n-body problem - Knowledge

7832: 3525:) maintaining the 60° spacing ahead of, and behind, the less massive body almost in its orbit (although in reality neither of the bodies are truly stationary, as they both orbit the center of mass of the whole system—about the barycenter). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrangian points. See figure below: 7820: 3367:) that the main reason, however, for failing to predict the forces for elliptical orbits was that his math model was for a body confined to a situation that hardly existed in the real world, namely, the motions of bodies attracted toward an unmoving center. Some present physics and astronomy textbooks do not emphasize the negative significance of Newton's assumption and end up teaching that his mathematical model is in effect reality. It is to be understood that the classical two-body problem solution above is a mathematical idealization. See also 3327:

some 30,000 miles above the solar surface! Other large planets also influence the center of mass of the solar system, however. In 1951, for example, the systems' center of mass was not far from the Sun's center because Jupiter was on the opposite side from Saturn, Uranus and Neptune. In the late 1950s, when all four of these planets were on the same side of the Sun, the system's center of mass was more than 330,000 miles from the solar surface, Dr. C. H. Cleminshaw of Griffith Observatory in Los Angeles has calculated.

612:– Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity. Having done so, he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits correctly or even very well. Newton realized that this was because gravitational interactive forces amongst all the planets were affecting all their orbits. 1153: 650:

and can be considered as two actions by which each attracts the other, yet inasmuch as they are between the same, two bodies they are not two but a simple operation between two termini. Two bodies can be drawn to each other by the contraction of rope between them. The cause of the action is twofold, namely the disposition of each of the two bodies; the action is likewise twofold, insofar as it is upon two bodies; but insofar as it is between two bodies it is single and one ...

114: 3529: 3076: 8645: 853: 5192:-body problem similarly (see References): "Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?" Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces first before the motions can be determined. 3642:, mainly to model spacecraft trajectories in the Earth-Moon system with the addition of the gravitational attraction of the Sun. The former formulation of the bicircular restricted four-body problem can be problematic when modelling other systems than the Earth-Moon-Sun, so the formulation was generalized by Negri and Prado to expand the application range and improve the accuracy without loss of simplicity. 7844: 2822: 1491: 3332: 7393: 40: 4150: 5145:-body problem, especially Ms. Kovalevskaya's 1868–1888 twenty-year complex-variables approach, failure; Section 1: "The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics" (Chapter 1, "The motion of a rigid body about a fixed point (Euler and Poisson equations)"; Chapter 2, "Mathematical Exterior Ballistics"), good precursor background to the 1148:{\displaystyle \mathbf {F} _{ij}={\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}}}\cdot {\frac {\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}={\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}},} 1242: 1828: 3071:{\displaystyle {\begin{aligned}m_{1}\mathbf {a} _{1}&={\frac {Gm_{1}m_{2}}{r_{12}^{3}}}(\mathbf {r} _{2}-\mathbf {r} _{1})&&\quad {\text{Sun–Earth}}\\m_{2}\mathbf {a} _{2}&={\frac {Gm_{1}m_{2}}{r_{21}^{3}}}(\mathbf {r} _{1}-\mathbf {r} _{2})&&\quad {\text{Earth–Sun}}\end{aligned}}} 4684:

There are a number of techniques to reduce errors in numerical integration. Local coordinate systems are used to deal with widely differing scales in some problems, for example an Earth–Moon coordinate system in the context of a solar system simulation. Variational methods and perturbation theory can

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in the 1990s. Since the structure of singularities is more complicated, Wang had to leave out completely the questions of singularities. The central point of his approach is to transform, in an appropriate manner, the equations to a new system, such that the interval of existence for the solutions of

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Inspired by the circular restricted three-body problem, the four-body problem can be greatly simplified by considering a smaller body to have a small mass compared to the other three massive bodies, which in turn are approximated to describe circular orbits. This is known as the bicircular restricted

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Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the

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Rudolf Kurth has an extensive discussion in his book (see References) on planetary perturbations. An aside: these mathematically undefined planetary perturbations (wobbles) still exist undefined even today and planetary orbits have to be constantly updated, usually yearly. See Astronomical Ephemeris

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The aforementioned revelation strikes directly at the core of what the n-body issue physically is: as Newton understood, it is not enough to just provide the beginning location and velocity, or even three orbital positions, in order to establish a planet's actual orbit; one must also be aware of the

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The Sun contains 98 per cent of the mass in the solar system, with the superior planets beyond Mars accounting for most of the rest. On the average, the center of the mass of the Sun–Jupiter system, when the two most massive objects are considered alone, lies 462,000 miles from the Sun's center, or

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And hence it is that the attractive force is found in both bodies. The Sun attracts Jupiter and the other planets, Jupiter attracts its satellites and similarly the satellites act on one another. And although the actions of each of a pair of planets on the other can be distinguished from each other

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the first steps in the study of the problem of the movements of three bodies subject to their mutual gravitational attractions, but his efforts resulted in verbal descriptions and geometrical sketches; see especially Book 1, Proposition 66 and its corollaries (Newton, 1687 and 1999 (transl.), see

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Note: The fact a parabolic orbit has zero energy arises from the assumption the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value to the potential energy in the state of infinite separation. That state is assumed to have zero potential

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discovered two classes of periodic solution, each for three bodies of any masses. In one class, the bodies lie on a rotating straight line. In the other class, the bodies lie at the vertices of a rotating equilateral triangle. In either case, the paths of the bodies will be conic sections. Those

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loads are the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, under a

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take advantage of the fact that the multipole-expanded forces from distant particles are similar for particles close to each other, and uses local expansions of far-field forces to reduce computational effort. It is claimed that this further approximation reduces the complexity to

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is the fundamental differential equation for the two-body problem Bernoulli solved in 1734. Notice for this approach forces have to be determined first, then the equation of motion resolved. This differential equation has elliptic, or parabolic or hyperbolic solutions.

1687: 2130: 1627: 1486:{\displaystyle m_{i}{\frac {d^{2}\mathbf {q} _{i}}{dt^{2}}}=\sum _{j=1 \atop j\neq i}^{n}{\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}}=-{\frac {\partial U}{\partial \mathbf {q} _{i}}}} 5690:, Chapter 3: "Curvilinear Motion in a Plane", and specifically paragraphs 3–9, "Planetary Motion"; pp. 83–96. Lindsay presentation goes a long way in explaining these latter comments for the fixed two-body problem; i.e., when the Sun is assumed fixed. 4991:

of a form similar to that of the gravitational potential: a sum of kernel functions over all pairs of objects, where the kernel function depends on the distance between the objects in parameter space. Example problems that fit into this form include

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is a point within the asteroid belt. It has to be realized for this model, this whole Sun-Jupiter diagram is rotating about its barycenter. The restricted three-body problem solution predicted the Trojan planetoids before they were first seen. The

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orbits in the case of the planetary problem restricted to the plane. In the KAM theory, chaotic planetary orbits would be bounded by quasiperiodic KAM tori. Arnold's result was extended to a more general theorem by Féjoz and Herman in 2004.

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See Bate, Mueller, and White, Chapter 1: "Two-Body Orbital Mechanics", pp. 1–49. These authors were from the Department of Astronautics and Computer Science, United States Air Force Academy. Their textbook is not filled with advanced

4681:, which means that even small errors in integration may grow exponentially in time. Third, a simulation may be over large stretches of model time (e.g. millions of years) and numerical errors accumulate as integration time increases. 3478:, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in 1964: 3688:

The existence of resonances and small denominators led to the important question of stability in the planetary problem: do planets, in nearly circular orbits around a star, remain in stable or bounded orbits over time? In 1963,

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Given the quasi-steady orbital properties (instantaneous position, velocity and time) of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future

3629:. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. 2371: 701:, even though he did not solve the original problem. (The first version of his contribution even contained a serious error.) The version finally printed contained many important ideas which led to the development of 4145:{\displaystyle {\frac {d^{2}\mathbf {x} _{i}(t)}{dt^{2}}}=G\sum _{k=1 \atop k\neq i}^{n}{\frac {m_{k}\left(\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right)}{\left|\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right|^{3}}},} 5149:-body problem; Section 2: "Celestial Mechanics" (Chapter 1, "The Uniformization of the Three-body Problem (Restricted Three-body Problem)"; Chapter 2, "Capture in the Three-Body Problem"; Chapter 3, "Generalized 2819:(1667–1748) by classical theory (and not by Newton) by assuming the main point-mass was fixed; this is outlined here. Consider then the motion of two bodies, say the Sun and the Earth, with the Sun fixed, then: 2028: 3866:-body problem" (via employing the above approach). However, care must be taken when discussing the 'impossibility' of a solution, as this refers only to the method of first integrals (compare the theorems by 3303:(the Sun) as fixed in space when applying Newton's law of universal gravitation, and to do so leads to erroneous results. The fixed point for two isolated gravitationally interacting bodies is their mutual 4863:

and uses direct methods with softened potentials at close range. PM-tree methods instead use tree codes at close range. As with particle mesh methods, adaptive meshes can increase computational efficiency.

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in which the configuration remains an isometry of the initial configuration, as if the configuration was a rigid body. Central configurations have played an important role in understanding the

5061:, with vorticity taking the place of electrical current. In the context of particle-laden turbulent multiphase flows, determining an overall disturbance field generated by all particles is an 5065:-body problem. If the particles translating within the flow are much smaller than the flow's Kolmogorov scale, their linear Stokes disturbance fields can be superposed, yielding a system of 3 4758:, are spatially-hierarchical methods used when distant particle contributions do not need to be computed to high accuracy. The potential of a distant group of particles is computed using a 2626: 4430:

does not remain finite. In this scenario, bodies diverge to infinity in a finite time, while at the same time tending towards zero separation (an imaginary collision occurs "at infinity").

