition to the 4553: 1397:, one customarily uses all three 984:This section has multiple issues. 11442:Supersymmetric quantum mechanics 10477:Geometric algebra for physicists 9930:Noether's theorem states that a 9870: 9859: 9824: 9816: 9765: 9754: 9731: 9720: 9669: 9658: 9650: 9624: 9616: 9608: 9582: 9546: 9527: 9511: 9488: 9469: 9453: 9416: 9356: 9339: 9294: 9277: 9210: 9202: 9179: 9171: 9163: 9137: 8709: 8682: 8622: 8396: 8083: 7808: 7774: 7756: 7712: 7336: 7305: 7289: 7258: 7189: 7181: 7165: 7162: 7152: 7013:additive separation of variables 6976: 6955: 6904: 6878: 6853: 6845: 6738: 6712: 6701:the generalized momenta become: 6665: 6638: 6427: 6400: 6397: 6382: 6355: 6352: 6301: 6293: 6248: 6240: 6217: 6209: 6184: 6176: 6131: 6123: 6100: 6092: 6067: 6059: 6014: 6006: 5983: 5975: 5950: 5942: 5897: 5889: 5866: 5858: 5617: 5614: 5604: 5498: 5495: 5485: 5308: 5305: 5295: 5276: 5173: 4862: 4854: 4753: 4233: 4208: 4183: 4158: 4130: 4122: 3901: 3884: 3787: 3779: 3668: 3604: 3601: 3464: 3437: 3434: 3419: 3392: 3389: 3320: 3317: 3307: 3291: 3288: 3278: 3261: 3253: 3035: 3032: 3008: 2860: 2811:{\displaystyle \mathbb {R} ^{N}} 2750: 2659: 2628: 2625: 2527: 2524: 2514: 2489: 2486: 2476: 2451: 2448: 2438: 2322: 2314: 2306: 2145: 2128:of the system, and the notation 2097: 2066: 2063: 1885: 1738: 1735: 1608: 1521: 1073: 1014: 973: 832:. Newtonian mechanics considers 738: 737: 724: 57: 11324:Stochastic variational calculus 11116:Ordinary differential equations 10854:Timeline of physics discoveries 10494: 10421: 10365: 10287:Advanced Dynamics for Engineers 9592:{\displaystyle G(\mathbf {v} )} 9147:{\displaystyle X(\mathbf {a} )} 8815: 8453: 7662: 7300: 6393: 5287: 3430: 2689:ordinary differential equations 1084:needs additional citations for 992:or discuss these issues on the 938:, and with some modifications, 11121:Partial differential equations 10994:Arbitrary-precision arithmetic 10351:. Vintage books. p. 474. 10338: 10319: 10303: 10241: 10198: 10054: 10051: 10045: 10027: 10021: 10015: 10006: 10003: 9991: 9973: 9961: 9955: 9883: 9874: 9863: 9837: 9828: 9820: 9758: 9724: 9654: 9612: 9586: 9578: 9542: 9521: 9515: 9484: 9463: 9457: 9431: 9410: 9379: 9360: 9352: 9349: 9343: 9317: 9298: 9290: 9287: 9281: 9255: 9242: 9206: 9167: 9141: 9133: 8988: 8965: 8719: 8705: 8692: 8678: 8632: 8554: 8533:{\displaystyle {\mathcal {H}}} 8406: 8392: 8172: 8149: 8093: 7957: 7216: 7177: 6986: 6972: 6959: 6951: 6863: 6841: 6822:is the Hamiltonian as before: 6675: 6661: 6648: 6634: 6603:{\displaystyle {\mathcal {S}}} 6311: 6289: 6258: 6236: 6227: 6205: 6194: 6172: 6141: 6119: 6110: 6088: 6077: 6055: 6024: 6002: 5993: 5971: 5960: 5938: 5907: 5885: 5876: 5854: 5753:, and is used for calculating 5728:{\displaystyle {\mathcal {C}}} 5696:{\displaystyle {\mathcal {C}}} 5629: 5600: 5510: 5481: 5314: 5291: 5280: 5222: 5199: 5196: 5177: 5166: 5119: 5107: 5081: 5078: 4949: 4872: 4850: 4763: 4749: 4589: 4583: 4568: 4559: 4541: 3914: 3911: 3905: 3894: 3888: 3880: 3791: 3775: 3332: 3303: 3271: 3249: 3216: 3177: 2870: 2864: 2539: 2510: 2501: 2472: 2463: 2434: 2341: 2332: 2326: 2318: 2260: 1983: 1926: 1819: 1747: 1573: 1528: 1: 11009:Interactive geometry software 10474:Doran, C; Lasenby, A (2003). 9913:about an axis defined by the 8748:The equations of motion are: 6610:) plus an arbitrary constant 6584:Hamilton's principal function 1334:determine the coordinates at 918:, a statement which connects 360:Koopman–von Neumann mechanics 6935:Another related function is 5765:Hamiltonian-Jacobi mechanics 1039:Knowledge's inclusion policy 803:. Analytical mechanics uses 428:Non-inertial reference frame 7: 11061:Computational number theory 11024:Numerical-analysis software 10818:Quantum information science 10178:Lanczos, Cornelius (1970). 10151:Lanczos, Cornelius (1970). 10078: 9079:in classical space and time 8275:of the Lagrangian density: 7849:is the acceleration of the 7499:Appell's equation of motion 7494:Appell's equation of motion 5060:If the kinetic energy is a 2374:is explicitly dependent on 2253:is a useful shorthand (see 1855:D'Alembert's principle is: 908:Appell's equation of motion 871:or dissipative forces like 355:Appell's equation of motion 225:Inertial frame of reference 10: 11721: 10649:Classical electromagnetism 8336:classical electromagnetism 7491: 7045: 5836:), in four possible ways: 5381:This is the basis for the 5374:{\displaystyle H=T+V=E\,.} 3997: 2964: 2842:) subject to the required 2410: 2264: 1841: 1386: 11659: 11621: 11605: 11567: 11519: 11467:Algebra of physical space 11332: 11290: 11084: 11046: 10934:Automated theorem proving 10902: 10826: 10763: 10691: 10607: 10579: 10551: 10217:10.1007/978-3-642-45943-6 7236:which leads to a set of 2 7036:Hamiltonian vector fields 7011:used to solve the HJE by 6453:canonical transformations 5771:Canonical transformations 5747:path integral formulation 5539:principle of least action 5394:Principle of least action 4895:canonical transformations 4512:of operators via Dirac's 1286:coordinate transformation 1216:of a body idealised as a 912:principle of least action 11260:Numerical linear algebra 10755:Condensed matter physics 10125:Non-autonomous mechanics 10105:Hamilton–Jacobi equation 9565:Galilean transformations 9261:{\displaystyle U(t_{0})} 9077:Symmetry transformations 8351:Hamiltonian field theory 6573:Hamilton–Jacobi equation 6331:With the restriction on 5711:) tracing out a path in 3711:-space"; the set of all 3488:which are now a set of 2 2577:Euler–Lagrange equations 2417:Euler–Lagrange equations 518:Rotating reference frame 350:Hamilton–Jacobi equation 10999:Finite element analysis 10949:Constraint satisfaction 10130:Udwadia–Kalaba equation 7893:Lagrangian field theory 7081:with conjugate momenta 5402:As the system evolves, 4902:constants of the motion 4790:exactly the same motion 2973:Legendre transformation 1495:generalized coordinates 1467:coordinates equals the 1435:generalized coordinates 1412:generalized coordinates 1405:, to refer to a body's 1389:Generalized coordinates 1282:generalized coordinates 896:Legendre transformation 865:generalized coordinates 459:Newton's laws of motion 319:Newton's laws of motion 11554:Mathematical economics 11228:Multivariable calculus 11111:Differential equations 10954:Constraint programming 10944:Computational geometry 10839:Nobel Prize in Physics 10701:Relativistic mechanics 10061: 9890: 9844: 9801: 9773: 9739: 9705: 9677: 9635: 9593: 9554: 9496: 9438: 9386: 9324: 9262: 9218: 9187: 9148: 9120:Translational symmetry 9062: 8998: 8888:variational derivative 8880: 8740: 8658: 8534: 8507: 8322: 8271:The Lagrangian is the 8224: 8104: 7887:Classical field theory 7840: 7735: 7699: 7595: 7459: 7351: 7227: 7002: 6926: 6809: 6749: 6692: 6604: 6550: 6442: 6322: 5729: 5697: 5652: 5527: 5419: 5406:traces a path through 5375: 5337:total conserved energy 5325: 5050:Such coordinates are " 5044: 4883: 4774: 4600: 4514:canonical quantization 4483: 4383: 3984: 3845:, a particular curve ( 3823: 3698: 3628: 3578: 3479: 3362: 3361:{\displaystyle \cdot } 3339: 3227: 2937: 2834:, i.e. one particular 2812: 2780: 2674: 2573:calculus of variations 2546: 2348: 2247: 2112: 1990: 1903: 1844:D'Alembert's principle 1829: 1592:generalized velocities 1580: 1363:mathematical structure 1344:differential equations 1268:variational principles 1108:"Analytical mechanics" 932:relativistic mechanics 900:Hamilton–Jacobi theory 486:Simple harmonic motion 399:Euler's laws of motion 193:D'Alembert's principle 76: 11507:Supersymmetry algebra 11492:Representation theory 11487:Renormalization group 11133:Differential geometry 11014:Optimization software 10986:Mathematical software 10844:Philosophy of physics 10095:Theoretical mechanics 10090:Hamiltonian mechanics 10062: 9891: 9845: 9802: 9774: 9740: 9706: 9678: 9636: 9594: 9555: 9497: 9439: 9397:Rotational invariance 9387: 9325: 9263: 9219: 9188: 9149: 9063: 8999: 8881: 8741: 8638: 8535: 8508: 8323: 8225: 8105: 7841: 7736: 7679: 7596: 7484:equations of motion. 7460: 7352: 7228: 7003: 6927: 6810: 6750: 6693: 6605: 6551: 6443: 6323: 5738:From this principle, 5730: 5698: 5653: 5528: 5401: 5376: 5326: 5045: 4884: 4775: 4601: 4484: 4396:of one of these, say 4384: 3985: 3824: 3699: 3629: 3579: 3480: 3363: 3340: 3228: 2967:Hamiltonian mechanics 2961:Hamiltonian mechanics 2938: 2813: 2781: 2675: 2547: 2366:holonomic constraints 2349: 2267:Holonomic constraints 2248: 2113: 1991: 1904: 1830: 1581: 1399:Cartesian coordinates 1307:concept: nowadays, a 888:Hamiltonian mechanics 797:theoretical mechanics 340:Hamiltonian mechanics 158:Statistical mechanics 77: 11700:Mathematical physics 11559:Mathematical finance 11544:Social choice theory 11459:Algebraic structures 11408:in quantum mechanics 11344:Analytical mechanics 11310:Stochastic processes 11282:Variational calculus 11094:Approximation theory 11019:Statistical software 10803:Mathematical physics 10447:Arnolʹd, VI (1989). 10345:Penrose, R. (2007). 10266:Analytical Mechanics 10136:References and notes 10110:Hamilton's principle 10085:Lagrangian mechanics 9949: 9855: 9812: 9791: 9750: 9716: 9695: 9646: 9604: 9572: 9507: 9449: 9404: 9335: 9273: 9236: 9198: 9159: 9127: 9019: 8893: 8752: 8544: 8520: 8379: 8344:quantum field theory 8279: 8243:summation convention 8114: 7937: 7751: 7611: 7508: 7476:degrees of freedom. 