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Leimanis, E.; Minorsky, N. (1958). "Part I: "Some Recent Advances in the Dynamics of Rigid Bodies and Celestial Mechanics" (Leimanis); Part II: "The Theory of Oscillations" (Minorsky)".

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did not hold for the forces associated with elliptical orbits. In fact, Newton's Universal Law does not account for the orbit of Mercury, the asteroid belt's gravitational behavior, or

631:-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his 6837:

Krumscheid, Sebastian (2010). Benchmark of fast Coulomb Solvers for open and periodic boundary conditions (Report). Technical Report FZJ-JSC-IB-2010-01. Jülich Supercomputing Centre.

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If the sum of both energies is positive, then they both trace out hyperbolas. As the distance between the bodies tends to infinity, their relative speed tends to some positive number.

1823:{\displaystyle {\frac {d\mathbf {q} _{i}}{dt}}={\frac {\partial H}{\partial \mathbf {p} _{i}}}\qquad {\frac {d\mathbf {p} _{i}}{dt}}=-{\frac {\partial H}{\partial \mathbf {q} _{i}}},} 620:-body "problem" in the early 17th century. These gravitational attractive forces do conform to Newton's laws of motion and to his law of universal gravitation, but the many multiple ( 2701:

a characteristic size of the system (for example, the radius containing half the mass of the system), then the critical time for a system to settle down to a dynamic equilibrium is

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divide up simulation space into a three dimensional grid onto which the mass density of the particles is interpolated. Then calculating the potential becomes a matter of solving a

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are hybrid methods that use the particle mesh approximation for distant particles, but use more accurate methods for close particles (within a few grid intervals). PM stands for

3583:-circles and closed loops echo the electromagnetic fluxes issued from the Sun and Jupiter. It is conjectured, contrary to Richard H. Batin's conjecture (see References), the two 4520:

is singular; it goes to infinity as the distance between two particles goes to zero. The gravitational potential may be "softened" to remove the singularity at small distances:

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four-body problem (also known as bicircular model) and it can be traced back to 1960 in a NASA report written by Su-Shu Huang. This formulation has been highly relevant in the

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are gravity sinks, in and where gravitational forces are zero, and the reason the Trojan planetoids are trapped there. The total amount of mass of the planetoids is unknown.

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Dr. Clarence Cleminshaw calculated the approximate position of the Solar System's barycenter, a result achieved mainly by combining only the masses of Jupiter and the Sun.

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was found numerically by C. Moore in 1993 and generalized and proven by A. Chenciner and R. Montgomery in 2000. Since then, many other choreographies have been found for

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Gelman, Harry (1968). "The second orthogonality conditions in the theory of proper and improper rotations: Derivation of the conditions and of their main consequences".

5295:, "General Scholium", page 372, last paragraph. Newton was well aware that his mathematical model did not reflect physical reality. This edition referenced is from the 3101: 5646:

at the barycenter of the system. In the case of a hyperbola it has the branch at the side of that focus. The two conics will be in the same plane. The type of conic (

4516:. These methods numerically integrate the differential equations of motion. Numerical integration for this problem can be a challenge for several reasons. First, the 3410:

has been the most studied. Many earlier attempts to understand the three-body problem were quantitative, aiming at finding explicit solutions for special situations.

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In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to

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in a fluid domain discretized onto particles which are then advected with the velocity at their centers. Because the fluid velocity and vorticity are related via a

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As has been shown by Siegel, collisions which involve more than two bodies cannot be regularized analytically, hence Sundman's regularization cannot be generalized.

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Board, John A. Jr.; Humphres, Christopher W.; Lambert, Christophe G.; Rankin, William T.; Toukmaji, Abdulnour Y. (1999). "Ewald and Multipole Methods for Periodic

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If the sum of both energies is zero, then they both trace out parabolas. As the distance between the bodies tends to infinity, their relative speed tends to zero.

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system, where the mass of the Sun is about 1000 times larger than the masses of Jupiter or Saturn. An approximate solution to the problem is to decompose it into

4659:{\displaystyle U_{\varepsilon }=\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\sqrt {\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}+\varepsilon ^{2}}}}} 658:

that "according to this Law all bodies must attract each other." This last statement, which implies the existence of gravitational interactive forces, is key.

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in which all masses moves along Keplerian trajectories (elliptical, circular, parabolic, or hyperbolic), with all trajectories having the same eccentricity

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in the system, that is none of the ratios of unperturbed Kepler frequencies is a rational number. Resonances appear as small denominators in the expansion.

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when the bodies are far apart. (This potential energy is always a negative value; energy of rotation of the bodies about their axes is not counted here)

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and the American Ephemeris and Nautical Almanac, prepared jointly by the Nautical Almanac Offices of the United Kingdom and the United States of America.

2785:. The purpose of this section is to relate the real complexity in calculating any planetary forces. Note in this Section also, several subjects, such as 4964:

has the same form as the gravitational potential, except that charges may be positive or negative, leading to repulsive as well as attractive forces.

4416:(the bodies' positions) remains finite. (In this mathematical sense, a "collision" means that two pointlike bodies have identical positions in space.) 2704: 3368: 8681: 5960:

Chenciner, Alain; Montgomery, Richard (November 2000). "A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses".

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is an initial configuration such that if the particles were all released with zero velocity, they would all collapse toward the center of mass

3658:-body problem in the case that one of the masses is much larger than all the others. A prototypical example of a planetary problem is the Sun– 3614:

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, most notably by

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While there are analytic solutions available for the classical (i.e. nonrelativistic) two-body problem and for selected configurations with

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of one of the bodies is negligible. For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see

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See Meirovitch's book: Chapters 11: "Problems in Celestial Mechanics"; 12; "Problem in Spacecraft Dynamics"; and Appendix A: "Dyadics".

4918: 5249:. A popularization of the historical events and bickering between those parties, but more importantly about the results they produced. 8749: 6479:-Body Problems". In Deuflhard, Peter; Hermans, Jan; Leimkuhler, Benedict; Mark, Alan E.; Reich, Sebastian; Skeel, Robert D. (eds.). 2562: 2503:{\displaystyle I=\sum _{i=1}^{n}m_{i}\mathbf {q} _{i}\cdot \mathbf {q} _{i}=\sum _{i=1}^{n}m_{i}\left\|\mathbf {q} _{i}\right\|^{2}} 7729: 7401: 3513:

Moulton's solution may be easier to visualize (and definitely easier to solve) if one considers the more massive body (such as the

624:-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach. 5201:

A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary

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representing the mass density that is coupled to a self-consistent Poisson equation representing the potential. It is a type of

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ensures that the simulation obeys Hamilton's equations to a high degree of accuracy and in particular that energy is conserved.

8522: 7782: 5576: 4922: 4914: 3439: 60:

Notes should be rewritten in a more consistent and formal style, and specifically linked to the corresponding references using

7253: 7103: 7065: 7057: 7023: 6969: 6942: 6865: 6827: 6804: 6777: 6685: 6529: 6506: 6418: 6391: 6372: 2125:{\displaystyle \mathbf {C} ={\frac {\displaystyle \sum _{i=1}^{n}m_{i}\mathbf {q} _{i}}{\displaystyle \sum _{i=1}^{n}m_{i}}}} 6483:. Lecture Notes in Computational Science and Engineering. Vol. 4. Berlin & Heidelberg: Springer. pp. 459–471. 8582: 6961: 3497:

solution (see figure below). An aside, see Meirovitch's book, pages 413–414 for his restricted three-body problem solution.

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Negri, Rodolfo B.; Prado, Antonio F. B. A. (2020). "Generalizing the Bicircular Restricted Four-Body Problem".

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The latter ones are called Painlevé's conjecture (no-collisions singularities). Their existence has been conjectured for

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condition, a system's state is invariant to time; otherwise, the first derivatives and all higher derivatives are zero.

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Cleminshaw, C. H.: "The Coming Conjunction of Jupiter and Saturn", 7 1960, Saturn, Jupiter, observe, conjunction.

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is analytically solvable only for the Kepler problem, in which one mass is assumed to be much larger than the other.

1622:{\displaystyle U=-\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}.} 86: 1210: 8817: 8577: 8457: 7876: 7245: 7831: 7798: 6735:

Féjoz, J. (2004). "Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman)".

5663: 684:, established a prize for anyone who could find the solution to the problem. The announcement was quite specific: 669:-body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces. 8874: 8802: 8542: 8296: 4877: 68: 7166: 5726:

Cleminshaw, C. H.: "The Scale of The Solar System", 7 1959, Solar system, scale, Jupiter, sun, size, light.

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A number of approximate methods have been developed that reduce the time complexity relative to direct methods:

2801:) are discussed on other Knowledge pages. Here though, these subjects are discussed from the perspective of the 8879: 8667: 8615: 8254: 8245: 7982: 6819: 5213:

must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the

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yield approximate analytic trajectories upon which the numerical integration can be a correction. The use of a

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is readily obtained from the differences between these two equations and after canceling common terms gives:

7564:

Gelman, Harry (October 1971). "A Note on the time dependence of the effective axis and angle of a rotation".

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The structure of singularities is more complicated in this case: other types of singularities may occur (see

3442:, in which the two massive bodies describe a circular orbit and are only fixed in a synodic reference frame). 3435: 8869: 8562: 8032: 5720:

Cleminshaw, C. H.: "Celestial Speeds", 4 1953, equation, Kepler, orbit, comet, Saturn, Mars, velocity.

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at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of

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Also English translation of 3rd (1726) edition by I. Bernard Cohen and Anne Whitman (Berkeley, CA, 1999).

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Crandall, Richard E. (1996). "Chapter 2: "Exploratory Computation"; Project 2.4.1: "Classical Physics"".

7042:(in Latin). Londini : Jussu Societatis Regiæ ac Typis Josephi Streater. Prostat apud plures Bibliopolas. 761: 502: 435: 8659: 3438:

is the special case in which two of the bodies are fixed in space (this should not be confused with the

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did in arriving at his three famous equations was curve-fit the apparent motions of the planets using

8827: 8625: 7411: 5638:(i.e., the barycenter) of the two bodies is considered to be at rest, then each body travels along a 5001: 4755: 55: 7772: 6843: 5311:, 2002 edition; is a copy from Daniel Adee's 1848 addition. Cohen also has translated new editions: 19:

This article is about the problem in classical mechanics. For the problem in quantum mechanics, see

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motions, in which three bodies of any masses move proportionately along a fixed straight line. The

676:-body problem was considered very important and challenging. Indeed, in the late 19th century King 573:

is considerably more difficult to solve due to additional factors like time and space distortions.

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yield three more constants of the motion. The last general constant of the motion is given by the

641:-body problem is unsolvable because of those gravitational interactive forces. Newton said in his 8807: 8797: 8620: 7928: 6259:"A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles" 5141:

Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the

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are the electrostatic counterpart to fast multipole method simulators. These are often used with

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techniques. This can provide fast solutions at the cost of higher error for short-range forces.

681: 8812: 8792: 8482: 8084: 8004: 7992: 7795:– with links to the original papers of Euler and Lagrange, and to translations, with discussion 6838: 6484: 4825: 4467:

has shown that for 4 or fewer bodies, the set of initial data giving rise to singularities has

3902: 2286: 2019: 1160: 789: 495: 218: 5221:, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book 4446: 8843: 8754: 8711: 8605: 8547: 8517: 8305: 8182: 8150: 8120: 8079: 8064: 7943: 7783:

A java applet to simulate the 3D movement of set of particles under gravitational interaction

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The Gravitational Million-Body Problem, A Multidisciplinary Approach to Star Cluster Dynamics

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Xia, Zhihong (May 1992). "The Existence of Noncollision Singularities in Newtonian Systems".

5218: 5050: 4941:-body problem has been on the gravitational problem. But there exist other systems for which 4781: 4686: 3717: 3711: 3446: 1980: 1677: 655: 403: 238: 146: 7477:

Crandall, Richard E. (1996). "Chapter 5: "Nonlinear & Complex Systems"; paragraph 5.1: "

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was discovered by Lagrange in 1772 in which three bodies are situated at the vertices of an

3351:'s data, and not curve-fitting their true circular motions about the Sun (see Figure). Both 8774: 8630: 8452: 8236: 8125: 8094: 8022: 7997: 7972: 7933: 7914: 7869: 7681: 7508: 7334: 7306: 7218: 6880: 6653: 6622: 6587: 6552: 6447: 6317: 6304:

Torres, C. E.; Parishani, H.; Ayala, O.; Rossi, L. F.; Wang, L.-P. (2013-07-15).

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computations to evaluate the potential energy over all pairs of particles, and thus have a

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The Suppressed Scientific Discoveries of Stephen Gray and John Flamsteed, Newton's Tyranny

8: 8769: 8492: 8287: 8027: 7720: 7350:

Xia, Zhihong (1992). "The Existence of Noncollision Singularities in Newtonian Systems".

7118: 5348:

For details of the serious error in Poincare's first submission see the article by Diacu.

5082: 5053:, the velocity can be solved in the same manner as gravitation and electrostatics: as an 4759: 3794: 3678: 3479: 2192: 1233: 541:. Solving this problem has been motivated by the desire to understand the motions of the 534: 318: 156: 7685: 7512: 7310: 7222: 6657: 6626: 6591: 6556: 6451: 6321: 6274: 6163: 6034: 5983: 5930: 5846: 5801: 5662:) is determined by finding the sum of the combined kinetic energy of two bodies and the 5488: 5430: 5375: 3697:

a kind of stability of the planetary problem: there exists a set of positive measure of

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Any discussion of planetary interactive forces has always started historically with the

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from 1938 to 1958 and as director from 1958 to 1969. Some publications by Cleminshaw:

5508: 5450: 5395: 5260: 4957: 4902: 4873: 3779: 3626: 3549: 3486: 3380: 3312: 2798: 2312: 2015: 1959:{\displaystyle T=\sum _{i=1}^{n}{\frac {\left\|\mathbf {p} _{i}\right\|^{2}}{2m_{i}}}.} 677: 593: 570: 391: 266: 7751: 7666:"On the Manifolds of Total Collapse Orbits and of Completely Parabolic Orbits for the 6459: 4762:

or other approximation of the potential. This allows for a reduction in complexity to

3871: 2661:

is zero. Then on average the total kinetic energy is half the total potential energy,

8532: 8430: 8360: 8115: 8069: 7987: 7819: 7739: 7694: 7665: 7466: 7342: 7322: 7276: 7249: 7189: 7126: 7099: 7053: 7043: 7037: 7019: 6965: 6938: 6921: 6913: 6892: 6884: 6861: 6823: 6800: 6773: 6756: 6727: 6681: 6525: 6518: 6502: 6467: 6414: 6403: 6387: 6368: 6333: 6286: 6237: 6172: 6147: 6058: 6046: 5942: 5862: 5512: 5454: 5399: 5241: 5122: 4997: 4961: 4480: 3867: 3682: 2361: 1990: 306: 243: 122: 50: 20: 6132: 6007: 3615: 2268:{\displaystyle \mathbf {A} =\sum _{i=1}^{n}\mathbf {q} _{i}\times \mathbf {p} _{i},} 698: 8884: 8764: 8512: 8444: 8208: 8170: 8044: 8014: 7967: 7848: 7803:

Parallel GPU N-body simulation program with fast stackless particles tree traversal

7689: 7644: 7516: 7361: 7314: 7226: 7181: 7151: 6996: 6744: 6715: 6661: 6630: 6595: 6560: 6494: 6455: 6325: 6278: 6167: 6124: 6120: 6081: 6038: 5987: 5934: 5850: 5813: 5805: 5643: 5492: 5434: 5379: 5117: 4984: 4806: 4468: 3875: 3553: 3522: 3308: 2816: 2794: 2782: 2776: 2196: 586: 558: 476: 425: 190: 134: 24: 7076: 4889:

In astrophysical systems with strong gravitational fields, such as those near the

4442: 3146:{\displaystyle \mathbf {\alpha } +{\frac {\eta }{r^{3}}}\mathbf {r} =\mathbf {0} } 8889: 8759: 8744: 8552: 8145: 8049: 8039: 7938: 7862: 7767: 7746: 7330: 7268: 6498: 5500: 5442: 5387: 5304: 5210: 5092: 5058: 4973: 4719: 4464: 3690: 3360: 3340: 706: 481: 386: 296: 271: 7499:

Eisele, John A.; Mason, Robert M. (1970). "Applied Matrix and Tensor Analysis".

6108: 5938: 5125:, a method for numerically obtaining trajectories of bodies in an N-body system. 5057:-body summation over all vorticity-containing particles. The summation uses the 604:

Knowing three orbital positions of a planet's orbit – positions obtained by Sir

8648: 8600: 8587: 8472: 8467: 8398: 8378: 8369: 7962: 7948: 7924: 7919: 7894: 7836: 6857: 5635: 5300: 5005: 4949: 3674: 3427: 2690: 2023: 1871: 609: 453: 369: 286: 211: 205: 200: 7231: 7202: 7064:

Ram, Parikshit; Lee, Dongryeol; March, William B.; Gray, Alexander G. (2009).

6748: 6635: 6610: 6600: 6575: 6565: 6540: 6329: 6282: 8858: 8502: 8497: 8416: 8059: 7977: 7756: 7326: 7280: 7193: 7047: 6925: 6337: 6290: 6148:"A global existence theorem for the four-body problem of Newtonian mechanics" 6050: 5639: 5206: 5102: 4988: 4890: 3895: 3639: 3619: 2276: 458: 291: 248: 5818: 4847:

can be used to increase accuracy in regions with large numbers of particles.

8706: 8567: 8477: 8351: 8334: 8192: 8089: 7957: 7724: 7288: 7033: 6896: 5946: 5468: 5263:(1905). "Discovery of gravitation, A.D. 1666". In Johnson, Rossiter (ed.). 5097: 4678: 4377: 3622: 3608: 3506: 3415: 3352: 724: 702: 605: 176: 166: 161: 61: 7734: 7130: 6021:

Qiu-Dong, Wang (1990-03-01). "The global solution of the N-body problem".

5669:

If the sum of the energies is negative, then they both trace out ellipses.

616:

gravitational interaction forces. Thus came the awareness and rise of the

576:

The classical physical problem can be informally stated as the following:

23:. For engineering problems and simulations involving many components, see 8689: 8572: 8407: 8177: 8157: 8074: 3600: 3348: 768:

moving under the influence of mutual gravitational attraction. Each mass

589:

has been completely solved and is discussed below, as well as the famous

538: 340: 7778:

Java applet simulating a stable solution to the equi-mass 3-body problem

7094:

Rosenberg, Reinhardt M. (1977). "Chapter 19: About Celestial Problems".

5358:

Babadzanjanz, L. K. (1979), "Existence of the continuations in the

5069:

equations for 3 components of disturbance velocities at the location of

3343:, dragging the Solar System and Earth along with it. What mathematician 8140: 7854: 7656: 7625:

https://web.archive.org/web/19990221123102/http://ftp.cica.indiana.edu/

7373: 7318: 7156: 7001: 6980: 6917: 6888: 6719: 6428:

Blanchet, Luc (2001). "On the two-body problem in general relativity".