7379: 7251: 7142: 7024:symplectic manifolds 6945: 6829: 6769: 6708: 6621: 6590: 6508: 6346: 5843: 5715: 5683: 5661:where the departure 5549: 5436: 5389:directly obtains it. 5383:Schrödinger equation 5343: 5072: 5062:homogeneous function 4912: 4817: 4711: 4523: 4411: 4079: 3874: 3739: 3717:De Broglie relations 3651: 3595: 3521: 3383: 3374:Hamilton's equations 3352: 3243: 3004: 2853: 2828:set-builder notation 2793: 2733: 2586: 2428: 2413:Lagrangian mechanics 2407:Lagrangian mechanics 2302: 2257:for this notation). 2132: 2014: 1913: 1859: 1598: 1517: 1348:difference equations 1093:improve this article 944:quantum field theory 922:to their associated 880:Lagrangian mechanics 793:analytical mechanics 789:mathematical physics 563:Angular acceleration 555:Rotational frequency 335:Lagrangian mechanics 328:Analytical mechanics 84:Second law of motion 37: 11534:Operations research 11403:Perturbation theory 11201:Multilinear algebra 11172:Functional analysis 11029:Numerical libraries 10961:Computational logic 10778:Atmospheric physics 10617:Classical mechanics 10545:branches of physics 10348:The Road to Reality 10326:Classical Mechanics 10100:Classical mechanics 10075:will be conserved. 10060:{\displaystyle L=L} 9099:, i.e. symmetries. 8515:Hamiltonian density 8245:has been used. For 7726: 7488:Appellian mechanics 7056:cyclic coordinates 7028:symplectic topology 6464:generating function 5743:equations of motion 5596: 5477: 5429:of the Lagrangian: 5408:configuration space 4780:so each Lagrangian 4699:of any function of 3994:The Poisson bracket 2994:generalized momenta 2722:of the system, the 2719:configuration space 2683:which are a set of 1429:Difference between 1395:Newtonian mechanics 1324:elementary function 1230:gravitational field 1200:quantities such as 951:applied mathematics 884:configuration space 869:conservative forces 850:vectorial mechanics 830:Newtonian mechanics 819:equations of motion 801:classical mechanics 785:theoretical physics 415:Harmonic oscillator 393:Equations of motion 28:Classical mechanics 22:Part of a series on 11671:Mathematics portal 11568:Other applications 11292:Probability theory 11275:Validated numerics 11255:Numerical analysis 11148:Geometric analysis 11138:Differential forms 10971:Information theory 10834:History of physics 10120:Kinetics (physics) 10057: 9886: 9840: 9797: 9769: 9735: 9701: 9673: 9631: 9589: 9550: 9492: 9434: 9382: 9320: 9258: 9214: 9183: 9144: 9058: 8994: 8876: 8736: 8530: 8503: 8340:general relativity 8318: 8220: 8100: 7930:Lagrangian density 7836: 7731: 7710: 7591: 7455: 7347: 7223: 7048:Routhian mechanics 7042:Routhian mechanics 6998: 6922: 6805: 6745: 6688: 6600: 6546: 6470:th kind" or "type- 6438: 6318: 6316: 5759:general relativity 5725: 5693: 5648: 5568: 5523: 5449: 5420: 5371: 5321: 5040: 4879: 4770: 4596: 4498:Heisenberg picture 4479: 4379: 4377: 4259: 3980: 3819: 3694: 3624: 3574: 3475: 3372:, also leading to 3358: 3335: 3223: 2933: 2844:initial conditions 2808: 2776: 2670: 2542: 2344: 2243: 2108: 1998:generalized forces 1986: 1899: 1825: 1576: 1501:For a system with 1491:degrees of freedom 1359:three-body problem 1346:being replaced by 1340:computer modelling 1256:mechanical systems 1191:mathematical model 936:general relativity 904:Routhian mechanics 861:degrees of freedom 731:Physics portal 345:Routhian mechanics 220:Frame of reference 72: 11705:Dynamical systems 11687: 11686: 11521:Decision sciences 11515: 11514: 11497:Spacetime algebra 11189:Harmonic analysis 11155:Dynamical systems 11099:Clifford analysis 11076:Discrete geometry 11042: 11041: 10862: 10861: 10849:Physics education 10798:Materials science 10765:Interdisciplinary 10723:Quantum mechanics 10487:978-0-521-71595-9 10460:978-0-387-96890-2 10436:978-0-521-13138-4 10417:978-0-07-154382-8 10396:978-0-13-146100-0 10388:Quantum Mechanics 10358:978-0-679-77631-4 10274:978-0-521-57572-0 10226:978-3-540-02547-4 10191:978-0-486-65067-8 10042: 9988: 9925:Noether's theorem 9899: 9898: 9800:{\displaystyle T} 9704:{\displaystyle P} 9533: 9475: 9422: 8992: 8941: 8916: 8870: 8826: 8809: 8765: 8669: 8501: 8464: 8450: 8440: 8332:Newtonian gravity 8214: 8176: 7834: 7781: 7677: 7656: 7585: 7534: 7449: 7424: 7420: 7395: 7341: 7312: 7294: 7265: 7213: 7168: 6909: 6803: 6743: 6586:(also the action 6432: 6403: 6387: 6358: 6277: 6160: 6043: 5926: 5751:quantum mechanics 5703:, in other words 5620: 5501: 5387:quantum operators 5311: 5263: 5244: 5212: 5157: 5135: 5097: 5032: 5022: 4996: 4978: 4940: 4845: 4744: 4586: 4571: 4557: 4506:quantum operators 4502:quantum mechanics 4492:This equation in 4473: 4432: 4369: 4342: 4312: 4285: 4250: 4238: 4213: 4188: 4163: 3834:Cartesian product 3607: 3568: 3542: 3469: 3440: 3424: 3395: 3323: 3294: 3167: 3157: 3125: 3115: 3086: 3076: 3042: 3038: 2664: 2635: 2631: 2602: 2530: 2492: 2454: 2236: 2205: 2180: 2150: 2102: 2073: 2069: 2040: 1810: 1782: 1760: 1741: 1722: 1686: 1656: 1621: 1449:degree of freedom 1403:coordinate system 1278:coordinate system 1187: 1186: 1179: 1169: 1168: 1161: 1143: 1067: 1066: 1059: 1007: 940:quantum mechanics 920:conservation laws 916:Noether's theorem 781: 780: 528:Centrifugal force 523:Centripetal force 479:Euler's equations 464:Relative velocity 240:Moment of inertia 70: 44: 11712: 11472:Feynman integral 11455: 11454: 11415:Potential theory 11304:random variables 11194:Fourier analysis 11177:Operator algebra 11104:Clifford algebra 11056:Computer algebra 10982: 10981: 10889: 10882: 10875: 10866: 10865: 10788:Chemical physics 10728:Particle physics 10654:Classical optics 10537: 10530: 10523: 10514: 10513: 10508: 10498: 10492: 10491: 10471: 10465: 10464: 10444: 10438: 10425: 10419: 10409: 10398: 10385: 10376: 10369: 10363: 10362: 10342: 10336: 10323: 10317: 10307: 10301: 10300: 10282: 10276: 10263: 10248: 10245: 10239: 10238: 10202: 10196: 10195: 10175: 10169: 10168: 10148: 10066: 10064: 10063: 10058: 10044: 10043: 10035: 9990: 9989: 9981: 9936:conservation law 9895: 9893: 9892: 9887: 9873: 9862: 9849: 9847: 9846: 9841: 9827: 9819: 9806: 9804: 9803: 9798: 9778: 9776: 9775: 9770: 9768: 9757: 9744: 9742: 9741: 9736: 9734: 9723: 9710: 9708: 9707: 9702: 9682: 9680: 9679: 9674: 9672: 9661: 9653: 9640: 9638: 9637: 9632: 9627: 9619: 9611: 9598: 9596: 9595: 9590: 9585: 9559: 9557: 9556: 9551: 9549: 9535: 9534: 9526: 9514: 9501: 9499: 9498: 9493: 9491: 9477: 9476: 9468: 9456: 9443: 9441: 9440: 9435: 9424: 9423: 9415: 9391: 9389: 9388: 9383: 9378: 9377: 9359: 9342: 9329: 9327: 9326: 9321: 9316: 9315: 9297: 9280: 9267: 9265: 9264: 9259: 9254: 9253: 9229:Time translation 9223: 9221: 9220: 9215: 9213: 9205: 9192: 9190: 9189: 9184: 9182: 9174: 9166: 9153: 9151: 9150: 9145: 9140: 9102: 9101: 9067: 9065: 9064: 9059: 9046: 9045: 9039: 9038: 9037: 9003: 9001: 9000: 8995: 8993: 8991: 8987: 8986: 8977: 8976: 8957: 8955: 8954: 8942: 8940: 8939: 8938: 8922: 8917: 8915: 8914: 8913: 8897: 8885: 8883: 8882: 8877: 8871: 8869: 8868: 8867: 8854: 8853: 8852: 8842: 8834: 8833: 8828: 8827: 8819: 8810: 8808: 8807: 8806: 8793: 8792: 8791: 8781: 8773: 8772: 8767: 8766: 8758: 8745: 8743: 8742: 8737: 8731: 8730: 8712: 8704: 8703: 8685: 8677: 8676: 8671: 8670: 8662: 8657: 8652: 8625: 8611: 8610: 8598: 8597: 8579: 8578: 8566: 8565: 8553: 8552: 8539: 8537: 8536: 8531: 8529: 8528: 8512: 8510: 8509: 8504: 8502: 8500: 8492: 8491: 8490: 8477: 8472: 8471: 8466: 8465: 8457: 8451: 8449: 8448: 8447: 8442: 8441: 8433: 8425: 8424: 8423: 8413: 8399: 8391: 8390: 8327: 8325: 8324: 8319: 8306: 8305: 8299: 8298: 8297: 8229: 8227: 8226: 8221: 8215: 8213: 8212: 8211: 8198: 8197: 8196: 8186: 8181: 8177: 8175: 8171: 8170: 8161: 8160: 8144: 8143: 8142: 8132: 8126: 8125: 8109: 8107: 8106: 8101: 8086: 8072: 8071: 8062: 8061: 8049: 8048: 8039: 8038: 8020: 8019: 8004: 8003: 7982: 7981: 7969: 7968: 7956: 7955: 7946: 7945: 7869:, likewise each 7845: 7843: 7842: 7837: 7835: 7833: 7832: 7831: 7818: 7817: 7816: 7811: 7805: 7804: 7794: 7789: 7788: 7783: 7782: 7777: 7772: 7765: 7764: 7759: 7740: 7738: 7737: 7732: 7725: 7720: 7715: 7709: 7708: 7698: 7693: 7678: 7670: 7657: 7655: 7654: 7653: 7640: 7632: 7627: 7626: 7621: 7620: 7600: 7598: 7597: 7592: 7586: 7584: 7583: 7582: 7569: 7568: 7567: 7558: 7557: 7547: 7542: 7541: 7536: 7535: 7527: 7520: 7519: 7479:The coordinates 7464: 7462: 7461: 7456: 7450: 7448: 7447: 7438: 7430: 7425: 7423: 7422: 7421: 7413: 7406: 7398: 7396: 7394: 7383: 7356: 7354: 7353: 7348: 7342: 7340: 7339: 7330: 7322: 7314: 7313: 7308: 7303: 7295: 7293: 7292: 7283: 7275: 7267: 7266: 7261: 7256: 7232: 7230: 7229: 7224: 7215: 7214: 7206: 7200: 7192: 7184: 7170: 7169: 7161: 7155: 7007: 7005: 7004: 6999: 6979: 6971: 6970: 6958: 6931: 6929: 6928: 6923: 6921: 6917: 6910: 6908: 6907: 6898: 6897: 6896: 6886: 6881: 6856: 6848: 6814: 6812: 6811: 6806: 6804: 6802: 6794: 6793: 6792: 6782: 6754: 6752: 6751: 6746: 6744: 6742: 6741: 6732: 6731: 6730: 6720: 6715: 6697: 6695: 6694: 6689: 6668: 6660: 6659: 6641: 6633: 6632: 6609: 6607: 6606: 6601: 6599: 6598: 6555: 6553: 6552: 6547: 6536: 6535: 6523: 6522: 6455:, each function 6447: 6445: 6444: 6439: 6433: 6431: 6430: 6421: 6413: 6405: 6404: 6396: 6388: 6386: 6385: 6376: 6368: 6360: 6359: 6351: 6327: 6325: 6324: 6319: 6317: 6304: 6296: 6288: 6287: 6278: 6276: 6265: 6251: 6243: 6220: 6212: 6200: 6187: 6179: 6171: 6170: 6161: 6159: 6148: 6134: 6126: 6103: 6095: 6083: 6070: 6062: 6054: 6053: 6044: 6042: 6031: 6017: 6009: 5986: 5978: 5966: 5953: 5945: 5937: 5936: 5927: 5925: 5914: 5900: 5892: 5869: 5861: 5849: 5734: 5732: 5731: 5726: 5724: 5723: 5702: 5700: 5699: 5694: 5692: 5691: 5657: 5655: 5654: 5649: 5622: 5621: 5613: 5607: 5595: 5594: 5593: 