6093: 6042: 5999: 5713:

states Clarence Cleminshaw (1902–1985) served as assistant director of

5496: 5438: 5383: 4980: 4894: 4867: 4325:

which is also known, and the Taylor series is constructed iteratively.

3845: 3815: 3803:

curve without collisions are called choreographies. A choreography for

3694: 3604: 2790: 420: 376: 335: 113: 7520: 6442: 6257:

Ayala, Orlando; Grabowski, Wojciech W.; Wang, Lian-Ping (2007-07-01).

5854: 3528: 8557: 7909: 7185: 5974: 5659: 5046: 4840: 3501: 3431: 7648: 7365: 7139: 6085: 5991: 4496:-body problems must be solved or simulated using numerical methods. 3862:), sometimes reference is made to "the impossibility of solving the 6382:

Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996).

5898:

See Chierchia 2010 for animations illustrating homographic motions.

5809: 5655: 4697:

Direct methods using numerical integration require on the order of

3841: 3775: 3331: 665:'s non-Newtonian first and second Principles and to the nonlinear 533:

is the problem of predicting the individual motions of a group of

8462: 7802: 6644:

Cohen, I. Bernard (March 1980). "Newton's Discovery of Gravity".

6306:"Analysis and parallel implementation of a forced N-body problem" 5651: 5299:, Volume 34, which was translated by Andrew Motte and revised by 4953: 4744:

factor makes large-scale calculations especially time-consuming.

3659: 3518: 2786: 2311:

of degree 2 and −1, respectively, the equations of motion have a

523: 5917:

Moore, Cristopher (1993-06-14). "Braids in classical dynamics".

5413:

Babadzanjanz, L. K. (1993), "On the global solution of the

4945:-body mathematics and simulation techniques have proven useful. 3540:

math model figure above (after Moulton), the Lagrangian points L

3517:) to be stationary in space, and the less massive body (such as 3389:-body problem solution after simplifying assumptions were made. 7447:

An Introduction to The Mathematics and Methods of Astrodynamics

5647: 3663: 3344: 550: 6791:-body Problem of Stellar Dynamics". In Roy, A. E. (ed.). 2756:{\displaystyle t_{\mathrm {cr} }={\sqrt {\frac {GM}{R^{3}}}}.} 8267: 7886: 3474:

A major study of the Earth–Moon–Sun system was undertaken by

141: 7267:-body problem on a Hilbert space of analytic functions". In 6481:

Computational Molecular Dynamics: Challenges, Methods, Ideas

3677:, treating interactions among the planets as perturbations. 7740:

Applet demonstrating chaos in restricted three-body problem

3848:(unlike in the case of friction) gives a solution as well. 3596: 554: 546: 7494:(corrected 3rd ed.). Springer-Verlag. pp. 93–97. 7066:"Linear-time Algorithms for Pairwise Statistical Problems" 6520:

The Western Intellectual Tradition, from Leonardo to Hegel

6474: 4872:

approximate the system of particles with a time-dependent

6401:

Bate, Roger R.; Mueller, Donald D.; White, Jerry (1971).

6303: 5034:

algorithms, that have applicability to the gravitational

4851: 3607:. Specific solutions to the three-body problem result in 3514: 705:. The problem as stated originally was finally solved by 542: 7735:

Regular Keplerian motions in classical many-body systems

7451:

Employs energy methods rather than a Newtonian approach.

7174:

Lilliad - Université de Lille - Sciences et Technologies

5618: 4369:

Lastly, Sundman's result was generalized to the case of

4362: 4328: 3840:

For every solution of the problem, not only applying an

7752:

Applets demonstrating many different three-body motions

6232:

Cottet, Georges-Henri; Koumoutsakos, Petros D. (2000).

4456:

have been constructed by Xia and a heuristic model for

4419:

singularities in which a collision does not occur, but

3505:

Motion of three particles under gravity, demonstrating

6954:

Introduction to Hamiltonian Dynamical Systems and the

4388: 3878:

or higher by means of formulas only involving roots).

8691:

Numerical methods for ordinary differential equations

7808: 6109:"Noncollision Singularities: Do Four Bodies Suffice?" 5267:. Vol. XII. The National Alumni. pp. 51–65. 4526: 4333:

In order to generalize Sundman's result for the case

3911: 3104: 2825: 2707: 2565: 2374: 2279:. The three components of the total angular momentum 2205: 2087: 2042: 2031: 1879: 1840: 1690: 1512: 1245: 856: 557:. In the 20th century, understanding the dynamics of 5135: 7730:

More detailed information on the three-body problem

6978: 6430:

Comptes Rendus de l'Académie des Sciences, Série IV

5959: 5239:Clark, David H.; Clark, Stephen P. H. (2001). 3782:created by fixing the first integrals of a system. 3611:motion with no obvious sign of a repetitious path. 3521:) to orbit around it, with the equilibrium points ( 672:The problem of finding the general solution of the 7465: 6905: 6517: 6402: 6381: 6256: 6231: 5240: 4658: 4144: 3145: 3070: 2755: 2620: 2502: 2267: 2124: 1958: 1858: 1822: 1621: 1485: 1147: 7635:-body problem via Elementary Complex Variables". 7454: 7275:. New York: Gordon and Breach. pp. 569–578. 6854:Introduction to the Mechanics of the Solar System 5876: 5874: 5872: 5553: 5551: 5549: 2173:is the initial position. The constants of motion 8856: 7664:Saari, D. G.; Hulkower, N. D. (1981). 6874: 6515: 6400: 5752:See Leimanis and Minorsky's historical comments. 4405:collisions of two or more bodies, but for which 7552:Gelman, Harry (1969). "The Conjugacy Theorem". 7463: 7063: 6912:(3rd ed.). Princeton: D. Van Nostrand Co. 6675:The Birth of a New Physics, Revised and Updated 6608: 5317:Isaac Newton's Principia, with Variant Readings 4397:There can be two types of singularities of the 3748:. Central configurations may also give rise to 832:says that the gravitational force felt on mass 828:is equal to the sum of the forces on the mass. 7540:Gelman, Harry (1968). "The intrinsic vector". 6816:Exploring the Sun: Solar Science since Galileo 5869: 5611: 5609: 5607: 5605: 5603: 5601: 5599: 5597: 5546: 4976:techniques are used to speed up computations. 3625:. In the restricted problem, there exist five 3595:The restricted three-body problem assumes the 3385:This section relates a historically important 2621:{\displaystyle {\frac {d^{2}I}{dt^{2}}}=2T-U.} 8675: 7870: 7663: 7604:Murray, Carl D.; Dermott, Stanley F. (2000). 7603: 5577:"AST1100 Lecture Notes: 5 The virial theorem" 5156: 661:As shown below, the problem also conforms to 503: 7631:Saari, D. (1990). "A visit to the Newtonian 7098:. New York: Plenum Press. pp. 364–371. 7039:Philosophiae Naturalis Principia Mathematica 5412: 5357: 7581:. Vol. I, II pt 1, II pt 2. MIT Press. 7498: 7464:Brouwer, Dirk; Clemence, Gerald M. (1961). 7298:Celestial Mechanics and Dynamical Astronomy 6384:Chaos: An Introduction to Dynamical Systems 6023:Celestial Mechanics and Dynamical Astronomy 5594: 5477:Celestial Mechanics and Dynamical Astronomy 5419:Celestial Mechanics and Dynamical Astronomy 4733:. For simulations with many particles, the 3335:Real motion versus Kepler's apparent motion 3080:The equation describing the motion of mass 2296:-body problem has ten integrals of motion. 8682: 8668: 8644: 7877: 7863: 7262: 6952:Meyer, Kenneth Ray; Hall, Glen R. (2009). 6932: 6836: 6609:Chierchia, Luigi; Mather, John N. (2010). 6516:Bronowski, Jacob; Mazlish, Bruce (1986) . 5835:Journal of Guidance, Control, and Dynamics 5832: 5765:for its analytical and graphical solution. 5634:For the classical approach, if the common 5238: 5008:. Alternative optimizations to reduce the 3799:Solutions in which all masses move on the 2797:, etc.; and in the following Section too ( 510: 496: 16:Problem in physics and celestial mechanics 7788:Javascript Simulation of our Solar System 7702: 7693: 7483:Topics in Advanced Scientific Computation 7430:Learn how and when to remove this message 7239: 7230: 7200: 7164: 7155: 7093: 7000: 6951: 6842: 6787:Heggie, Douglas C. (1991). "Chaos in the 6763: 6634: 6599: 6573: 6564: 6488: 6441: 6171: 5973: 5817: 5303:. This same paragraph is on page 1160 in 4901:-body simulations must take into account 4880:approximation suitable for large systems. 3705: 3392:In the past not much was known about the 3355:and Newton were well aware that Newton's 3339:The Sun wobbles as it rotates around the 1187:is the magnitude of the distance between 87:Learn how and when to remove this message 7884: 7612: 7576: 7489: 7476: 7273:Analytic Methods in Mathematical Physics 7140:"Mémoire sur le problème de trois corps" 7096:Analytical Dynamics, of Discrete Systems 6822:, sponsored by the NASA History Office. 6813: 6538: 6427: 6236:. Cambridge, UK: Cambridge Univ. Press. 6020: 5331: 5329: 5259: 5162:See references cited for Heggie and Hut. 4928: 3881: 3527: 3500: 3330: 7599:. Vol. 1–2. John Wiley & Sons. 7585: 7180:. Paris: Gauthier-Villars Et Fils: 27. 7137: 7112: 7009: 6903: 6793:Predictability, Stability and Chaos in 6361:-body Simulations, Tools and Algorithms 6354: 4211:are given as initial conditions, every 2191:represent six integrals of the motion. 2132:moving with constant velocity, so that 8857: 8523:Transposition, docking, and extraction 7594: 7563: 7551: 7539: 7527: 7444: 7032: 7012:An Introduction to Celestial Mechanics 6786: 6386:. New York: Springer. pp. 46–48. 6106: 4923:two-body problem in general relativity 4915:parameterized post-Newtonian formalism 4809:on the grid, which can be computed in 3835: 3785: 3493:(see references) with its plot of the 3491:An Introduction to Celestial Mechanics 3440:circular restricted three-body problem 3369:Kepler's first law of planetary motion 3363:. Newton stated (in section 11 of the 730: 8663: 7858: 7630: 6962:Springer Science & Business Media 6851: 6734: 6694: 6672: 6643: 6145: 5916: 5783: 5737: 5326: 5265:The Great Events by Famous Historians 4905:; such simulations are the domain of 4884: 4329:A generalized Sundman global solution 3851:In the physical literature about the 3763:corresponds to homothetic motion and 3374: 3311:can be solved exactly, such as using 7619:-Body Simulation: 16 bit DOS version 7485:. Springer-Verlag. pp. 215–221. 7455:Boccaletti, D.; Pucacco, G. (1998). 7386: 7291:(1991). "The global solution of the 7287: 5471:(1991), "The global solution of the 5467: 4952:problems, such as the simulation of 3645: 3632: 33: 7773:Java applet simulating Solar System 7349: 7201:Trenti, Michele; Hut, Piet (2008). 6985:-body problem (Prize Announcement)" 6764:Heggie, Douglas; Hut, Piet (2003). 6541:"Computational celestial mechanics" 6234:Vortex Methods: Theory and Practice 6071: 5786:"Very Restricted Four-Body Problem" 5108:Numerical model of the Solar System 3901:We start by defining the system of 3874:about the impossibility of solving 3681:works well as long as there are no 2770: 1968:Hamilton's equations show that the 1228:Summing over all masses yields the 13: 7492:Projects in Scientific Computation 7382: 7242:Essays in the History of Mechanics 7165:Tisserand, François Félix (1894). 6666:10.1038/scientificamerican0381-166 5313:Introduction to Newton's Principia 5217:-body problem may be solved using 4919:Einstein–Infeld–Hoffmann equations 4508:-body problem can be solved using 3975: 3876:algebraic equations of degree five 2717: 2714: 1799: 1791: 1734: 1726: 1465: 1457: 1307: 792:says that mass times acceleration 14: 8901: 8583:Kepler's laws of planetary motion 7713: 7674:Journal of Differential Equations 7263:Van Winter, Clasine (1970). "The 6937:. New York: McGraw-Hill Book Co. 4504:For a small number of bodies, an 4449:). Examples of this behavior for 4354:) one has to face two obstacles: 3886:One way of solving the classical 3322:stated in reference to his work: 2000:initial position coordinates and 561:star systems became an important 282:Kepler's laws of planetary motion 8818:Backward differentiation formula 8643: 8578:Interplanetary Transport Network 8458:Collision avoidance (spacecraft) 7842: 7830: 7818: 7391: 6979:Mittag-Leffler, G. (1885–1886). 6877:Dynamics and Nonlinear Mechanics 6611:"Kolmogorov–Arnold–Moser Theory" 6310:Journal of Computational Physics 6263:Journal of Computational Physics 5686:For this approach see Lindsay's 5297:Great Books of the Western World 5188:R. M. Rosenberg states the 4861:particle–particle, particle–mesh 4619: 4604: 4107: 4083: 4050: 4026: 3927: 3139: 3131: 3042: 3027: 2961: 2923: 2908: 2842: 2765: 2480: 2429: 2414: 2252: 2237: 2207: 2076: 2033: 1916: 1804: 1761: 1739: 1699: 1599: 1584: 1470: 1425: 1410: 1386: 1371: 1271: 1119: 1104: 1080: 1065: 1012: 997: 975: 960: 925: 910: 859: 112: 38: 8543:Astronomical coordinate systems 8297:Longitude of the ascending node 6297: 6250: 6225: 6216: 6207: 6198: 6189: 6180: 6139: 6100: 6065: 6014: 5953: 5910: 5901: 5892: 5883: 5826: 5777: 5768: 5755: 5746: 5738:Brush, Stephen G., ed. (1983). 