5583: 5582: 5581: 5561: 5560: 5532: 5530: 5529: 5524: 5503: 5502: 5494: 5488: 5476: 5475: 5474: 5464: 5463: 5462: 5445: 5444: 5380: 5378: 5377: 5372: 5330: 5328: 5327: 5322: 5313: 5312: 5304: 5298: 5279: 5271: 5270: 5265: 5264: 5256: 5252: 5251: 5246: 5245: 5237: 5230: 5229: 5220: 5219: 5214: 5213: 5205: 5192: 5191: 5176: 5165: 5164: 5159: 5158: 5150: 5143: 5142: 5137: 5136: 5128: 5115: 5114: 5105: 5104: 5099: 5098: 5090: 5049: 5047: 5046: 5041: 5033: 5031: 5030: 5029: 5024: 5023: 5015: 5007: 4999: 4997: 4995: 4984: 4979: 4977: 4969: 4968: 4967: 4954: 4941: 4939: 4938: 4937: 4924: 4916: 4888: 4886: 4885: 4880: 4865: 4857: 4846: 4844: 4833: 4779: 4777: 4776: 4771: 4756: 4745: 4743: 4732: 4721: 4605: 4603: 4602: 4597: 4588: 4587: 4579: 4573: 4572: 4564: 4558: 4556: 4545: 4488: 4486: 4485: 4480: 4474: 4472: 4464: 4456: 4433: 4431: 4423: 4415: 4394:total derivative 4392:Calculating the 4388: 4386: 4385: 4380: 4378: 4370: 4368: 4367: 4366: 4353: 4345: 4343: 4341: 4340: 4339: 4326: 4318: 4313: 4311: 4310: 4309: 4296: 4288: 4286: 4284: 4283: 4282: 4269: 4261: 4258: 4243: 4239: 4237: 4236: 4227: 4219: 4214: 4212: 4211: 4202: 4194: 4189: 4187: 4186: 4177: 4169: 4164: 4162: 4161: 4152: 4144: 4135: 4134: 4133: 4125: 3989: 3987: 3986: 3981: 3975: 3974: 3962: 3934: 3933: 3925: 3904: 3887: 3828: 3826: 3825: 3820: 3811: 3810: 3802: 3790: 3782: 3768: 3767: 3758: 3757: 3748: 3747: 3703: 3701: 3700: 3695: 3686: 3685: 3680: 3671: 3660: 3659: 3633: 3631: 3630: 3625: 3619: 3618: 3609: 3608: 3600: 3583: 3581: 3580: 3575: 3569: 3567: 3559: 3551: 3543: 3541: 3533: 3525: 3484: 3482: 3481: 3476: 3470: 3468: 3467: 3458: 3450: 3442: 3441: 3433: 3425: 3423: 3422: 3413: 3405: 3397: 3396: 3388: 3367: 3365: 3364: 3359: 3344: 3342: 3341: 3336: 3325: 3324: 3316: 3310: 3296: 3295: 3287: 3281: 3264: 3256: 3232: 3230: 3229: 3224: 3215: 3214: 3202: 3201: 3189: 3188: 3173: 3169: 3168: 3166: 3165: 3164: 3159: 3158: 3150: 3142: 3134: 3126: 3124: 3123: 3122: 3117: 3116: 3108: 3100: 3092: 3087: 3085: 3084: 3083: 3078: 3077: 3069: 3061: 3053: 3043: 3041: 3040: 3039: 3031: 3024: 3016: 3011: 2942: 2940: 2939: 2934: 2928: 2927: 2915: 2887: 2886: 2881: 2863: 2826:space (see also 2817: 2815: 2814: 2809: 2807: 2806: 2801: 2785: 2783: 2782: 2777: 2768: 2767: 2762: 2753: 2742: 2741: 2679: 2677: 2676: 2671: 2665: 2663: 2662: 2653: 2645: 2640: 2636: 2634: 2633: 2632: 2624: 2617: 2609: 2603: 2601: 2590: 2569:potential energy 2551: 2549: 2548: 2543: 2532: 2531: 2523: 2517: 2494: 2493: 2485: 2479: 2456: 2455: 2447: 2441: 2390: 2379: 2373: 2363: 2357: 2353: 2351: 2350: 2345: 2325: 2317: 2309: 2297: 2293: 2287: 2252: 2250: 2249: 2244: 2242: 2238: 2237: 2235: 2234: 2233: 2217: 2206: 2204: 2203: 2202: 2186: 2181: 2179: 2178: 2177: 2161: 2151: 2149: 2148: 2136: 2117: 2115: 2114: 2109: 2103: 2101: 2100: 2091: 2083: 2078: 2074: 2072: 2071: 2070: 2062: 2055: 2047: 2041: 2039: 2028: 2023: 2022: 2005: 1995: 1993: 1992: 1987: 1982: 1981: 1976: 1975: 1959: 1958: 1953: 1952: 1942: 1941: 1936: 1935: 1922: 1921: 1908: 1906: 1905: 1900: 1888: 1877: 1876: 1834: 1832: 1831: 1826: 1818: 1817: 1812: 1811: 1803: 1790: 1789: 1784: 1783: 1775: 1768: 1767: 1762: 1761: 1753: 1743: 1742: 1734: 1728: 1724: 1723: 1721: 1713: 1712: 1711: 1698: 1687: 1685: 1677: 1676: 1675: 1662: 1657: 1655: 1647: 1646: 1645: 1632: 1622: 1620: 1612: 1611: 1602: 1585: 1583: 1582: 1577: 1572: 1571: 1553: 1552: 1540: 1539: 1524: 1480:× − (number of 1378:Intrinsic motion 1355:two-body problem 1182: 1175: 1164: 1157: 1153: 1150: 1144: 1142: 1101: 1077: 1069: 1062: 1055: 1051: 1048: 1042: 1018: 1017: 1010: 999: 977: 976: 969: 914:. One result is 815:potential energy 773: 766: 759: 746: 741: 740: 733: 729: 728: 634:Johann Bernoulli 629:Daniel Bernoulli 550:Tangential speed 454: 430: 405:Fictitious force 400: 252:Mechanical power 242: 183:Angular momentum 81: 79: 78: 73: 71: 69: 61: 60: 51: 46: 45: 19: 18: 11720: 11719: 11715: 11714: 11713: 11711: 11710: 11709: 11690: 11689: 11688: 11683: 11655: 11617: 11601: 11563: 11511: 11477:Poisson algebra 11453: 11335: 11328: 11286: 11182:Operator theory 11080: 11038: 11004:Tensor software 10980: 10929:Automata theory 10898: 10893: 10863: 10858: 10822: 10808:Medical physics 10759: 10718:Nuclear physics 10687: 10681:Non-equilibrium 10603: 10575: 10547: 10541: 10511: 10499: 10495: 10488: 10472: 10468: 10461: 10445: 10441: 10426: 10422: 10410: 10401: 10386: 10379: 10370: 10366: 10359: 10343: 10339: 10324: 10320: 10308: 10304: 10297: 10283: 10279: 10264: 10251: 10246: 10242: 10227: 10203: 10199: 10192: 10176: 10172: 10165: 10149: 10142: 10138: 10081: 10034: 10033: 9980: 9979: 9950: 9947: 9946: 9911:rotation matrix 9869: 9858: 9856: 9853: 9852: 9823: 9815: 9813: 9810: 9809: 9792: 9789: 9788: 9764: 9753: 9751: 9748: 9747: 9730: 9719: 9717: 9714: 9713: 9696: 9693: 9692: 9668: 9657: 9649: 9647: 9644: 9643: 9623: 9615: 9607: 9605: 9602: 9601: 9581: 9573: 9570: 9569: 9545: 9525: 9524: 9510: 9508: 9505: 9504: 9487: 9467: 9466: 9452: 9450: 9447: 9446: 9414: 9413: 9405: 9402: 9401: 9373: 9369: 9355: 9338: 9336: 9333: 9332: 9311: 9307: 9293: 9276: 9274: 9271: 9270: 9249: 9245: 9237: 9234: 9233: 9209: 9201: 9199: 9196: 9195: 9178: 9170: 9162: 9160: 9157: 9156: 9136: 9128: 9125: 9124: 9105:Transformation 9073: 9041: 9040: 9033: 9032: 9028: 9020: 9017: 9016: 8982: 8978: 8972: 8968: 8961: 8956: 8950: 8946: 8934: 8930: 8926: 8921: 8909: 8905: 8901: 8896: 8894: 8891: 8890: 8863: 8859: 8855: 8848: 8847: 8843: 8841: 8829: 8818: 8817: 8816: 8802: 8798: 8794: 8787: 8786: 8782: 8780: 8768: 8757: 8756: 8755: 8753: 8750: 8749: 8726: 8725: 8708: 8699: 8695: 8681: 8672: 8661: 8660: 8659: 8653: 8642: 8621: 8606: 8602: 8593: 8589: 8574: 8570: 8561: 8557: 8548: 8547: 8545: 8542: 8541: 8524: 8523: 8521: 8518: 8517: 8493: 8486: 8482: 8478: 8476: 8467: 8456: 8455: 8454: 8443: 8432: 8431: 8430: 8426: 8419: 8418: 8414: 8412: 8395: 8386: 8382: 8380: 8377: 8376: 8365: 8354: 8301: 8300: 8293: 8292: 8288: 8280: 8277: 8276: 8273:volume integral 8235: 8207: 8203: 8199: 8192: 8191: 8187: 8185: 8166: 8162: 8156: 8152: 8145: 8138: 8137: 8133: 8131: 8127: 8121: 8117: 8115: 8112: 8111: 8082: 8067: 8063: 8057: 8053: 8044: 8040: 8034: 8030: 8015: 8011: 7999: 7995: 7977: 7973: 7964: 7960: 7951: 7950: 7941: 7940: 7938: 7935: 7934: 7909: 7896: 7889: 7881: 7875: 7867: 7861: 7827: 7823: 7819: 7812: 7807: 7806: 7800: 7796: 7795: 7793: 7784: 7773: 7771: 7770: 7769: 7760: 7755: 7754: 7752: 7749: 7748: 7721: 7716: 7711: 7704: 7700: 7694: 7683: 7669: 7649: 7645: 7641: 7633: 7631: 7622: 7616: 7615: 7614: 7612: 7609: 7608: 7578: 7574: 7570: 7563: 7559: 7553: 7549: 7548: 7546: 7537: 7526: 7525: 7524: 7515: 7511: 7509: 7506: 7505: 7496: 7490: 7443: 7439: 7431: 7429: 7412: 7411: 7407: 7399: 7397: 7387: 7382: 7380: 7377: 7376: 7335: 7331: 7323: 7321: 7304: 7302: 7301: 7288: 7284: 7276: 7274: 7257: 7255: 7254: 7252: 7249: 7248: 7205: 7204: 7196: 7188: 7180: 7160: 7159: 7151: 7143: 7140: 7139: 7129: 7123: 7116: 7104: 7098: 7091: 7079: 7073: 7066: 7050: 7044: 7032:integral curves 6975: 6966: 6965: 6954: 6946: 6943: 6942: 6903: 6899: 6892: 6891: 6887: 6885: 6877: 6876: 6872: 6852: 6844: 6830: 6827: 6826: 6795: 6788: 6787: 6783: 6781: 6770: 6767: 6766: 6737: 6733: 6726: 6725: 6721: 6719: 6711: 6709: 6706: 6705: 6664: 6655: 6654: 6637: 6628: 6624: 6622: 6619: 6618: 6594: 6593: 6591: 6588: 6587: 6531: 6527: 6518: 6514: 6509: 6506: 6505: 6460: 6426: 6422: 6414: 6412: 6395: 6394: 6381: 6377: 6369: 6367: 6350: 6349: 6347: 6344: 6343: 6315: 6314: 6300: 6292: 6283: 6279: 6269: 6264: 6247: 6239: 6216: 6208: 6198: 6197: 6183: 6175: 6166: 6162: 6152: 6147: 6130: 6122: 6099: 6091: 6081: 6080: 6066: 6058: 6049: 6045: 6035: 6030: 6013: 6005: 5982: 5974: 5964: 5963: 5949: 5941: 5932: 5928: 5918: 5913: 5896: 5888: 5865: 5857: 5846: 5844: 5841: 5840: 5767: 5719: 5718: 5716: 5713: 5712: 5687: 5686: 5684: 5681: 5680: 5674: 5667: 5612: 5611: 5603: 5589: 5585: 5584: 5577: 5573: 5572: 5556: 5555: 5550: 5547: 5546: 5493: 5492: 5484: 5470: 5466: 5465: 5458: 5454: 5453: 5440: 5439: 5437: 5434: 5433: 5396: 5344: 5341: 5340: 5303: 5302: 5294: 5275: 5266: 5255: 5254: 5253: 5247: 5236: 5235: 5234: 5225: 5221: 5215: 5204: 5203: 5202: 5187: 5183: 5172: 5160: 5149: 5148: 5147: 5138: 5127: 5126: 5125: 5110: 5106: 5100: 5089: 5088: 5087: 5073: 5070: 5069: 5025: 5014: 5013: 5012: 5008: 5000: 4998: 4988: 4983: 4970: 4963: 4959: 4955: 4953: 4933: 4929: 4925: 4917: 4915: 4913: 4910: 4909: 4861: 4853: 4837: 4832: 4818: 4815: 4814: 4752: 4736: 4731: 4714: 4712: 4709: 4708: 4697:time derivative 4645: 4634: 4623: 4612: 4578: 4577: 4563: 4562: 4549: 4544: 4524: 4521: 4520: 4465: 4457: 4455: 4424: 4416: 4414: 4412: 4409: 4408: 4376: 4375: 4362: 4358: 4354: 4346: 4344: 4335: 4331: 4327: 4319: 4317: 4305: 4301: 4297: 4289: 4287: 4278: 4274: 4270: 4262: 4260: 4254: 4241: 4240: 4232: 4228: 4220: 4218: 4207: 4203: 4195: 4193: 4182: 4178: 4170: 4168: 4157: 4153: 4145: 4143: 4136: 4129: 4121: 4120: 4116: 4082: 4080: 4077: 4076: 4070:Poisson bracket 4002: 4000:Poisson bracket 3996: 3970: 3969: 3958: 3926: 3921: 3920: 3900: 3883: 3875: 3872: 3871: 3803: 3798: 3797: 3786: 3778: 3763: 3762: 3753: 3752: 3743: 3742: 3740: 3737: 3736: 3681: 3676: 3675: 3667: 3655: 3654: 3652: 3649: 3648: 3614: 3613: 3599: 3598: 3596: 3593: 3592: 3560: 3552: 3550: 3534: 3526: 3524: 3522: 3519: 3518: 3508: 3497: 3463: 3459: 3451: 3449: 3432: 3431: 3418: 3414: 3406: 3404: 3387: 3386: 3384: 3381: 3380: 3353: 3350: 3349: 3315: 3314: 3306: 3286: 3285: 3277: 