5731: 5703: 5693: 5680: 5628: 5569: 5560: 5537: 5528: 5519: 5461: 5406: 5351: 5342: 5045:called Vortex Methods sees the 4913:is extremely challenging and a 4878:smoothed-particle hydrodynamics 3756:. For elliptical trajectories, 3058: 2939: 2634:, the longterm time average of 1752: 723:by L. K. Babadzanjanz and 8616:Retrograde and prograde motion 7613:Quadling, Henley (June 1994). 7588:The Mechanics of Guided Bodies 7481:-body problems & chaos"". 7468:Methods of Celestial Mechanics 6935:Methods of Analytical Dynamics 6904:Lindsay, Robert Bruce (1961). 6820:Johns Hopkins University Press 6708:The Mathematical Intelligencer 6125:10.1080/10586458.2003.10504491 5281: 5271: 5252: 5232: 5195: 5182: 5165: 5088:Gravitational two-body problem 4692: 4630: 4598: 4123: 4117: 4099: 4093: 4066: 4060: 4042: 4036: 3943: 3937: 3450:solutions led to the study of 3052: 3022: 2933: 2903: 2693:for gravitational systems. If 2490: 2475: 2022:of the problem results in the 1926: 1911: 1678:Hamilton's equations of motion 1610: 1578: 1436: 1404: 1130: 1098: 1023: 991: 936: 904: 1: 7637:American Mathematical Monthly 7608:. Cambridge University Press. 7414:and help improve the section. 7167:"Traité de Mécanique Céleste" 6737:Ergodic Theory Dynam. Systems 6539:Celletti, Alessandra (2008). 6460:10.1016/s1296-2147(01)01267-7 6405:Fundamentals of Astrodynamics 6347: 5763:Restricted Three-body Problem 5113:Stability of the Solar System 5030:have been developed, such as 4909:. Numerically simulating the 4499: 4474: 3538:restricted three-body problem 3532:Restricted three-body problem 3495:restricted three-body problem 2689:, which is an example of the 1972:-body problem is a system of 8563:Equatorial coordinate system 7695:10.1016/0022-0396(81)90051-6 7586:Korenev, G. V. (1967). 7566:J. Res. NBS 72B (Math. Sci.) 7554:J. Res. NBS 72B (Math. Sci.) 7542:J. Res. NBS 72B (Math. Sci.) 7530:J. Res. NBS 72B (Math. Sci.) 7240:Truesdell, Clifford (1968). 7138:Sundman, K. F. (1912). 7010:Moulton, Forest Ray (1970). 6933:Meirovitch, Leonard (1970). 6499:10.1007/978-3-642-58360-5_27 6173:10.1016/0022-0396(77)90100-0 5247:. W. H. Freeman and Co. 5043:Computational fluid dynamics 4972:on the region simulated and 4970:periodic boundary conditions 3357:Law of Universal Gravitation 3315:relative to the barycenter. 3294:It is incorrect to think of 1631:Defining the momentum to be 537:interacting with each other 7: 8803:List of Runge–Kutta methods 7623:nbody*.zip is available at 7595:Meriam, J. L. (1978). 7445:Battin, Richard H. (1987). 7271:; Newton, Roger G. (eds.). 7075:: 1527–1535. Archived from 6679:W. W. Norton & Co. 6355:Aarseth, Sverre J. (2003). 5939:10.1103/PhysRevLett.70.3675 5076: 4956:and cellular assemblies in 4921:, is used if possible. The 3844:or a time shift but also a 3772:relative equilibrium motion 3489:published his now classic, 2815:) was completely solved by 2164:is the linear velocity and 2018:that simplify the problem. 2014:-body problem yield global 764:in three dimensional space 436:Tsiolkovsky rocket equation 58:. The specific problem is: 10: 8906: 8315:Longitude of the periapsis 7757:On the integration of the 7703:Szebehely, Victor (1967). 7115:The Nature of the Universe 6799:. New York: Plenum Press. 6770:Cambridge University Press 6673:Cohen, I. Bernard (1985). 6524:. New York: Dorset Press. 6365:Cambridge University Press 6107:Gerver, Joseph L. (2003). 5711:The Nature of the Universe 5319:, 1972. Cajori also wrote 5309:On the Shoulders of Giants 5209:, but in practice such an 4478: 4246:is known. Differentiating 3792: 3744:. Such a motion is called 3709: 3679:Perturbative approximation 3603:; for binary systems, see 3436:Euler's three-body problem 3378: 3176:is the vector position of 2774: 2331:is a solution, then so is 599: 405:Engineering and efficiency 224:Bi-elliptic transfer orbit 18: 8836: 8783: 8730: 8697: 8639: 8626:Specific angular momentum 8531: 8443: 8387: 8323: 8276: 8216: 8207: 8103: 8013: 7902: 7893: 7232:10.4249/scholarpedia.3930 6749:10.1017/S0143385704000410 6636:10.4249/scholarpedia.2123 6601:10.4249/scholarpedia.2111 6574:Chenciner, Alain (2007). 6566:10.4249/scholarpedia.4079 6330:10.1016/j.jcp.2013.03.008 6283:10.1016/j.jcp.2006.11.016 6152:J. Differential Equations 6146:Saari, Donald G. (1977). 5962:The Annals of Mathematics 5740:Maxwell on Saturn's Rings 5229:listed in the References. 5002:kernel density estimation 4514:particle–particle methods 3814:in the rotating frame. A 2368:-body system is given by 2007:initial momentum values. 654:Newton concluded via his 8865:Concepts in astrophysics 7113:Gallant, Roy A. (1968). 5205:can be approximated via 5129: 4911:Einstein field equations 4845:Adaptive mesh refinement 3552:planetoids resided (see 762:inertial reference frame 739:-body problem considers 627:After Newton's time the 431:Propellant mass fraction 330:Gravitational influences 8808:Linear multistep method 8621:Specific orbital energy 6814:Hufbauer, Karl (1991). 6797:-Body Dynamical Systems 5919:Physical Review Letters 5335:See I. Bernard Cohen's 5038:-body problem as well. 4666:Second, in general for 4518:gravitational potential 3476:Charles-Eugène Delaunay 2557:Lagrange–Jacobi formula 830:Newton's law of gravity 663:Jean Le Rond D'Alembert 302:Specific orbital energy 8875:Computational problems 8813:General linear methods 8793:Exponential integrator 8033:Geostationary transfer 7244:. Berlin; Heidelberg: 7117:. In partnership with 6852:Kurth, Rudolf (1959). 5784:Huang, Su-Shu (1960). 4937:Most work done on the 4826:fast Fourier transform 4782:Fast multipole methods 4660: 4146: 4005: 3903:differential equations 3890:-body problem is "the 3706:Central configurations 3533: 3510: 3452:central configurations 3423:also Tisserand, 1894). 3336: 3329: 3147: 3072: 2808:The two-body problem ( 2757: 2697:is the total mass and 2622: 2504: 2462: 2401: 2287:conservation of energy 2269: 2234: 2126: 2108: 2063: 2020:Translational symmetry 1981:differential equations 1960: 1906: 1860: 1824: 1623: 1487: 1337: 1161:gravitational constant 1149: 777:has a position vector 695: 652: 583: 219:Hohmann transfer orbit 8880:Computational physics 8844:Symplectic integrator 8828:Gauss–Legendre method 8606:Orbital state vectors 8548:Characteristic energy 8518:Trans-lunar injection 8306:Argument of periapsis 7983:Prograde / Retrograde 7944:Hyperbolic trajectory 7627:: see external links. 7606:Solar System Dynamics 7597:Engineering Mechanics 7577:Hagihara, Y. (1970). 7353:Annals of Mathematics 6881:John Wiley & Sons 6697:"The solution of the 5700:energy by convention. 5543:Meyer 2009, pp. 28–29 5525:Meyer 2009, pp. 27–28 5219:numerical integration 4994:all-nearest-neighbors 4802:Particle mesh methods 4756:Barnes–Hut simulation 4687:symplectic integrator 4661: 4389:Singularities of the 4147: 3970: 3882:Power series solution 3718:central configuration 3712:Central configuration 3673:pairs of star–planet 3531: 3504: 3334: 3324: 3148: 3073: 2758: 2623: 2505: 2442: 2381: 2309:homogeneous functions 2270: 2214: 2195:results in the total 2127: 2088: 2043: 1961: 1886: 1861: 1859:{\displaystyle H=T+U} 1825: 1684:-body problem become 1624: 1488: 1302: 1150: 686: 647: 578: 415:Preflight engineering 147:Argument of periapsis 8785:Higher-order methods 8775:Leapfrog integration 8732:Second-order methods 8453:Bi-elliptic transfer 7973:Parabolic trajectory 6576:"Three body problem" 5715:Griffith Observatory 5582:. University of Oslo 4966:Fast Coulomb solvers 4907:numerical relativity 4524: 3909: 3812:equilateral triangle 3102: 2823: 2705: 2563: 2372: 2203: 2029: 1877: 1838: 1832:Hamiltonian function 1688: 1510: 1243: 854: 707:Karl Fritiof Sundman 682:Gösta Mittag-Leffler 471:Propulsive maneuvers 69:improve this article 54:to meet Knowledge's 29:Multibody simulation 8870:Classical mechanics 8798:Runge–Kutta methods 8770:Newmark-beta method 8717:Semi-implicit Euler 8699:First-order methods 8493:Low-energy transfer 7793:The Lagrange Points 7686:1981JDE....41...27S 7579:Celestial Mechanics 7513:1972PhT....25l..55E 7311:1991CeMDA..50...73W 7223:2008SchpJ...3.3930T 7121:. Garden City, NY: 6658:1981SciAm.244c.166C 6646:Scientific American 6627:2010SchpJ...5.2123C 6592:2007SchpJ...2.2111C 6557:2008SchpJ...3.4079C 6452:2001CRASP...2.1343B 6322:2013JCoPh.245..235T 6275:2007JCoPh.225...51A 6164:1977JDE....26...80S 6035:1990CeMDA..50...73W 5984:2000math.....11268C 5931:1993PhRvL..70.3675M 5847:2020JGCD...43.1173N 5802:1960AJ.....65S.347H 5489:1991CeMDA..50...73W 5431:1993CeMDA..56..427B 5376:1979CeMec..20...43B 5364:Celestial Mechanics 5337:Scientific American 5293:System of the World 5083:Celestial mechanics 5019:time complexity to 4987:, some models have 4917:(PPN), such as the 4760:multipole expansion 4447:Painlevé conjecture 4381:this new system is 3836:Analytic approaches 3795:n-body choreography 3780:invariant manifolds 3750:homographic motions 3480:perturbation theory 3019: 2900: 2632:dynamic equilibrium 2193:Rotational symmetry 2016:integrals of motion 1234:equations of motion 790:Newton's second law 731:General formulation 716:and generalized to 691:converges uniformly 656:third law of motion 565:-body problem. The 448:Efficiency measures 351:Sphere of influence 320:Celestial mechanics 102:Part of a series on 8755:Beeman's algorithm 8740:Verlet integration 8488:Inclination change 8136:Distant retrograde 7766:2016-10-30 at the 7745:2009-10-17 at the 7721:Three-Body Problem 7459:. Springer-Verlag. 7319:10.1007/BF00048987 7269:Gilbert, Robert P. 7207:-body simulations" 7157:10.1007/bf02422379 7016:Dover Publications 7002:10.1007/BF02402191 6908:Physical Mechanics 6720:10.1007/bf03024313 6695:Diacu, F. (1996). 6411:Dover Publications 6043:10.1007/BF00048987 5709:Science Program's 5688:Physical Mechanics 5497:10.1007/BF00048987 5439:10.1007/BF00691812 5384:10.1007/BF01236607 5323:, which is online. 5321:History of Science 5051:Poisson's equation 4958:structural biology 4903:general relativity 4885:Strong gravitation 4874:Boltzmann equation 4656: 4567: 4142: 3789:-body choreography 3683:orbital resonances 3627:equilibrium points 3534: 3511: 3487:Forest Ray Moulton 3464:for some constant 3396:-body problem for 3381:Three-body problem 3375:Three-body problem 3337: 3313:Jacobi coordinates 3143: 3068: 3066: 3005: 2886: 2799:Three-body problem 2753: 2618: 2500: 2313:scaling invariance 2265: 2122: 2119: 2086: 2010:Symmetries in the 1991:initial conditions 1956: 1856: 1820: 1619: 1549: 1483: 1145: 678:Oscar II of Sweden 594:three-body problem 571:general relativity 267:Dynamical friction 8852: 8851: 8722:Exponential Euler 8657: 8656: 8631:Two-line elements 8439: 8438: 8361:Eccentric anomaly 8203: 8202: 8070:Orbit of the Moon 7929:Highly elliptical 7707:. Academic Press. 7521:10.1063/1.3071146 7472:. Academic Press. 7440: 7439: 7432: 7255:978-3-642-86649-4 7105:978-0-306-31014-0 7058:978-0-520-08817-7 7025:978-0-486-62563-8 6971:978-0-387-09724-4 6944:978-0-07-041455-6 6867:978-0-08-009141-9 6829:978-0-8018-4098-2 6806:978-0-306-44034-2 6779:978-0-521-77303-4 6687:978-0-393-30045-1 6531:978-0-88029-069-2 6508:978-3-540-63242-9 6420:978-0-486-60061-1 6393:978-0-387-94677-1 6374:978-0-521-43272-6 6076:. Second Series. 5925:(24): 3675–3679. 5855:10.2514/1.G004848 5566:Meyer 2009, p. 34 5534:Meyer 2009, p. 28 5227:-Body Simulations 5123:N-body simulation 4998:manifold learning 4962:Coulomb potential 4752:Tree code methods 4677:-body problem is 4654: 4653: 4540: 4481:n-body simulation 4137: 3998: 3962: 3894:-body problem by 3818:choreography for 3652:planetary problem 3646:Planetary problem 3633:Four-body problem 3523:Lagrangian points 3418:published in the 3128: 3089:relative to mass 3062: 3020: 2943: 2901: 2748: 2747: 2598: 2362:moment of inertia 2120: 1951: 1815: 1780: 1750: 1718: 1614: 1522: 1481: 1446: 1330: 1297: 1211:metric induced by 1140: 1027: 946: 841:by a single mass 569:-body problem in 535:celestial objects 520: 519: 370:Lagrangian points 307:Vis-viva equation 277:Kepler's equation 124:Orbital mechanics 97: 96: 89: 56:quality standards 47:This article may 21:Many-body problem 8897: 8750:Trapezoidal rule 8684: 8677: 8670: 8661: 8660: 8647: 8646: 8588:Lagrangian point 8483:Hohmann transfer 8428: 8414: 8405: 8396: 8376: 8367: 8358: 8349: 8345: 8341: 8332: 8312: 8303: 8294: 8285: 8265: 8261: 8252: 8243: 8234: 8214: 8213: 8183:Heliosynchronous 8132:Lagrange points 8085:Transatmospheric 7900: 7899: 7879: 7872: 7865: 7856: 7855: 7847: 7846: 7845: 7835: 7834: 7823: 7822: 7814: 7760: 7708: 7705:Theory of Orbits 7699: 7697: 7669: 7660: 7634: 7622: 7618: 7609: 7600: 7591: 7582: 7573: 7561: 7549: 7537: 7524: 7495: 7486: 7480: 7473: 7471: 7460: 7457:Theory of Orbits 7450: 7435: 7428: 7424: 7421: 7415: 7410:Please read the 7406:may need cleanup 7395: 7394: 7387: 7377: 7346: 7295:-body problem". 7294: 7284: 7266: 7259: 7236: 7234: 7206: 7197: 7171: 7161: 7159: 7144:Acta Mathematica 7134: 7109: 7090: 7088: 7087: 7081: 7070: 7051: 7029: 7006: 7004: 6989:Acta Mathematica 6984: 6975: 6957: 6948: 6929: 6911: 6900: 6871: 6848: 6846: 6833: 6810: 6796: 6790: 6783: 6760: 6743:(5): 1521–1582. 6731: 6705: 6700: 6691: 6669: 6640: 6638: 6605: 6603: 6570: 6568: 6535: 6523: 6512: 6492: 6478: 6471: 6445: 6436:(9): 1343–1352. 6424: 6408: 6397: 6378: 6360: 6342: 6341: 6301: 6295: 6294: 6254: 6248: 6247: 6229: 6223: 6220: 6214: 6211: 6205: 6202: 6196: 6193: 6187: 6184: 6178: 6177: 6175: 6143: 6137: 6136: 6104: 6098: 6097: 6069: 6063: 6062: 6018: 6012: 6011: 5977: 5957: 5951: 5950: 5914: 5908: 5905: 5899: 5896: 5890: 5887: 5881: 5878: 5867: 5866: 5841:(6): 1173–1179. 5830: 5824: 5823: 5821: 5819:2060/19890068606 5781: 5775: 5772: 5766: 5759: 5753: 5750: 5744: 5743: 5735: 5729: 5707: 5701: 5697: 5691: 5684: 5678: 5664:potential energy 5632: 5626: 5622: 5616: 5613: 5592: 5591: 5589: 5587: 5581: 5573: 5567: 5564: 5558: 5555: 5544: 5541: 5535: 5532: 5526: 5523: 5517: 5515: 5475:-body problem", 5465: 5459: 5457: 5417:-body problem", 5410: 5404: 5402: 5362:-body problem", 5355: 5349: 5346: 5340: 5333: 5324: 5285: 5279: 5275: 5269: 5268: 5256: 5250: 5248: 5246: 5236: 5230: 5226: 5216: 5204: 5199: 5193: 5191: 5186: 5180: 5169: 5163: 5160: 5154: 5153:-body Problem"). 5152: 5148: 5144: 5139: 5118:Few-body systems 5072: 5068: 5064: 5056: 5037: 5029: 5018: 4985:machine learning 4944: 4940: 4932: 4900: 4838: 4823: 4807:Poisson equation 4796: 4776: 4743: 4732: 4717: 4713: 4711: 4710: 4707: 4704: 4676: 4672: 4665: 4663: 4662: 4657: 4655: 4652: 4651: 4639: 4638: 4633: 4629: 4628: 4627: 4622: 4613: 4612: 4607: 4595: 4594: 4593: 4592: 4583: 4582: 4569: 4566: 4536: 4535: 4507: 4495: 4491: 4462: 4455: 4440: 4429: 4415: 4400: 4392: 4384: 4375: 4353: 4346: 4339: 4324: 4315: 4314: 4312: 4311: 4306: 4303: 4280: 4279: 4277: 4276: 4271: 4268: 4245: 4244: 4242: 4241: 4236: 4233: 4210: 4209: 4207: 4206: 4201: 4198: 4172: 4151: 4149: 4148: 4143: 4138: 4136: 4135: 4130: 4126: 4116: 4115: 4110: 4092: 4091: 4086: 4074: 4073: 4069: 4059: 4058: 4053: 4035: 4034: 4029: 4018: 4017: 4007: 4004: 3999: 3997: 3986: 3963: 3961: 3960: 3959: 3946: 3936: 3935: 3930: 3924: 3923: 3913: 3893: 3889: 3865: 3861: 3854: 3846:reversal of time 3831: 3824: 3809: 3788: 3769: 3762: 3755: 3743: 3737: 3672: 3657: 3591: 3582: 3573: 3564: 3554:Lagrangian point 3470: 3463: 3409: 3402: 3395: 3388: 3309:two-body problem 3302: 3289: 3284: 3282: 3281: 3276: 3273: 3255: 3228: 3227: 3225: 3224: 3219: 3216: 3199: 3193: 3184: 3175: 3152: 3150: 3149: 3144: 3142: 3134: 3129: 3127: 3126: 3114: 3109: 3097: 3088: 3077: 3075: 3074: 3069: 3067: 3063: 3061:Earth–Sun 3060: 3056: 3051: 3050: 3045: 3036: 3035: 3030: 3021: 3018: 3013: 3004: 3003: 3002: 2993: 2992: 2979: 2970: 2969: 2964: 2958: 2957: 2944: 2942:Sun–Earth 2941: 2937: 2932: 2931: 2926: 2917: 2916: 2911: 2902: 2899: 2894: 2885: 2884: 2883: 2874: 2873: 2860: 2851: 2850: 2845: 2839: 2838: 2817:Johann Bernoulli 2814: 2804: 2783:two-body problem 2777:Two-body problem 2771:Two-body problem 2762: 2760: 2759: 2754: 2749: 2746: 2745: 2736: 2728: 2727: 2722: 2721: 2720: 2700: 2696: 2688: 2682: 2680: 2679: 2676: 2673: 2660: 2658: 2656: 2655: 2650: 2647: 2627: 2625: 2624: 2619: 2599: 2597: 2596: 2595: 2582: 2578: 2577: 2567: 2554: 2553: 2551: 2550: 2545: 2542: 2534: 2532: 2531: 2528: 2525: 2509: 2507: 2506: 2501: 2499: 2498: 2493: 2489: 2488: 2483: 2472: 2471: 2461: 2456: 2438: 2437: 2432: 2423: 2422: 2417: 2411: 2410: 2400: 2395: 2367: 2356: 2349: 2330: 2306: 2302: 2295: 2291: 2284: 2274: 2272: 2271: 2266: 2261: 2260: 2255: 2246: 2245: 2240: 2233: 2228: 2210: 