3260: 3252: 3244: 3241: 3240: 3210: 3206: 3197: 3193: 3184: 3180: 3160: 3149: 3148: 3147: 3143: 3135: 3133: 3118: 3107: 3106: 3105: 3101: 3093: 3091: 3079: 3068: 3067: 3066: 3062: 3054: 3052: 3051: 3047: 3030: 3029: 3025: 3017: 3015: 3007: 3005: 3002: 3001: 2969: 2963: 2923: 2922: 2911: 2882: 2877: 2876: 2859: 2854: 2851: 2850: 2802: 2797: 2796: 2794: 2791: 2790: 2763: 2758: 2757: 2749: 2737: 2736: 2734: 2731: 2730: 2696: 2691:, one for each 2658: 2654: 2646: 2644: 2623: 2622: 2618: 2610: 2608: 2604: 2594: 2589: 2587: 2584: 2583: 2522: 2521: 2513: 2484: 2483: 2475: 2446: 2445: 2437: 2429: 2426: 2425: 2419: 2411:Main articles: 2409: 2381: 2375: 2369: 2359: 2355: 2321: 2313: 2305: 2303: 2300: 2299: 2295: 2289: 2283: 2281:position vector 2277: 2265:Main articles: 2263: 2255:matrix calculus 2229: 2225: 2221: 2216: 2198: 2194: 2190: 2185: 2173: 2169: 2165: 2160: 2159: 2155: 2144: 2140: 2135: 2133: 2130: 2129: 2096: 2092: 2084: 2082: 2061: 2060: 2056: 2048: 2046: 2042: 2032: 2027: 2018: 2017: 2015: 2012: 2011: 2001: 1977: 1971: 1970: 1969: 1954: 1948: 1947: 1946: 1937: 1931: 1930: 1929: 1917: 1916: 1914: 1911: 1910: 1884: 1872: 1871: 1860: 1857: 1856: 1846: 1840: 1813: 1802: 1801: 1800: 1785: 1774: 1773: 1772: 1763: 1752: 1751: 1750: 1733: 1732: 1714: 1707: 1703: 1699: 1697: 1678: 1671: 1667: 1663: 1661: 1648: 1641: 1637: 1633: 1631: 1630: 1626: 1613: 1607: 1603: 1601: 1599: 1596: 1595: 1588:time derivative 1567: 1563: 1548: 1544: 1535: 1531: 1520: 1518: 1515: 1514: 1499: 1493:) = (number of 1486: 1445: 1438: 1425:= 1, 2, 3...). 1419: 1391: 1385: 1380: 1183: 1172: 1171: 1170: 1165: 1154: 1148: 1145: 1102: 1100: 1090: 1078: 1063: 1052: 1046: 1043: 1029:Please help by 1028: 1019: 1015: 978: 974: 967: 953:, particularly 777: 736: 723: 722: 715: 714: 713: 588: 580: 579: 559: 513:Circular motion 507: 497: 496: 495: 452: 422: 419: 398: 377: 369: 368: 365: 364: 322: 312: 304: 303: 302: 261: 257:Mechanical work 250: 234: 172: 164: 163: 162: 117: 109: 86: 62: 56: 52: 50: 41: 40: 38: 35: 34: 17: 12: 11: 5: 11718: 11708: 11707: 11702: 11685: 11684: 11682: 11681: 11668: 11660: 11657: 11656: 11654: 11653: 11648: 11643: 11638: 11637: 11636: 11625: 11623: 11619: 11618: 11616: 11615: 11609: 11607: 11603: 11602: 11600: 11599: 11592: 11587: 11582: 11577: 11571: 11569: 11565: 11564: 11562: 11561: 11556: 11551: 11546: 11541: 11536: 11531: 11525: 11523: 11517: 11516: 11513: 11512: 11510: 11509: 11504: 11499: 11494: 11489: 11484: 11479: 11474: 11469: 11463: 11461: 11452: 11451: 11450: 11449: 11444: 11434: 11433: 11432: 11427: 11417: 11412: 11411: 11410: 11400: 11399: 11398: 11393: 11388: 11383: 11378: 11373: 11368: 11358: 11357: 11356: 11351: 11340: 11338: 11330: 11329: 11327: 11326: 11321: 11316: 11307: 11296: 11294: 11288: 11287: 11285: 11284: 11279: 11278: 11277: 11272: 11267: 11262: 11252: 11251: 11250: 11245: 11240: 11235: 11225: 11224: 11223: 11218: 11213: 11208: 11198: 11197: 11196: 11186: 11185: 11184: 11179: 11169: 11168: 11167: 11165:Control theory 11162: 11152: 11151: 11150: 11145: 11140: 11130: 11129: 11128: 11123: 11118: 11108: 11107: 11106: 11096: 11090: 11088: 11082: 11081: 11079: 11078: 11073: 11068: 11063: 11058: 11052: 11050: 11044: 11043: 11040: 11039: 11037: 11036: 11031: 11026: 11021: 11016: 11011: 11006: 11001: 10996: 10990: 10988: 10979: 10978: 10973: 10968: 10963: 10958: 10957: 10956: 10946: 10941: 10936: 10931: 10926: 10925: 10924: 10919: 10908: 10906: 10900: 10899: 10892: 10891: 10884: 10877: 10869: 10860: 10859: 10857: 10856: 10851: 10846: 10841: 10836: 10830: 10828: 10824: 10823: 10821: 10820: 10815: 10810: 10805: 10800: 10795: 10790: 10785: 10780: 10775: 10769: 10767: 10761: 10760: 10758: 10757: 10752: 10751: 10750: 10745: 10740: 10730: 10725: 10720: 10715: 10714: 10713: 10708: 10697: 10695: 10689: 10688: 10686: 10685: 10684: 10683: 10678: 10671:Thermodynamics 10668: 10667: 10666: 10661: 10651: 10646: 10641: 10640: 10639: 10634: 10629: 10624: 10613: 10611: 10605: 10604: 10602: 10601: 10600: 10599: 10589: 10583: 10581: 10577: 10576: 10574: 10573: 10572: 10571: 10561: 10555: 10553: 10549: 10548: 10540: 10539: 10532: 10525: 10517: 10510: 10509: 10493: 10486: 10466: 10459: 10439: 10420: 10399: 10377: 10364: 10357: 10337: 10318: 10302: 10295: 10277: 10249: 10240: 10225: 10197: 10190: 10170: 10163: 10139: 10137: 10134: 10133: 10132: 10127: 10122: 10117: 10112: 10107: 10102: 10097: 10092: 10087: 10080: 10077: 10056: 10053: 10050: 10047: 10041: 10038: 10032: 10029: 10026: 10023: 10020: 10017: 10014: 10011: 10008: 10005: 10002: 9999: 9996: 9993: 9987: 9984: 9978: 9975: 9972: 9969: 9966: 9963: 9960: 9957: 9954: 9928: 9927: 9897: 9896: 9885: 9882: 9879: 9876: 9872: 9868: 9865: 9861: 9850: 9839: 9836: 9833: 9830: 9826: 9822: 9818: 9807: 9796: 9786: 9780: 9779: 9767: 9763: 9760: 9756: 9745: 9733: 9729: 9726: 9722: 9711: 9700: 9690: 9684: 9683: 9671: 9667: 9664: 9660: 9656: 9652: 9641: 9630: 9626: 9622: 9618: 9614: 9610: 9599: 9588: 9584: 9580: 9577: 9567: 9561: 9560: 9548: 9544: 9541: 9538: 9532: 9529: 9523: 9520: 9517: 9513: 9502: 9490: 9486: 9483: 9480: 9474: 9471: 9465: 9462: 9459: 9455: 9444: 9433: 9430: 9427: 9421: 9418: 9412: 9409: 9399: 9393: 9392: 9381: 9376: 9372: 9368: 9365: 9362: 9358: 9354: 9351: 9348: 9345: 9341: 9330: 9319: 9314: 9310: 9306: 9303: 9300: 9296: 9292: 9289: 9286: 9283: 9279: 9268: 9257: 9252: 9248: 9244: 9241: 9231: 9225: 9224: 9212: 9208: 9204: 9193: 9181: 9177: 9173: 9169: 9165: 9154: 9143: 9139: 9135: 9132: 9122: 9116: 9115: 9112: 9109: 9106: 9081: 9080: 9072: 9069: 9057: 9053: 9050: 9044: 9036: 9031: 9027: 9024: 8990: 8985: 8981: 8975: 8971: 8967: 8964: 8960: 8953: 8949: 8945: 8937: 8933: 8929: 8925: 8920: 8912: 8908: 8904: 8900: 8875: 8866: 8862: 8858: 8851: 8846: 8840: 8837: 8832: 8825: 8822: 8814: 8805: 8801: 8797: 8790: 8785: 8779: 8776: 8771: 8764: 8761: 8735: 8729: 8724: 8721: 8718: 8715: 8711: 8707: 8702: 8698: 8694: 8691: 8688: 8684: 8680: 8675: 8668: 8665: 8656: 8651: 8648: 8645: 8641: 8637: 8634: 8631: 8628: 8624: 8620: 8617: 8614: 8609: 8605: 8601: 8596: 8592: 8588: 8585: 8582: 8577: 8573: 8569: 8564: 8560: 8556: 8551: 8527: 8499: 8496: 8489: 8485: 8481: 8475: 8470: 8463: 8460: 8446: 8439: 8436: 8429: 8422: 8417: 8411: 8408: 8405: 8402: 8398: 8394: 8389: 8385: 8363: 8360:scalar fields 8353: 8348: 8317: 8313: 8310: 8304: 8296: 8291: 8287: 8284: 8233: 8219: 8210: 8206: 8202: 8195: 8190: 8184: 8180: 8174: 8169: 8165: 8159: 8155: 8151: 8148: 8141: 8136: 8130: 8124: 8120: 8099: 8095: 8092: 8089: 8085: 8081: 8078: 8075: 8070: 8066: 8060: 8056: 8052: 8047: 8043: 8037: 8033: 8029: 8026: 8023: 8018: 8014: 8010: 8007: 8002: 7998: 7994: 7991: 7988: 7985: 7980: 7976: 7972: 7967: 7963: 7959: 7954: 7949: 7944: 7907: 7895: 7890: 7888: 7885: 7879: 7873: 7865: 7857: 7847: 7846: 7830: 7826: 7822: 7815: 7810: 7803: 7799: 7792: 7787: 7780: 7776: 7768: 7763: 7758: 7742: 7741: 7730: 7724: 7719: 7714: 7707: 7703: 7697: 7692: 7689: 7686: 7682: 7676: 7673: 7668: 7665: 7661: 7652: 7648: 7644: 7639: 7636: 7630: 7625: 7619: 7602: 7601: 7590: 7581: 7577: 7573: 7566: 7562: 7556: 7552: 7545: 7540: 7533: 7530: 7523: 7518: 7514: 7492:Main article: 7489: 7486: 7466: 7465: 7454: 7446: 7442: 7437: 7434: 7428: 7419: 7416: 7410: 7405: 7402: 7393: 7390: 7386: 7358: 7357: 7346: 7338: 7334: 7329: 7326: 7320: 7317: 7311: 7307: 7299: 7291: 7287: 7282: 7279: 7273: 7270: 7264: 7260: 7234: 7233: 7222: 7218: 7212: 7209: 7203: 7199: 7195: 7191: 7187: 7183: 7179: 7176: 7173: 7167: 7164: 7158: 7154: 7150: 7147: 7127: 7121: 7114: 7102: 7096: 7089: 7077: 7071: 7064: 7046:Main article: 7043: 7040: 7009: 7008: 6997: 6994: 6991: 6988: 6985: 6982: 6978: 6974: 6969: 6964: 6961: 6957: 6953: 6950: 6933: 6932: 6920: 6916: 6913: 6906: 6902: 6895: 6890: 6884: 6880: 6875: 6871: 6868: 6865: 6862: 6859: 6855: 6851: 6847: 6843: 6840: 6837: 6834: 6816: 6815: 6801: 6798: 6791: 6786: 6780: 6777: 6774: 6756: 6755: 6740: 6736: 6729: 6724: 6718: 6714: 6699: 6698: 6687: 6683: 6680: 6677: 6674: 6671: 6667: 6663: 6658: 6653: 6650: 6647: 6644: 6640: 6636: 6631: 6627: 6597: 6576: 6575: 6557: 6556: 6545: 6542: 6539: 6534: 6530: 6526: 6521: 6517: 6513: 6477:The choice of 6458: 6449: 6448: 6437: 6429: 6425: 6420: 6417: 6411: 6408: 6402: 6399: 6392: 6384: 6380: 6375: 6372: 6366: 6363: 6357: 6354: 6329: 6328: 6313: 6310: 6307: 6303: 6299: 6295: 6291: 6286: 6282: 6275: 6272: 6268: 6263: 6260: 6257: 6254: 6250: 6246: 6242: 6238: 6235: 6232: 6229: 6226: 6223: 6219: 6215: 6211: 6207: 6204: 6201: 6199: 6196: 6193: 6190: 6186: 6182: 6178: 6174: 6169: 6165: 6158: 6155: 6151: 6146: 6143: 6140: 6137: 6133: 6129: 6125: 6121: 6118: 6115: 6112: 6109: 6106: 6102: 6098: 6094: 6090: 6087: 6084: 6082: 6079: 6076: 6073: 6069: 6065: 6061: 6057: 6052: 6048: 6041: 6038: 6034: 6029: 6026: 6023: 6020: 6016: 6012: 6008: 6004: 6001: 5998: 5995: 5992: 5989: 5985: 5981: 5977: 5973: 5970: 5967: 5965: 5962: 5959: 5956: 5952: 5948: 5944: 5940: 5935: 5931: 5924: 5921: 5917: 5912: 5909: 5906: 5903: 5899: 5895: 5891: 5887: 5884: 5881: 5878: 5875: 5872: 5868: 5864: 5860: 5856: 5853: 5850: 5848: 5774: 5773: 5766: 5763: 5722: 5690: 5677:time evolution 5672: 5665: 5659: 5658: 5647: 5643: 5640: 5637: 5634: 5631: 5628: 5625: 5619: 5616: 5610: 5606: 5602: 5599: 5592: 5588: 5580: 5576: 5571: 5567: 5564: 5559: 5554: 5534: 5533: 5522: 5518: 5515: 5512: 5509: 5506: 5500: 5497: 5491: 5487: 5483: 5480: 5473: 5469: 5461: 5457: 5452: 5448: 5443: 5395: 5392: 5391: 5390: 5370: 5366: 5363: 5360: 5357: 5354: 5351: 5348: 5320: 5316: 5310: 5307: 5301: 5297: 5293: 5290: 5286: 5282: 5278: 5274: 5269: 5262: 5259: 5250: 5243: 5240: 5233: 5228: 5224: 5218: 5211: 5208: 5201: 5198: 5195: 5190: 5186: 5182: 5179: 5175: 5171: 5168: 5163: 5156: 5153: 5146: 5141: 5134: 5131: 5124: 5121: 5118: 5113: 5109: 5103: 5096: 5093: 5086: 5083: 5080: 5077: 5058: 5055: 5039: 5036: 5028: 5021: 5018: 5011: 5006: 5003: 4994: 4991: 4987: 4982: 4976: 4973: 4966: 4962: 4958: 4951: 4947: 4944: 4936: 4932: 4928: 4923: 4920: 4898: 4878: 4874: 4871: 4868: 4864: 4860: 4856: 4852: 4849: 4843: 4840: 4836: 4831: 4828: 4825: 4822: 4793: 4769: 4765: 4762: 4759: 4755: 4751: 4748: 4742: 4739: 4735: 4730: 4727: 4724: 4720: 4717: 4688: 4643: 4640:) and momenta 4632: 4629:), velocities 4621: 4611: 4608: 4607: 4606: 4595: 4591: 4585: 4582: 4576: 4570: 4567: 4561: 4555: 4552: 4548: 4543: 4540: 4537: 4534: 4531: 4528: 4490: 4489: 4478: 4471: 4468: 4463: 4460: 4454: 4451: 4448: 4445: 4442: 4439: 4436: 4430: 4427: 4422: 4419: 4390: 4389: 4374: 4365: 4361: 4357: 4352: 4349: 4338: 4334: 4330: 4325: 4322: 4316: 4308: 4304: 4300: 4295: 4292: 4281: 4277: 4273: 4268: 4265: 4257: 4253: 4249: 4246: 4244: 4242: 4235: 4231: 4226: 4223: 4217: 4210: 4206: 4201: 4198: 4192: 4185: 4181: 4176: 4173: 4167: 4160: 4156: 4151: 4148: 4142: 4139: 4137: 4132: 4128: 4124: 4119: 4115: 4112: 4109: 4106: 4103: 4100: 4097: 4094: 4091: 4088: 4085: 4084: 3998:Main article: 3995: 3992: 3991: 3990: 3979: 3973: 3968: 3965: 3961: 3957: 3954: 3951: 3948: 3945: 3942: 3938: 3932: 3929: 3924: 3919: 3916: 3913: 3910: 3907: 3903: 3899: 3896: 3893: 3890: 3886: 3882: 3879: 3864:phase portrait 3830: 3829: 3818: 3814: 3809: 3806: 3801: 3796: 3793: 3789: 3785: 3781: 3777: 3774: 3771: 3766: 3761: 3756: 3751: 3746: 3705: 3704: 3693: 3689: 3684: 3679: 3674: 3670: 3666: 3663: 3658: 3641:momentum space 3635: 3634: 3623: 3617: 3612: 3606: 3603: 3585: 3584: 3573: 3566: 3563: 3558: 3555: 3549: 3546: 3540: 3537: 3532: 3529: 3506: 3495: 3486: 3485: 3474: 3466: 3462: 3457: 3454: 3448: 3445: 3439: 3436: 3429: 3421: 3417: 3412: 3409: 3403: 3400: 3394: 3391: 3357: 3346: 3345: 3334: 3331: 3328: 3322: 3319: 3313: 3309: 3305: 3302: 3299: 3293: 3290: 3284: 3280: 3276: 3273: 3270: 3267: 3263: 3259: 3255: 3251: 3248: 3234: 3233: 3222: 3218: 3213: 3209: 3205: 3200: 3196: 3192: 3187: 3183: 3179: 3176: 3172: 3163: 3156: 3153: 3146: 3141: 3138: 3132: 3129: 3121: 3114: 3111: 3104: 3099: 3096: 3090: 3082: 3075: 3072: 3065: 3060: 3057: 3050: 3046: 3037: 3034: 3028: 3023: 3020: 3014: 3010: 2965:Main article: 2962: 2959: 2955:tangent bundle 2944: 2943: 2932: 2926: 2921: 2918: 2914: 2910: 2907: 2904: 2901: 2898: 2895: 2891: 2885: 2880: 2875: 2872: 2869: 2866: 2862: 2858: 2805: 2800: 2787: 2786: 2775: 2771: 2766: 2761: 2756: 2752: 2748: 2745: 2740: 2711:kinetic energy 2694: 2681: 2680: 2669: 2661: 2657: 2652: 2649: 2643: 2639: 2630: 2627: 2621: 2616: 2613: 2607: 2600: 2597: 2593: 2561:kinetic energy 2553: 2552: 2541: 2538: 2535: 2529: 2526: 2520: 2516: 2512: 2509: 2506: 2503: 2500: 2497: 2491: 2488: 2482: 2478: 2474: 2471: 2468: 2465: 2462: 2459: 2453: 2450: 2444: 2440: 2436: 2433: 2408: 2405: 2343: 2340: 2337: 2334: 2331: 2328: 2324: 2320: 2316: 2312: 2308: 2262: 2259: 2241: 2232: 2228: 2224: 2220: 2215: 2212: 2209: 2201: 2197: 2193: 2189: 2184: 2176: 2172: 2168: 2164: 2158: 2154: 2147: 2143: 2139: 2126:kinetic energy 2107: 2099: 2095: 2090: 2087: 2081: 2077: 2068: 2065: 2059: 2054: 2051: 2045: 2038: 2035: 2031: 2026: 2021: 1985: 1980: 1974: 1968: 1965: 1962: 1957: 1951: 1945: 1940: 1934: 1928: 1925: 1920: 1898: 1894: 1891: 1887: 1883: 1880: 1875: 1870: 1867: 1864: 1842:Main article: 1839: 1836: 1824: 1821: 1816: 1809: 1806: 1799: 1796: 1793: 1788: 1781: 1778: 1771: 1766: 1759: 1756: 1749: 1746: 1740: 1737: 1731: 1727: 1720: 1717: 1710: 1706: 1702: 1696: 1693: 1690: 1684: 1681: 1674: 1670: 1666: 1660: 1654: 1651: 1644: 1640: 1636: 1629: 1625: 1619: 1616: 1610: 1606: 1575: 1570: 1566: 1562: 1559: 1556: 1551: 1547: 1543: 1538: 1534: 1530: 1527: 1523: 1487: 1478: 1443: 1437: 1427: 1417: 1401:, or other 3D 1387:Main article: 1384: 1381: 1379: 1376: 1185: 1184: 1167: 1166: 1081: 1079: 1072: 1065: 1064: 1022: 1020: 1013: 1008: 982: 981: 979: 972: 966: 963: 811:kinetic energy 779: 778: 776: 775: 768: 761: 753: 750: 749: 748: 747: 734: 717: 716: 712: 711: 706: 701: 696: 691: 686: 681: 676: 671: 666: 661: 656: 651: 646: 641: 636: 631: 626: 621: 616: 611: 606: 601: 596: 590: 589: 586: 585: 582: 581: 578: 577: 558: 557: 552: 547: 542: 540:Coriolis force 537: 536: 535: 525: 520: 515: 509: 508: 503: 502: 499: 498: 494: 493: 488: 483: 482: 481: 476: 466: 461: 456: 449: 438: 437: 436: 431: 418: 417: 412: 407: 402: 395: 390: 385: 379: 378: 375: 374: 371: 370: 367: 366: 363: 362: 357: 352: 347: 342: 337: 331: 325: 323: 316: 313: 310: 309: 306: 305: 301: 300: 295: 290: 285: 280: 275: 270: 265: 259: 254: 248: 243: 232: 227: 222: 217: 212: 211: 210: 205: 195: 190: 185: 180: 174: 173: 170: 169: 166: 165: 161: 160: 155: 150: 145: 140: 135: 130: 125: 119: 118: 115: 114: 111: 110: 108: 107: 102: 97: 91: 88: 87: 82: 68: 65: 59: 55: 49: 31: 30: 24: 23: 15: 9: 6: 4: 3: 2: 11717: 11706: 11703: 11701: 11698: 11697: 11695: 11680: 11676: 11672: 11669: 11667: 11666: 11662: 11661: 11658: 11652: 11649: 11647: 11644: 11642: 11639: 11635: 11632: 11631: 11630: 11627: 11626: 11624: 11622:Organizations 11620: 11614: 11611: 11610: 11608: 11604: 11597: 11593: 11591: 11588: 11586: 11583: 11581: 11578: 11576: 11573: 11572: 11570: 11566: 11560: 11557: 11555: 11552: 11550: 11547: 11545: 11542: 11540: 11537: 11535: 11532: 11530: 11527: 11526: 11524: 11522: 11518: 11508: 11505: 11503: 11500: 11498: 11495: 11493: 11490: 11488: 11485: 11483: 11482:Quantum group 11480: 11478: 11475: 11473: 11470: 11468: 11465: 11464: 11462: 11460: 11456: 11448: 11445: 11443: 11440: 11439: 11438: 11437:Supersymmetry 11435: 11431: 11428: 11426: 11423: 11422: 11421: 11420:String theory 11418: 11416: 11413: 11409: 11406: 11405: 11404: 11401: 11397: 11394: 11392: 11389: 11387: 11384: 11382: 11379: 11377: 11374: 11372: 11369: 11367: 11364: 11363: 11362: 11359: 11355: 11352: 11350: 11347: 11346: 11345: 11342: 11341: 11339: 11337: 11331: 11325: 11322: 11320: 11319:Path integral 11317: 11315: 11311: 11308: 11305: 11301: 11300:Distributions 11298: 11297: 11295: 11293: 11289: 11283: 11280: 11276: 11273: 11271: 11268: 11266: 11263: 11261: 11258: 11257: 11256: 11253: 11249: 11246: 11244: 11241: 11239: 11236: 11234: 11231: 11230: 11229: 11226: 11222: 11219: 11217: 11214: 11212: 11209: 11207: 11204: 11203: 11202: 11199: 11195: 11192: 11191: 11190: 11187: 11183: 11180: 11178: 11175: 11174: 11173: 11170: 11166: 11163: 11161: 11158: 11157: 11156: 11153: 11149: 11146: 11144: 11141: 11139: 11136: 11135: 11134: 11131: 11127: 11124: 11122: 11119: 11117: 11114: 11113: 11112: 11109: 11105: 11102: 11101: 11100: 11097: 11095: 11092: 11091: 11089: 11087: 11083: 11077: 11074: 11072: 11069: 11067: 11066:Combinatorics 11064: 11062: 11059: 11057: 11054: 11053: 11051: 11049: 11045: 11035: 11032: 11030: 11027: 11025: 11022: 11020: 11017: 11015: 11012: 11010: 11007: 11005: 11002: 11000: 10997: 10995: 10992: 10991: 10989: 10987: 10983: 10977: 10974: 10972: 10969: 10967: 10964: 10962: 10959: 10955: 10952: 10951: 10950: 10947: 10945: 10942: 10940: 10939:Coding theory 10937: 10935: 10932: 10930: 10927: 10923: 10920: 10918: 10915: 10914: 10913: 10910: 10909: 10907: 10905: 10904:Computational 10901: 10897: 10890: 10885: 10883: 10878: 10876: 10871: 10870: 10867: 10855: 10852: 10850: 10847: 10845: 10842: 10840: 10837: 10835: 10832: 10831: 10829: 10825: 10819: 10816: 10814: 10813:Ocean physics 10811: 10809: 10806: 10804: 10801: 10799: 10796: 10794: 10791: 10789: 10786: 10784: 10781: 10779: 10776: 10774: 10771: 10770: 10768: 10766: 10762: 10756: 10753: 10749: 10748:Modern optics 10746: 10744: 10741: 10739: 10736: 10735: 10734: 10731: 10729: 10726: 10724: 10721: 10719: 10716: 10712: 10709: 10707: 10704: 10703: 10702: 10699: 10698: 10696: 10694: 10690: 10682: 10679: 10677: 10674: 10673: 10672: 10669: 10665: 10662: 10660: 10657: 10656: 10655: 10652: 10650: 10647: 10645: 10642: 10638: 10635: 10633: 10630: 10628: 10625: 10623: 10620: 10619: 10618: 10615: 10614: 10612: 10610: 10606: 10598: 10597:Computational 10595: 10594: 10593: 10590: 10588: 10585: 10584: 10582: 10578: 10570: 10567: 10566: 10565: 10562: 10560: 10557: 10556: 10554: 10550: 10546: 10538: 10533: 10531: 10526: 10524: 10519: 10518: 10515: 10507: 10506:0-7167-0344-0 10503: 10497: 10489: 10483: 10479: 10478: 10470: 10462: 10456: 10452: 10451: 10443: 10437: 10433: 10429: 10424: 10418: 10414: 10408: 10406: 10404: 10397: 10393: 10389: 10384: 10382: 10374: 10368: 10360: 10354: 10350: 10349: 10341: 10335: 10334:0-07-084018-0 10331: 10327: 10322: 10316: 10315:0-07-051400-3 10312: 10306: 10298: 10296:0-03-063366-4 10292: 10288: 10281: 10275: 10271: 10267: 10262: 10260: 10258: 10256: 10254: 10244: 10236: 10232: 10228: 10222: 10218: 10214: 10210: 10209: 10201: 10193: 10187: 10183: 10182: 10174: 10166: 10164:0-486-65067-7 10160: 10156: 10155: 10147: 10145: 10140: 10131: 10128: 10126: 10123: 10121: 10118: 10116: 10113: 10111: 10108: 10106: 10103: 10101: 10098: 10096: 10093: 10091: 10088: 10086: 10083: 10082: 10076: 10074: 10070: 10048: 10039: 10036: 10030: 10024: 10018: 10012: 10009: 10000: 9997: 9994: 9985: 9982: 9976: 9970: 9967: 9964: 9958: 9952: 9944: 9941: 9937: 9933: 9926: 9923: 9922: 9921: 9920:and angle θ. 9919: 9916: 9912: 9908: 9904: 9880: 9877: 9866: 9851: 9834: 9831: 9808: 9794: 9787: 9785: 9782: 9781: 9761: 9746: 9727: 9712: 9698: 9691: 9689: 9686: 9685: 9665: 9662: 9642: 9628: 9620: 9600: 9575: 9568: 9566: 9563: 9562: 9539: 9536: 9518: 9503: 9481: 9478: 9460: 9445: 9428: 9425: 9407: 9400: 9398: 9395: 9394: 9374: 9370: 9366: 9363: 9346: 9331: 9312: 9308: 9304: 9301: 9284: 9269: 9250: 9246: 9239: 9232: 9230: 9227: 9226: 9194: 9175: 9155: 9130: 9123: 9121: 9118: 9117: 9113: 9110: 9107: 9104: 9103: 9100: 9098: 9094: 9090: 9086: 9078: 9075: 9074: 9068: 9055: 9051: 9048: 9029: 9025: 9022: 9013: 9011: 9007: 8983: 8979: 8973: 8951: 8943: 8935: 8931: 8918: 8910: 8906: 8902: 8898: 8889: 8873: 8864: 8860: 8856: 8844: 8838: 8835: 8830: 8823: 8820: 8812: 8803: 8799: 8795: 8783: 8777: 8774: 8769: 8762: 8759: 8746: 8733: 8722: 8716: 8713: 8700: 8696: 8689: 8686: 8673: 8666: 8663: 8654: 8649: 8646: 8643: 8639: 8635: 8629: 8626: 8618: 8615: 8612: 8607: 8603: 8599: 8594: 8590: 8586: 8583: 8580: 8575: 8571: 8567: 8562: 8558: 8516: 8497: 8487: 8483: 8473: 8468: 8461: 8458: 8444: 8437: 8434: 8409: 8403: 8400: 8387: 8383: 8374: 8370: 8366: 8359: 8352: 8347: 8345: 8341: 8337: 8333: 8328: 8315: 8311: 8308: 8289: 8285: 8282: 8274: 8269: 8267: 8266:spinor fields 8263: 8262:tensor fields 8259: 8258:vector fields 8254: 8252: 8248: 8244: 8240: 8236: 8217: 8208: 8204: 8182: 8178: 8167: 8163: 8157: 8128: 8122: 8097: 8090: 8087: 8079: 8076: 8073: 8068: 8064: 8058: 8050: 8045: 8041: 8035: 8027: 8024: 8021: 8016: 8012: 8005: 8000: 7996: 7989: 7986: 7983: 7978: 7974: 7970: 7965: 7961: 7947: 7932: 7931: 7926: 7922: 7918: 7914: 7910: 7904: 7903:scalar fields 7901: 7894: 7884: 7882: 7872: 7868: 7860: 7856: 7852: 7828: 7824: 7820: 7813: 7801: 7797: 7790: 7785: 7778: 7766: 7761: 7747: 7746: 7745: 7728: 7722: 7717: 7705: 7701: 7695: 7690: 7687: 7684: 7680: 7674: 7671: 7666: 7663: 7659: 7650: 7646: 7637: 7628: 7623: 7607: 7606: 7605: 7588: 7579: 7575: 7571: 7564: 7560: 7554: 7550: 7543: 7538: 7531: 7528: 7521: 7516: 7512: 7504: 7503: 7502: 7500: 7495: 7485: 7482: 7477: 7475: 7471: 7452: 7435: 7426: 7417: 7403: 7391: 7388: 7384: 7375: 7374: 7373: 7371: 7367: 7363: 7344: 7327: 7318: 7315: 7309: 7297: 7280: 7271: 7268: 7262: 7247: 7246: 7245: 7243: 7239: 7220: 7210: 7201: 7193: 7185: 7174: 7171: 7156: 7148: 7145: 7138: 7137: 7136: 7134: 7130: 7120: 7113: 7109: 7105: 7095: 7088: 7084: 7080: 7070: 7063: 7059: 7055: 7049: 7039: 7037: 7033: 7029: 7025: 7020: 7018: 7014: 6995: 6992: 6989: 6983: 6980: 6962: 6948: 6941: 6940: 6939: 6938: 6918: 6914: 6911: 6882: 6873: 6869: 6866: 6860: 6857: 6849: 6838: 6835: 6832: 6825: 6824: 6823: 6821: 6799: 6778: 6775: 6772: 6765: 6764: 6763: 6761: 6716: 6704: 6703: 6702: 6685: 6681: 6678: 6672: 6669: 6651: 6645: 6642: 6629: 6625: 6617: 6616: 6615: 6613: 6585: 6581: 6574: 6570: 6569: 6568: 6566: 6562: 6543: 6540: 6532: 6528: 6524: 6519: 6515: 6504: 6503: 6502: 6500: 6496: 6492: 6488: 6484: 6480: 6475: 6473: 6469: 6465: 6461: 6454: 6435: 6418: 6409: 6406: 6390: 6373: 6364: 6361: 6342: 6341: 6340: 6338: 6334: 6308: 6305: 6297: 6284: 6280: 6273: 6261: 6255: 6252: 6244: 6233: 6230: 6224: 6221: 6213: 6202: 6191: 6188: 6180: 6167: 6163: 6156: 6144: 6138: 6135: 6127: 6116: 6113: 6107: 6104: 6096: 6085: 6074: 6071: 6063: 6050: 6046: 6039: 6027: 6021: 6018: 6010: 5999: 5996: 5990: 5987: 5979: 5968: 5957: 5954: 5946: 5933: 5929: 5922: 5910: 5904: 5901: 5893: 5882: 5879: 5873: 5870: 5862: 5851: 5839: 5838: 5837: 5835: 5831: 5827: 5823: 5819: 5815: 5811: 5807: 5803: 5799: 5795: 5791: 5787: 5783: 5779: 5772: 5769: 5768: 5762: 5760: 5756: 5752: 5748: 5744: 5741: 5736: 5710: 5706: 5678: 5671: 5664: 5645: 5641: 5638: 5635: 5632: 5626: 5623: 5608: 5597: 5590: 5586: 5578: 5574: 5569: 5565: 5562: 5552: 5545: 5544: 5543: 5541: 5540: 5520: 5516: 5513: 5507: 5504: 5489: 5478: 5471: 5467: 5459: 5455: 5450: 5446: 5432: 5431: 5430: 5428: 5424: 5417: 5413: 5409: 5405: 5400: 5388: 5384: 5368: 5364: 5361: 5358: 5355: 5352: 5349: 5346: 5338: 5334: 5318: 5299: 5288: 5284: 5272: 5267: 5260: 5257: 5248: 5241: 5238: 5231: 5226: 5216: 5209: 5206: 5193: 5188: 5184: 5180: 5169: 5161: 5154: 5151: 5144: 5139: 5132: 5129: 5122: 5116: 5111: 5101: 5094: 5091: 5084: 5075: 5067: 5063: 5059: 5056: 5053: 5037: 5034: 5026: 5019: 5016: 5004: 4992: 4989: 4985: 4980: 4974: 4971: 4964: 4960: 4956: 4945: 4942: 4934: 4930: 4921: 4907: 4903: 4899: 4896: 4892: 4876: 4869: 4866: 4858: 4847: 4841: 4829: 4826: 4823: 4820: 4812: 4808: 4804: 4800: 4799: 4794: 4791: 4787: 4785: 4781: 4767: 4760: 4757: 4746: 4740: 4737: 4733: 4728: 4725: 4722: 4718: 4715: 4704: 4698: 4695: 4694: 4689: 4686: 4682: 4678: 4674: 4670: 4666: 4662: 4658: 4654: 4650: 4646: 4639: 4635: 4628: 4624: 4617: 4616: 4615: 4593: 4580: 4574: 4565: 4550: 4546: 4535: 4532: 4529: 4519: 4518: 4517: 4515: 4511: 4507: 4503: 4499: 4495: 4476: 4469: 4461: 4452: 4446: 4443: 4440: 4434: 4428: 4425: 4420: 4417: 4407: 4406: 4405: 4403: 4399: 4395: 4372: 4363: 4359: 4350: 4336: 4332: 4323: 4314: 4306: 4302: 4293: 4279: 4275: 4266: 4255: 4251: 4247: 4245: 4224: 4215: 4199: 4190: 4174: 4165: 4149: 4140: 4138: 4126: 4113: 4110: 4107: 4101: 4095: 4092: 4089: 4075: 4074: 4073: 4071: 4067: 4063: 4059: 4055: 4051: 4047: 4043: 4039: 4035: 4031: 4027: 4023: 4019: 4015: 4011: 4007: 4001: 3977: 3966: 3955: 3952: 3949: 3946: 3943: 3940: 3936: 3930: 3927: 3917: 3908: 3897: 3891: 3870: 3869: 3868: 3866: 3865: 3860: 3856: 3852: 3848: 3844: 3843: 3837: 3835: 3832:that is, the 3816: 3807: 3804: 3794: 3783: 3769: 3759: 3749: 3735: 3734: 3733: 3731: 3730: 3724: 3722: 3718: 3714: 3710: 3691: 3682: 3672: 3661: 3647: 3646: 3645: 3643: 3642: 3621: 3610: 3591: 3590: 3589: 3571: 3564: 3556: 3547: 3544: 3538: 3535: 3530: 3527: 3517: 3516: 3515: 3513: 3509: 3502: 3498: 3491: 3472: 3455: 3446: 3443: 3427: 3410: 3401: 3398: 3379: 3378: 3377: 3375: 3371: 3355: 3329: 3326: 3311: 3300: 3297: 3282: 3274: 3268: 3265: 3257: 3246: 3239: 3238: 3237: 3220: 3211: 3207: 3203: 3198: 3194: 3190: 3185: 3181: 3174: 3170: 3161: 3154: 3151: 3139: 3130: 3127: 3119: 3112: 3109: 3097: 3088: 3080: 3073: 3070: 3058: 3048: 3044: 3021: 3012: 3000: 2999: 2998: 2996: 2995: 2990: 2986: 2982: 2978: 2974: 2968: 2958: 2956: 2952: 2949: 2930: 2919: 2908: 2905: 2902: 2899: 2896: 2893: 2889: 2883: 2873: 2867: 2849: 2848: 2847: 2845: 2841: 2837: 2833: 2829: 2825: 2822:-dimensional 2821: 2803: 2773: 2764: 2754: 2743: 2729: 2728: 2727: 2725: 2721: 2720: 2714: 2712: 2708: 2707:time integral 2703: 2701: 2697: 2690: 2687:second-order 2686: 2667: 2650: 2641: 2637: 2614: 2605: 2598: 2595: 2591: 2582: 2581: 2580: 2578: 2574: 2570: 2567:is the total 2566: 2562: 2559:is the total 2558: 2536: 2533: 2518: 2507: 2504: 2498: 2495: 2480: 2469: 2466: 2460: 2457: 2442: 2431: 2424: 2423: 2422: 2418: 2414: 2404: 2402: 2401: 2396: 2395: 2388: 2384: 2378: 2372: 2367: 2362: 2338: 2335: 2329: 2310: 2298:in the form: 2292: 2286: 2282: 2276: 2272: 2268: 2258: 2256: 2239: 2230: 2226: 2213: 2210: 2207: 2199: 2195: 2182: 2174: 2170: 2156: 2152: 2127: 2124:is the total 2123: 2118: 2105: 2088: 2079: 2075: 2052: 2043: 2036: 2033: 2029: 2024: 2009: 2008:Newton's laws 2004: 1999: 1978: 1966: 1963: 1960: 1955: 1943: 1938: 1923: 1896: 1892: 1889: 1881: 1878: 1868: 1865: 1862: 1853: 1852: 1845: 1835: 1822: 1814: 1807: 1804: 1797: 1794: 1791: 1786: 1779: 1776: 1769: 1764: 1757: 1754: 1744: 1729: 1725: 1718: 1715: 1708: 1704: 1700: 1694: 1691: 1688: 1682: 1679: 1672: 1668: 1664: 1658: 1652: 1649: 1642: 1638: 1634: 1627: 1623: 1617: 1614: 1604: 1593: 1589: 1568: 1564: 1560: 1557: 1554: 1549: 1545: 1541: 1536: 1532: 1525: 1512: 1508: 1504: 1498: 1496: 1492: 1489:= (number of 1485: 1483: 1477: 1474: 1470: 1466: 1462: 1461:configuration 1458: 1454: 1450: 1446: 1436: 1432: 1426: 1424: 1420: 1413: 1408: 1404: 1400: 1396: 1390: 1375: 1371: 1367: 1364: 1360: 1356: 1351: 1349: 1345: 1341: 1337: 1333: 1329: 1325: 1321: 1317: 1313: 1310: 1306: 1302: 1298: 1293: 1291: 1287: 1283: 1279: 1275: 1274: 1269: 1264: 1262: 1257: 1252: 1250: 1246: 1242: 1241:his third law 1238: 1233: 1231: 1227: 1223: 1219: 1215: 1211: 1207: 1203: 1199: 1194: 1192: 1181: 1178: 1163: 1160: 1152: 1149:February 2023 1141: 1138: 1134: 1131: 1127: 1124: 1120: 1117: 1113: 1110: – 1109: 1105: 1104:Find sources: 1098: 1094: 1088: 1087: 1082:This section 1080: 1076: 1071: 1070: 1061: 1058: 1050: 1047:February 2023 1040: 1036: 1032: 1026: 1023:This section 1021: 1012: 1011: 1006: 1004: 997: 996: 991: 990: 985: 980: 971: 970: 962: 958: 956: 952: 947: 945: 941: 937: 933: 927: 925: 921: 917: 913: 909: 905: 901: 897: 893: 889: 885: 881: 876: 874: 870: 866: 862: 858: 853: 851: 847: 843: 839: 838:accelerations 835: 831: 826: 824: 820: 816: 812: 808: 807: 802: 798: 794: 790: 786: 774: 769: 767: 762: 760: 755: 754: 752: 751: 745: 735: 732: 727: 721: 720: 719: 718: 710: 707: 705: 702: 700: 697: 695: 692: 690: 687: 685: 682: 680: 677: 675: 672: 670: 667: 665: 662: 660: 657: 655: 652: 650: 647: 645: 642: 640: 637: 635: 632: 630: 627: 625: 622: 620: 617: 615: 612: 610: 607: 605: 602: 600: 597: 595: 592: 591: 584: 583: 576: 572: 568: 564: 561: 560: 556: 553: 551: 548: 546: 543: 541: 538: 534: 531: 530: 529: 526: 524: 521: 519: 516: 514: 511: 510: 506: 501: 500: 492: 489: 487: 484: 480: 477: 475: 472: 471: 470: 467: 465: 462: 460: 457: 455: 450: 447: 443: 440: 439: 435: 432: 429: 425: 421: 420: 416: 413: 411: 408: 406: 403: 401: 396: 394: 391: 389: 386: 384: 381: 380: 373: 372: 361: 358: 356: 353: 351: 348: 346: 343: 341: 338: 336: 333: 332: 330: 329: 324: 321: 320: 315: 314: 308: 307: 299: 296: 294: 291: 289: 286: 284: 281: 279: 276: 274: 271: 269: 266: 264: 260: 258: 255: 253: 249: 247: 244: 241: 237: 233: 231: 228: 226: 223: 221: 218: 216: 213: 209: 206: 204: 201: 200: 199: 196: 194: 191: 189: 186: 184: 181: 179: 176: 175: 168: 167: 159: 156: 154: 151: 149: 146: 144: 141: 139: 136: 134: 131: 129: 126: 124: 121: 120: 113: 112: 106: 103: 101: 98: 96: 93: 92: 90: 89: 85: 66: 63: 53: 47: 33: 32: 29: 26: 25: 21: 20: 11677: / 11673: / 11663: 11539:Optimization 11502:Superalgebra 11361:Field theory 11343: 11334:Mathematical 11312: / 11160:Chaos theory 11143:Gauge theory 11071:Graph theory 10966:Cryptography 10773:Astrophysics 10626: 10587:Experimental 10496: 10476: 10469: 10449: 10442: 10427: 10423: 10387: 10367: 10347: 10340: 10325: 10321: 10305: 10286: 10280: 10265: 10243: 10207: 10200: 10180: 10173: 10153: 10072: 10068: 9942: 9929: 9917: 9909:, θ) is the 9906: 9902: 9900: 9096: 9092: 9088: 9087:or momentum 9084: 9082: 9014: 9009: 9005: 8747: 8514: 8372: 8368: 8361: 8357: 8355: 8329: 8270: 8255: 8250: 8246: 8237:denotes the 8231: 7928: 7924: 7923:= 1, 2, ... 7920: 7916: 7912: 7905: 7899: 7897: 7877: 7870: 7863: 7858: 7854: 7850: 7848: 7743: 7603: 7497: 7480: 7478: 7473: 7469: 7467: 7369: 7365: 7361: 7359: 7241: 7237: 7235: 7132: 7125: 7118: 7111: 7107: 7100: 7093: 7086: 7082: 7075: 7068: 7061: 7057: 7053: 7051: 7021: 7016: 7010: 6936: 6934: 6819: 6817: 6759: 6757: 6700: 6611: 6583: 6579: 6577: 6564: 6560: 6558: 6498: 6494: 6490: 6486: 6482: 6478: 6476: 6471: 6467: 6462:is called a 6456: 6452: 6450: 6336: 6332: 6330: 5833: 5829: 5825: 5821: 5817: 5813: 5809: 5805: 5801: 5797: 5793: 5792:and momenta 5789: 5785: 5781: 5777: 5775: 5739: 5737: 5708: 5704: 5669: 5668:and arrival 5662: 5660: 5537: 5535: 5421: 5415: 5411: 5403: 5385:, inserting 5336: 5332: 5065: 4897:(see below). 4890: 4810: 4806: 4802: 4796: 4789: 4783: 4706: 4702: 4700: 4691: 4684: 4680: 4676: 4672: 4668: 4664: 4660: 4656: 4652: 4648: 4641: 4637: 4630: 4626: 4619: 4613: 4493: 4491: 4401: 4397: 4391: 4069: 4065: 4061: 4057: 4053: 4049: 4045: 4041: 4037: 4033: 4029: 4025: 4021: 4017: 4013: 4009: 4005: 4003: 3862: 3858: 3854: 3850: 3846: 3840: 3838: 3831: 3727: 3725: 3713:wave vectors 3708: 3706: 3639:generalized 3638: 3636: 3586: 3511: 3504: 3500: 3493: 3489: 3487: 3368:denotes the 3347: 3235: 2992: 2988: 2984: 2980: 2976: 2970: 2945: 2839: 2835: 2831: 2819: 2788: 2717: 2715: 2704: 2699: 2692: 2684: 2682: 2564: 2556: 2554: 2420: 2398: 2392: 2386: 2382: 2376: 2370: 2365: 2360: 2290: 2284: 2278: 2271:Scleronomous 2121: 2119: 2002: 1851:virtual work 1849: 1847: 1591: 1506: 1502: 1500: 1494: 1490: 1488: 1481: 1479: 1472: 1464: 1456: 1452: 1441: 1439: 1431:curvillinear 1422: 1415: 1411: 1392: 1372: 1368: 1352: 1335: 1331: 1327: 1319: 1315: 1311: 1305:well-defined 1300: 1296: 1294: 1272: 1265: 1260: 1253: 1234: 1218:"mass point" 1210:acceleration 1195: 1188: 1173: 1155: 1146: 1136: 1129: 1122: 1115: 1103: 1091:Please help 1086:verification 1083: 1053: 1044: 1031:spinning off 1024: 1000: 993: 987: 986:Please help 983: 959: 955:chaos theory 948: 928: 877: 856: 854: 849: 827: 804: 796: 792: 782: 573: / 569: / 567:displacement 565: / 426: / 388:Displacement 327: 326: 317: 311:Formulations 298:Virtual work 238: / 178:Acceleration 171:Fundamentals 11679:topics list 11613:Mathematics 11529:Game theory 11430:Topological 11396:Topological 11391:Statistical 11354:Hamiltonian 10676:Statistical 10592:Theoretical 10569:Engineering 10373:R.G. Lerner 9915:unit vector 4904:, i.e. are 4813:, that is: 4707:, that is: 4012:, and time 4008:, momentum 3729:phase space 3370:dot product 2948:topological 2394:scleronomic 2364:are called 2261:Constraints 1482:constraints 1473:generalized 1465:curvilinear 892:phase space 857:constraints 709:von Neumann 376:Core topics 11694:Categories 11585:Psychology 11549:Statistics 11349:Lagrangian 10976:Statistics 10912:Algorithms 10793:Geophysics 10783:Biophysics 10627:Analytical 10580:Approaches 10115:Kinematics 9932:continuous 9784:T-symmetry 8886:where the 8239:4-gradient 6563:= 1, 2,... 5757:motion in 5427:functional 4510:commutator 3842:phase path 3715:(given by 2275:Rheonomous 1414:, denoted 1290:invariance 1249:solid body 1119:newspapers 1035:relocating 989:improve it 924:symmetries 644:d'Alembert 624:Maupertuis 587:Scientists 469:Rigid body 143:Kinematics 11590:Sociology 11580:Chemistry 11376:Effective 11371:Conformal 11366:Classical 11238:Geometric 11211:Geometric 10743:Molecular 10644:Acoustics 10637:Continuum 10632:Celestial 10622:Newtonian 10609:Classical 10552:Divisions 10235:165699220 10040:˙ 9986:˙ 9940:parameter 9878:− 9867:− 9864:→ 9832:− 9821:→ 9762:− 9759:→ 9728:− 9725:→ 9655:→ 9613:→ 9540:θ 9531:^ 9516:→ 9482:θ 9473:^ 9458:→ 9429:θ 9420:^ 9353:→ 9291:→ 9207:→ 9168:→ 9114:Momentum 9111:Position 9108:Operator 9030:∫ 8980:ϕ 8974:μ 8970:∂ 8963:∂ 8959:∂ 8952:μ 8948:∂ 8944:− 8932:ϕ 8928:∂ 8924:∂ 8907:ϕ 8903:δ 8899:δ 8861:ϕ 8857:δ 8845:δ 8839:− 8824:˙ 8821:π 8800:π 8796:δ 8784:δ 8763:˙ 8760:ϕ 8723:− 8697:π 8667:˙ 8664:ϕ 8640:∑ 8616:… 8604:π 8591:π 8584:… 8572:ϕ 8559:ϕ 8495:∂ 8484:ϕ 8480:∂ 8474:≡ 8462:˙ 8459:ϕ 8438:˙ 8435:ϕ 8428:∂ 8416:∂ 8384:π 8290:∫ 8205:ϕ 8201:∂ 8189:∂ 8164:ϕ 8158:μ 8154:∂ 8147:∂ 8135:∂ 8123:μ 8119:∂ 8077:… 8065:ϕ 8055:∂ 8042:ϕ 8032:∂ 8025:… 8013:ϕ 8009:∇ 7997:ϕ 7993:∇ 7987:… 7975:ϕ 7962:ϕ 7779:¨ 7681:∑ 7647:α 7643:∂ 7635:∂ 7532:¨ 7513:α 7445:ζ 7441:∂ 7433:∂ 7418:˙ 7415:ζ 7409:∂ 7401:∂ 7333:∂ 7325:∂ 7319:− 7310:˙ 7286:∂ 7278:∂ 7263:˙ 7211:˙ 7208:ζ 7198:ζ 7172:− 7166:˙ 7157:⋅ 6901:∂ 6889:∂ 6797:∂ 6785:∂ 6779:− 6735:∂ 6723:∂ 6424:∂ 6416:∂ 6401:˙ 6379:∂ 6371:∂ 6365:− 6356:˙ 6271:∂ 6267:∂ 6154:∂ 6150:∂ 6037:∂ 6033:∂ 5920:∂ 5916:∂ 5618:˙ 5570:∫ 5566:δ 5553:δ 5499:˙ 5451:∫ 5309:˙ 5261:˙ 5242:˙ 5210:˙ 5185:λ 5155:˙ 5145:λ 5133:˙ 5123:λ 5095:˙ 5085:λ 5020:˙ 5010:∂ 5002:∂ 4950:→ 4927:∂ 4919:∂ 4906:conserved 4839:∂ 4835:∂ 4788:describe 4584:^ 4569:^ 4554:ℏ 4542:→ 4467:∂ 4459:∂ 4356:∂ 4348:∂ 4329:∂ 4321:∂ 4315:− 4299:∂ 4291:∂ 4272:∂ 4264:∂ 4252:∑ 4248:≡ 4230:∂ 4222:∂ 4216:⋅ 4205:∂ 4197:∂ 4191:− 4180:∂ 4172:∂ 4166:⋅ 4155:∂ 4147:∂ 4102:≡ 3967:⊆ 3956:∈ 3944:≥ 3918:∈ 3795:∈ 3760:× 3673:∈ 3605:˙ 3562:∂ 3554:∂ 3548:− 3461:∂ 3453:∂ 3438:˙ 3416:∂ 3408:∂ 3402:− 3393:˙ 3356:⋅ 3321:˙ 3298:− 3292:˙ 3283:⋅ 3204:⋯ 3155:˙ 3145:∂ 3137:∂ 3131:⋯ 3113:˙ 3103:∂ 3095:∂ 3074:˙ 3064:∂ 3056:∂ 3036:˙ 3027:∂ 3019:∂ 2951:manifolds 2920:⊆ 2909:∈ 2897:≥ 2874:∈ 2755:∈ 2656:∂ 2648:∂ 2629:˙ 2620:∂ 2612:∂ 2528:˙ 2505:− 2490:˙ 2452:˙ 2400:rheonomic 2368:. Vector 2294:and time 2223:∂ 2219:∂ 2211:… 2192:∂ 2188:∂ 2167:∂ 2163:∂ 2142:∂ 2138:∂ 2094:∂ 2086:∂ 2080:− 2067:˙ 2058:∂ 2050:∂ 1964:… 1882:δ 1879:⋅ 1863:δ 1808:˙ 1795:… 1780:˙ 1758:˙ 1739:˙ 1730:≡ 1692:… 1558:… 1469:dimension 1455:= 1, 2... 1447:for each 1245:rotations 995:talk page 823:variation 689:Liouville 571:frequency 491:Vibration 208:potential 133:Continuum 128:Celestial 105:Textbooks 11665:Category 11314:analysis 11233:Exterior 11206:Exterior 11086:Analysis 11048:Discrete 10922:analysis 10079:See also 8241:and the 7919:) where 7133:Routhian 6559:for all 6466:of the " 5755:geodesic 4719:′ 2983:) with ( 2953:and the 1996:are the 1586:and the 1407:position 1309:function 1261:a priori 1222:particle 1206:velocity 873:friction 744:Category 669:Hamilton 654:Lagrange 649:Clairaut 614:Horrocks 575:velocity 545:Pendulum 533:reactive 505:Rotation 474:dynamics 424:Inertial 410:Friction 293:Velocity 268:Momentum 148:Kinetics 138:Dynamics 116:Branches 100:Timeline 11675:outline 11606:Related 11575:Biology 11425:Bosonic 11386:Quantum 11336:physics 11302: ( 11034:Solvers 10827:Related 10711:General 10706:Special 10564:Applied 8375:) are: 7124:, ..., 4798:partial 2358:, then 1133:scholar 842:momenta 704:Koopman 664:Poisson 659:Laplace 604:Huygens 599:Galileo 444: ( 383:Damping 236:Inertia 230:Impulse 203:kinetic 153:Statics 123:Applied 95:History 11248:Vector 11243:Tensor 11221:Vector 11216:Tensor 10917:design 10738:Atomic 10693:Modern 10543:Major 10504: 10484: 10457: 10434: 10415: 10394: 10355: 10332: 10313: 10293: 10272: 10233: 10223: 10188: 10161: 9901:where 9688:Parity 8342:, and 8264:, and 8230:where 7927:, the 7744:where 7099:, ... 7074:, ... 6818:where 5816:) and 5784:, and 5423:Action 5331:where 5052:cyclic 4679:) and 4052:) and 4036:). If 3503:) and 3348:where 2789:where 2555:where 2273:, and 2120:where 1909:where 1273:action 1220:or a " 1214:motion 1198:vector 1135: 1128: 1121: 1114: 1106: 906:, and 886:) and 846:forces 834:vector 817:. The 806:scalar 742: 694:Appell 679:Cauchy 674:Jacobi 619:Halley 609:Newton 594:Kepler 446:linear 442:Motion 288:Torque 263:Moment 198:Energy 188:Couple 11381:Gauge 7128:N − s 4693:total 3721:waves 1511:tuple 1288:, an 1247:of a 1237:fluid 1226:Earth 1202:force 1140:JSTOR 1126:books 795:, or 699:Gibbs 684:Routh 639:Euler 278:Speed 273:Space 215:Force 10664:Wave 10559:Pure 10502:ISBN 10482:ISBN 10455:ISBN 10432:ISBN 10413:ISBN 10392:ISBN 10353:ISBN 10330:ISBN 10311:ISBN 10291:ISBN 10270:ISBN 10231:OCLC 10221:ISBN 10186:ISBN 10159:ISBN 7360:and 7026:and 6758:and 6571:The 6493:and 6481:and 6335:and 4809:and 4784:and 4703:and 2971:The 2824:real 2563:and 2415:and 1433:and 1112:news 942:and 934:and 813:and 787:and 283:Time 246:Mass 10659:Ray 10213:doi 9095:or 7034:of 5749:of 5740:all 5066:and 4663:), 4500:of 2818:is 2724:set 2709:of 2702:). 1393:In 1228:'s 1095:by 1033:or 825:. 783:In 11696:: 10402:^ 10380:^ 10252:^ 10229:. 10219:. 10143:^ 9945:: 9918:n̂ 9907:n̂ 8371:, 8338:, 8334:, 8268:. 8260:, 7915:, 7883:. 7472:− 7372:. 7364:− 7244:, 7135:: 7117:, 7110:= 7092:, 7085:= 7067:, 7060:= 7038:. 7019:. 6614:: 6497:→ 6489:→ 5832:, 5828:, 5820:= 5812:, 5808:, 5800:= 5780:, 5761:. 5542:: 5418:). 4805:, 4701:q' 4631:q̇ 4516:: 4404:: 4064:, 4060:, 4048:, 4044:, 4032:, 4028:, 4020:= 3867:: 3853:), 3732:: 3723:) 3644:: 3376:: 2987:, 2981:q̇ 2979:, 2957:. 2579:; 2403:. 2269:, 1594:: 1513:: 1350:. 1208:, 1204:, 998:. 957:. 946:. 926:. 902:, 844:, 840:, 791:, 11598:" 11594:" 11306:) 10888:e 10881:t 10874:v 10536:e 10529:t 10522:v 10490:. 10463:. 10361:. 10299:. 10237:. 10215:: 10194:. 10167:. 10073:q 10069:s 10055:] 10052:) 10049:t 10046:( 10037:q 10031:, 10028:) 10025:t 10022:( 10019:q 10016:[ 10013:L 10010:= 10007:] 10004:) 10001:t 9998:, 9995:s 9992:( 9983:q 9977:, 9974:) 9971:t 9968:, 9965:s 9962:( 9959:q 9956:[ 9953:L 9943:s 9905:( 9903:R 9884:) 9881:t 9875:( 9871:p 9860:p 9838:) 9835:t 9829:( 9825:r 9817:r 9795:T 9766:p 9755:p 9732:r 9721:r 9699:P 9670:v 9666:m 9663:+ 9659:p 9651:p 9629:t 9625:v 9621:+ 9617:r 9609:r 9587:) 9583:v 9579:( 9576:G 9547:p 9543:) 9537:, 9528:n 9522:( 9519:R 9512:p 9489:r 9485:) 9479:, 9470:n 9464:( 9461:R 9454:r 9432:) 9426:, 9417:n 9411:( 9408:R 9380:) 9375:0 9371:t 9367:+ 9364:t 9361:( 9357:p 9350:) 9347:t 9344:( 9340:p 9318:) 9313:0 9309:t 9305:+ 9302:t 9299:( 9295:r 9288:) 9285:t 9282:( 9278:r 9256:) 9251:0 9247:t 9243:( 9240:U 9211:p 9203:p 9180:a 9176:+ 9172:r 9164:r 9142:) 9138:a 9134:( 9131:X 9097:p 9093:r 9089:p 9085:r 9056:. 9052:V 9049:d 9043:H 9035:V 9026:= 9023:H 9010:N 9006:N 8989:) 8984:i 8966:( 8936:i 8919:= 8911:i 8874:, 8865:i 8850:H 8836:= 8831:i 8813:, 8804:i 8789:H 8778:+ 8775:= 8770:i 8734:. 8728:L 8720:) 8717:t 8714:, 8710:r 8706:( 8701:i 8693:) 8690:t 8687:, 8683:r 8679:( 8674:i 8655:N 8650:1 8647:= 8644:i 8636:= 8633:) 8630:t 8627:, 8623:r 8619:, 8613:, 8608:2 8600:, 8595:1 8587:, 8581:, 8576:2 8568:, 8563:1 8555:( 8550:H 8526:H 8498:t 8488:i 8469:i 8445:i 8421:L 8410:= 8407:) 8404:t 8401:, 8397:r 8393:( 8388:i 8373:t 8369:r 8367:( 8364:i 8362:φ 8358:N 8316:. 8312:V 8309:d 8303:L 8295:V 8286:= 8283:L 8251:N 8247:N 8234:μ 8232:∂ 8218:, 8209:i 8194:L 8183:= 8179:) 8173:) 8168:i 8150:( 8140:L 8129:( 8098:. 8094:) 8091:t 8088:, 8084:r 8080:, 8074:, 8069:2 8059:t 8051:, 8046:1 8036:t 8028:, 8022:, 8017:2 8006:, 8001:1 7990:, 7984:, 7979:2 7971:, 7966:1 7958:( 7953:L 7948:= 7943:L 7925:N 7921:i 7917:t 7913:r 7911:( 7908:i 7906:φ 7900:N 7880:r 7878:q 7874:k 7871:r 7866:r 7864:α 7859:k 7855:a 7851:k 7829:2 7825:t 7821:d 7814:k 7809:r 7802:2 7798:d 7791:= 7786:k 7775:r 7767:= 7762:k 7757:a 7729:, 7723:2 7718:k 7713:a 7706:k 7702:m 7696:N 7691:1 7688:= 7685:k 7675:2 7672:1 7667:= 7664:S 7660:, 7651:r 7638:S 7629:= 7624:r 7618:Q 7589:, 7580:2 7576:t 7572:d 7565:r 7561:q 7555:2 7551:d 7544:= 7539:r 7529:q 7522:= 7517:r 7481:q 7474:s 7470:N 7453:. 7436:R 7427:= 7404:R 7392:t 7389:d 7385:d 7370:ζ 7366:s 7362:N 7345:, 7337:q 7328:R 7316:= 7306:p 7298:, 7290:p 7281:R 7272:+ 7269:= 7259:q 7242:q 7238:s 7221:, 7217:) 7202:, 7194:, 7190:p 7186:, 7182:q 7178:( 7175:L 7163:q 7153:p 7149:= 7146:R 7126:ζ 7122:1 7119:ζ 7115:1 7112:ζ 7108:ζ 7103:s 7101:p 7097:2 7094:p 7090:1 7087:p 7083:p 7078:s 7076:q 7072:2 7069:q 7065:1 7062:q 7058:q 7054:s 7017:H 6996:t 6993:E 6990:+ 6987:) 6984:t 6981:, 6977:q 6973:( 6968:S 6963:= 6960:) 6956:q 6952:( 6949:W 6919:) 6915:t 6912:, 6905:q 6894:S 6883:, 6879:q 6874:( 6870:H 6867:= 6864:) 6861:t 6858:, 6854:p 6850:, 6846:q 6842:( 6839:H 6836:= 6833:H 6820:H 6800:t 6790:S 6776:= 6773:H 6760:P 6739:q 6728:S 6717:= 6713:p 6686:, 6682:C 6679:+ 6676:) 6673:t 6670:, 6666:q 6662:( 6657:S 6652:= 6649:) 6646:t 6643:, 6639:q 6635:( 6630:2 6626:G 6612:C 6596:S 6580:K 6565:N 6561:i 6544:1 6541:= 6538:} 6533:i 6529:P 6525:, 6520:i 6516:Q 6512:{ 6499:P 6495:p 6491:Q 6487:q 6483:P 6479:Q 6472:n 6468:n 6459:n 6457:G 6436:, 6428:P 6419:K 6410:+ 6407:= 6398:Q 6391:, 6383:Q 6374:K 6362:= 6353:P 6337:Q 6333:P 6312:) 6309:t 6306:, 6302:P 6298:, 6294:p 6290:( 6285:4 6281:G 6274:t 6262:+ 6259:) 6256:t 6253:, 6249:p 6245:, 6241:q 6237:( 6234:H 6231:= 6228:) 6225:t 6222:, 6218:P 6214:, 6210:Q 6206:( 6203:K 6195:) 6192:t 6189:, 6185:Q 6181:, 6177:p 6173:( 6168:3 6164:G 6157:t 6145:+ 6142:) 6139:t 6136:, 6132:p 6128:, 6124:q 6120:( 6117:H 6114:= 6111:) 6108:t 6105:, 6101:P 6097:, 6093:Q 6089:( 6086:K 6078:) 6075:t 6072:, 6068:P 6064:, 6060:q 6056:( 6051:2 6047:G 6040:t 6028:+ 6025:) 6022:t 6019:, 6015:p 6011:, 6007:q 6003:( 6000:H 5997:= 5994:) 5991:t 5988:, 5984:P 5980:, 5976:Q 5972:( 5969:K 5961:) 5958:t 5955:, 5951:Q 5947:, 5943:q 5939:( 5934:1 5930:G 5923:t 5911:+ 5908:) 5905:t 5902:, 5898:p 5894:, 5890:q 5886:( 5883:H 5880:= 5877:) 5874:t 5871:, 5867:P 5863:, 5859:Q 5855:( 5852:K 5834:t 5830:p 5826:q 5824:( 5822:P 5818:P 5814:t 5810:p 5806:q 5804:( 5802:Q 5798:Q 5794:p 5790:q 5786:t 5782:q 5778:p 5721:C 5709:t 5707:( 5705:q 5689:C 5673:2 5670:t 5666:1 5663:t 5646:, 5642:0 5639:= 5636:t 5633:d 5630:) 5627:t 5624:, 5615:q 5609:, 5605:q 5601:( 5598:L 5591:2 5587:t 5579:1 5575:t 5563:= 5558:S 5521:. 5517:t 5514:d 5511:) 5508:t 5505:, 5496:q 5490:, 5486:q 5482:( 5479:L 5472:2 5468:t 5460:1 5456:t 5447:= 5442:S 5416:q 5412:S 5404:q 5369:. 5365:E 5362:= 5359:V 5356:+ 5353:T 5350:= 5347:H 5333:λ 5319:, 5315:) 5306:q 5300:, 5296:q 5292:( 5289:L 5285:, 5281:) 5277:q 5273:, 5268:k 5258:q 5249:j 5239:q 5232:, 5227:2 5223:) 5217:i 5207:q 5200:( 5197:( 5194:T 5189:2 5181:= 5178:) 5174:q 5170:, 5167:) 5162:k 5152:q 5140:j 5130:q 5120:( 5117:, 5112:2 5108:) 5102:i 5092:q 5082:( 5079:( 5076:T 5038:0 5035:= 5027:j 5017:q 5005:L 4993:t 4990:d 4986:d 4981:= 4975:t 4972:d 4965:j 4961:p 4957:d 4946:0 4943:= 4935:j 4931:q 4922:L 4891:K 4889:( 4877:, 4873:) 4870:t 4867:, 4863:p 4859:, 4855:q 4851:( 4848:G 4842:t 4830:+ 4827:H 4824:= 4821:K 4811:t 4807:p 4803:q 4786:L 4782:L 4768:, 4764:) 4761:t 4758:, 4754:q 4750:( 4747:F 4741:t 4738:d 4734:d 4729:+ 4726:L 4723:= 4716:L 4705:t 4685:t 4683:( 4681:p 4677:t 4675:( 4673:q 4669:t 4667:( 4665:p 4661:t 4659:( 4657:q 4653:t 4649:t 4647:( 4644:i 4642:p 4638:t 4636:( 4633:i 4627:t 4625:( 4622:i 4620:q 4594:. 4590:] 4581:B 4575:, 4566:A 4560:[ 4551:i 4547:1 4539:} 4536:B 4533:, 4530:A 4527:{ 4494:A 4477:. 4470:t 4462:A 4453:+ 4450:} 4447:H 4444:, 4441:A 4438:{ 4435:= 4429:t 4426:d 4421:A 4418:d 4402:A 4398:A 4373:, 4364:k 4360:q 4351:B 4337:k 4333:p 4324:A 4307:k 4303:p 4294:B 4280:k 4276:q 4267:A 4256:k 4234:q 4225:B 4209:p 4200:A 4184:p 4175:B 4159:q 4150:A 4141:= 4131:p 4127:, 4123:q 4118:} 4114:B 4111:, 4108:A 4105:{ 4099:} 4096:B 4093:, 4090:A 4087:{ 4066:t 4062:p 4058:q 4056:( 4054:B 4050:t 4046:p 4042:q 4040:( 4038:A 4034:t 4030:p 4026:q 4024:( 4022:A 4018:A 4014:t 4010:p 4006:q 3978:, 3972:P 3964:} 3960:R 3953:t 3950:, 3947:0 3941:t 3937:: 3931:N 3928:2 3923:R 3915:) 3912:) 3909:t 3906:( 3902:p 3898:, 3895:) 3892:t 3889:( 3885:q 3881:( 3878:{ 3859:t 3857:( 3855:p 3851:t 3849:( 3847:q 3817:, 3813:} 3808:N 3805:2 3800:R 3792:) 3788:p 3784:, 3780:q 3776:( 3773:{ 3770:= 3765:M 3755:C 3750:= 3745:P 3709:k 3692:. 3688:} 3683:N 3678:R 3669:p 3665:{ 3662:= 3657:M 3622:. 3616:Q 3611:= 3602:p 3572:, 3565:t 3557:L 3545:= 3539:t 3536:d 3531:H 3528:d 3512:t 3510:( 3507:i 3505:p 3501:t 3499:( 3496:i 3494:q 3490:N 3473:, 3465:p 3456:H 3447:+ 3444:= 3435:q 3428:, 3420:q 3411:H 3399:= 3390:p 3333:) 3330:t 3327:, 3318:q 3312:, 3308:q 3304:( 3301:L 3289:q 3279:p 3275:= 3272:) 3269:t 3266:, 3262:p 3258:, 3254:q 3250:( 3247:H 3221:, 3217:) 3212:N 3208:p 3199:2 3195:p 3191:, 3186:1 3182:p 3178:( 3175:= 3171:) 3162:N 3152:q 3140:L 3128:, 3120:2 3110:q 3098:L 3089:, 3081:1 3071:q 3059:L 3049:( 3045:= 3033:q 3022:L 3013:= 3009:p 2989:p 2985:q 2977:q 2931:, 2925:C 2917:} 2913:R 2906:t 2903:, 2900:0 2894:t 2890:: 2884:N 2879:R 2871:) 2868:t 2865:( 2861:q 2857:{ 2840:t 2838:( 2836:q 2820:N 2804:N 2799:R 2774:, 2770:} 2765:N 2760:R 2751:q 2747:{ 2744:= 2739:C 2700:t 2698:( 2695:i 2693:q 2685:N 2668:, 2660:q 2651:L 2642:= 2638:) 2626:q 2615:L 2606:( 2599:t 2596:d 2592:d 2565:V 2557:T 2540:) 2537:t 2534:, 2525:q 2519:, 2515:q 2511:( 2508:V 2502:) 2499:t 2496:, 2487:q 2481:, 2477:q 2473:( 2470:T 2467:= 2464:) 2461:t 2458:, 2449:q 2443:, 2439:q 2435:( 2432:L 2389:) 2387:t 2385:( 2383:q 2377:t 2371:r 2361:q 2356:t 2342:) 2339:t 2336:, 2333:) 2330:t 2327:( 2323:q 2319:( 2315:r 2311:= 2307:r 2296:t 2291:q 2285:r 2240:) 2231:N 2227:q 2214:, 2208:, 2200:2 2196:q 2183:, 2175:1 2171:q 2157:( 2153:= 2146:q 2122:T 2106:, 2098:q 2089:T 2076:) 2064:q 2053:T 2044:( 2037:t 2034:d 2030:d 2025:= 2020:Q 2003:q 1984:) 1979:N 1973:Q 1967:, 1961:, 1956:2 1950:Q 1944:, 1939:1 1933:Q 1927:( 1924:= 1919:Q 1897:, 1893:0 1890:= 1886:q 1874:Q 1869:= 1866:W 1823:. 1820:) 1815:N 1805:q 1798:, 1792:, 1787:2 1777:q 1770:, 1765:1 1755:q 1748:( 1745:= 1736:q 1726:) 1719:t 1716:d 1709:N 1705:q 1701:d 1695:, 1689:, 1683:t 1680:d 1673:2 1669:q 1665:d 1659:, 1653:t 1650:d 1643:1 1639:q 1635:d 1628:( 1624:= 1618:t 1615:d 1609:q 1605:d 1574:) 1569:N 1565:q 1561:, 1555:, 1550:2 1546:q 1542:, 1537:1 1533:q 1529:( 1526:= 1522:q 1509:- 1507:N 1503:N 1497:) 1484:) 1457:N 1453:i 1444:i 1442:q 1423:i 1421:( 1418:i 1416:q 1336:t 1332:t 1328:t 1322:( 1320:t 1316:t 1314:( 1312:f 1301:t 1297:t 1180:) 1174:( 1162:) 1156:( 1151:) 1147:( 1137:· 1130:· 1123:· 1116:· 1089:. 1060:) 1054:( 1049:) 1045:( 1041:. 1027:. 1005:) 1001:( 772:e 765:t 758:v 448:) 67:t 64:d 58:p 54:d 48:= 43:F

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