2197:angular momentum 2190: 2181: 2172: 2163: 2154: 2131: 2129: 2128: 2123: 2121: 2118: 2117: 2107: 2102: 2085: 2084: 2079: 2073: 2072: 2062: 2057: 2041: 2036: 2013: 2006: 1999: 1989: 1978: 1971: 1965: 1963: 1962: 1957: 1952: 1950: 1949: 1948: 1935: 1934: 1929: 1925: 1924: 1919: 1908: 1905: 1900: 1869: 1865: 1863: 1862: 1857: 1829: 1827: 1826: 1821: 1816: 1814: 1813: 1812: 1807: 1797: 1789: 1781: 1779: 1771: 1770: 1769: 1764: 1754: 1751: 1749: 1748: 1747: 1742: 1732: 1724: 1719: 1717: 1709: 1708: 1707: 1702: 1692: 1683: 1675: 1674: 1672: 1671: 1666: 1663: 1628: 1626: 1625: 1620: 1615: 1613: 1609: 1608: 1607: 1602: 1593: 1592: 1587: 1576: 1575: 1574: 1565: 1564: 1551: 1548: 1499: 1492: 1490: 1489: 1484: 1482: 1480: 1479: 1478: 1473: 1463: 1455: 1447: 1445: 1444: 1439: 1435: 1434: 1433: 1428: 1419: 1418: 1413: 1401: 1400: 1396: 1395: 1394: 1389: 1380: 1379: 1374: 1363: 1362: 1353: 1352: 1339: 1336: 1331: 1329: 1318: 1298: 1296: 1295: 1294: 1281: 1280: 1279: 1274: 1268: 1267: 1257: 1255: 1254: 1231: 1222: 1208: 1197: 1186: 1184: 1158: 1154: 1152: 1151: 1146: 1141: 1139: 1138: 1133: 1129: 1128: 1127: 1122: 1113: 1112: 1107: 1095: 1094: 1090: 1089: 1088: 1083: 1074: 1073: 1068: 1057: 1056: 1047: 1046: 1033: 1028: 1026: 1022: 1021: 1020: 1015: 1006: 1005: 1000: 989: 985: 984: 983: 978: 969: 968: 963: 952: 947: 945: 944: 939: 935: 934: 933: 928: 919: 918: 913: 901: 900: 899: 890: 889: 876: 871: 870: 862: 849: 840: 827: 826: 824: 823: 818: 815: 787: 776: 767: 759: 742: 738: 722: 715: 675: 668: 645:, paragraph 21: 640: 630: 623: 619: 608:from astronomer 587:two-body problem 568: 564: 559:globular cluster 530: 512: 505: 498: 477:Orbital maneuver 426:Payload fraction 406: 387:Lissajous orbits 321: 292:Orbital velocity 239:Hyperbolic orbit 135:Orbital elements 125: 116: 99: 98: 92: 85: 81: 78: 72: 42: 41: 34: 25:Multibody system 8905: 8904: 8900: 8899: 8898: 8896: 8895: 8894: 8855: 8854: 8853: 8848: 8832: 8779: 8760:Midpoint method 8745:Velocity Verlet 8726: 8693: 8688: 8658: 8653: 8635: 8553:Escape velocity 8534: 8527: 8508:Rocket equation 8435: 8427: 8421: 8412: 8403: 8394: 8383: 8374: 8365: 8356: 8347: 8343: 8339: 8330: 8319: 8310: 8301: 8292: 8283: 8272: 8263: 8259: 8255:Semi-minor axis 8250: 8246:Semi-major axis 8241: 8232: 8226: 8199: 8121:Areosynchronous 8105: 8099: 8080:Sun-synchronous 8065:Near-equatorial 8009: 7889: 7883: 7853: 7843: 7841: 7829: 7817: 7809: 7807: 7768:Wayback Machine 7761:-body equations 7758: 7747:Wayback Machine 7716: 7711: 7667: 7649:10.2307/2323910 7632: 7616: 7562: 7550: 7538: 7478: 7436: 7425: 7419: 7416: 7409: 7402:Further reading 7396: 7392: 7385: 7383:Further reading 7380: 7366:10.2307/2946572 7292: 7264: 7256: 7246:Springer-Verlag 7204: 7169: 7119:Science Service 7106: 7085: 7083: 7079: 7068: 7026: 6982: 6972: 6955: 6945: 6868: 6844:10.1.1.163.3549 6830: 6807: 6794: 6788: 6780: 6703: 6698: 6688: 6532: 6509: 6476: 6421: 6394: 6375: 6358: 6350: 6345: 6302: 6298: 6255: 6251: 6244: 6230: 6226: 6221: 6217: 6212: 6208: 6204:Krumscheid 2010 6203: 6199: 6194: 6190: 6185: 6181: 6144: 6140: 6105: 6101: 6086:10.2307/2946572 6070: 6066: 6019: 6015: 5992:10.2307/2661357 5958: 5954: 5915: 5911: 5906: 5902: 5897: 5893: 5888: 5884: 5879: 5870: 5831: 5827: 5782: 5778: 5773: 5769: 5760: 5756: 5751: 5747: 5736: 5732: 5708: 5704: 5698: 5694: 5685: 5681: 5633: 5629: 5623: 5619: 5614: 5595: 5585: 5583: 5579: 5575: 5574: 5570: 5565: 5561: 5556: 5547: 5542: 5538: 5533: 5529: 5524: 5520: 5466: 5462: 5411: 5407: 5356: 5352: 5347: 5343: 5334: 5327: 5305:Stephen Hawkins 5286: 5282: 5276: 5272: 5261:Brewster, David 5257: 5253: 5237: 5233: 5224: 5214: 5211:infinite series 5202: 5200: 5196: 5189: 5187: 5183: 5170: 5166: 5161: 5157: 5150: 5146: 5142: 5140: 5136: 5132: 5093:Jacobi integral 5079: 5070: 5066: 5062: 5059:Biot-Savart law 5054: 5041:A technique in 5035: 5020: 5009: 5006:kernel machines 4974:Ewald summation 4948:In large scale 4942: 4938: 4935: 4930: 4898: 4887: 4857:PM-tree methods 4829: 4810: 4787: 4763: 4734: 4723: 4720:time complexity 4708: 4705: 4702: 4701: 4699: 4698: 4695: 4674: 4667: 4647: 4643: 4634: 4623: 4618: 4617: 4608: 4603: 4602: 4601: 4597: 4596: 4588: 4584: 4578: 4574: 4570: 4568: 4544: 4531: 4527: 4525: 4522: 4521: 4505: 4502: 4493: 4486: 4483: 4477: 4465:Donald G. Saari 4457: 4450: 4435: 4420: 4406: 4401:-body problem: 4398: 4395: 4390: 4382: 4370: 4348: 4341: 4334: 4331: 4323: 4317: 4307: 4304: 4297: 4286: 4285: 4283: 4282: 4272: 4269: 4262: 4251: 4250: 4248: 4247: 4237: 4234: 4227: 4216: 4215: 4213: 4212: 4202: 4199: 4196: 4189: 4178: 4177: 4175: 4174: 4170: 4163: 4155: 4131: 4111: 4106: 4105: 4087: 4082: 4081: 4080: 4076: 4075: 4054: 4049: 4048: 4030: 4025: 4024: 4023: 4019: 4013: 4009: 4008: 4006: 4000: 3987: 3976: 3974: 3955: 3951: 3947: 3931: 3926: 3925: 3919: 3915: 3914: 3912: 3910: 3907: 3906: 3891: 3887: 3884: 3863: 3856: 3855:-body problem ( 3852: 3838: 3826: 3819: 3804: 3797: 3791: 3786: 3764: 3757: 3753: 3739: 3735: 3726: 3720: 3714: 3708: 3691:Vladimir Arnold 3675:Kepler problems 3667: 3655: 3648: 3635: 3590: 3584: 3580: 3577: 3572: 3566: 3565:is the Sun and 3563: 3557: 3547: 3543: 3465: 3455: 3404: 3397: 3393: 3386: 3383: 3377: 3341:Galactic Center 3320:Science Program 3301: 3295: 3277: 3274: 3269: 3268: 3266: 3261: 3253: 3246: 3232: 3220: 3217: 3209: 3208: 3206: 3205: 3197: 3192: 3186: 3183: 3177: 3174: 3167: 3157: 3138: 3130: 3122: 3118: 3113: 3105: 3103: 3100: 3099: 3096: 3090: 3087: 3081: 3065: 3064: 3059: 3055: 3046: 3041: 3040: 3031: 3026: 3025: 3014: 3009: 2998: 2994: 2988: 2984: 2980: 2978: 2971: 2965: 2960: 2959: 2953: 2949: 2946: 2945: 2940: 2936: 2927: 2922: 2921: 2912: 2907: 2906: 2895: 2890: 2879: 2875: 2869: 2865: 2861: 2859: 2852: 2846: 2841: 2840: 2834: 2830: 2826: 2824: 2821: 2820: 2809: 2805:-body problem. 2802: 2779: 2773: 2768: 2741: 2737: 2729: 2726: 2713: 2712: 2708: 2706: 2703: 2702: 2698: 2694: 2677: 2674: 2671: 2670: 2668: 2662: 2651: 2648: 2640: 2639: 2637: 2635: 2630:For systems in 2591: 2587: 2583: 2573: 2569: 2568: 2566: 2564: 2561: 2560: 2546: 2543: 2538: 2537: 2535: 2529: 2526: 2523: 2522: 2520: 2515: 2494: 2484: 2479: 2478: 2474: 2473: 2467: 2463: 2457: 2446: 2433: 2428: 2427: 2418: 2413: 2412: 2406: 2402: 2396: 2385: 2373: 2370: 2369: 2365: 2351: 2343: 2332: 2324: 2316: 2304: 2300: 2293: 2292:. Hence, every 2289: 2280: 2275:where × is the 2256: 2251: 2250: 2241: 2236: 2235: 2229: 2218: 2206: 2204: 2201: 2200: 2199:being constant 2189: 2183: 2180: 2174: 2171: 2165: 2162: 2156: 2153: 2143: 2133: 2113: 2109: 2103: 2092: 2080: 2075: 2074: 2068: 2064: 2058: 2047: 2040: 2032: 2030: 2027: 2026: 2011: 2001: 1994: 1984: 1973: 1969: 1944: 1940: 1936: 1930: 1920: 1915: 1914: 1910: 1909: 1907: 1901: 1890: 1878: 1875: 1874: 1867: 1839: 1836: 1835: 1808: 1803: 1802: 1798: 1790: 1788: 1772: 1765: 1760: 1759: 1755: 1753: 1743: 1738: 1737: 1733: 1725: 1723: 1710: 1703: 1698: 1697: 1693: 1691: 1689: 1686: 1685: 1681: 1667: 1664: 1662: 1651: 1650: 1648: 1646: 1640: 1632: 1603: 1598: 1597: 1588: 1583: 1582: 1581: 1577: 1570: 1566: 1560: 1556: 1552: 1550: 1526: 1511: 1508: 1507: 1497: 1494: 1474: 1469: 1468: 1464: 1456: 1454: 1440: 1429: 1424: 1423: 1414: 1409: 1408: 1407: 1403: 1402: 1390: 1385: 1384: 1375: 1370: 1369: 1368: 1364: 1358: 1354: 1348: 1344: 1340: 1338: 1332: 1319: 1308: 1306: 1290: 1286: 1282: 1275: 1270: 1269: 1263: 1259: 1258: 1256: 1250: 1246: 1244: 1241: 1240: 1229: 1221: 1215: 1207: 1199: 1196: 1188: 1183: 1174: 1166: 1164: 1156: 1134: 1123: 1118: 1117: 1108: 1103: 1102: 1101: 1097: 1096: 1084: 1079: 1078: 1069: 1064: 1063: 1062: 1058: 1052: 1048: 1042: 1038: 1034: 1032: 1016: 1011: 1010: 1001: 996: 995: 994: 990: 979: 974: 973: 964: 959: 958: 957: 953: 951: 940: 929: 924: 923: 914: 909: 908: 907: 903: 902: 895: 891: 885: 881: 877: 875: 863: 858: 857: 855: 852: 851: 847: 842: 838: 833: 819: 816: 814: 803: 802: 800: 798: 793: 786: 778: 774: 769: 765: 749: 744: 740: 736: 733: 717: 710: 673: 666: 638: 628: 621: 617: 602: 566: 562: 539:gravitationally 528: 516: 487: 486: 482:Orbit insertion 472: 464: 463: 449: 441: 440: 416: 408: 404: 397: 396: 392:Lyapunov orbits 383: 382: 366: 356: 355: 331: 323: 319: 312: 311: 297:Surface gravity 272:Escape velocity 262: 254: 253: 234:Parabolic orbit 230: 229: 196: 194: 191:two-body orbits 182: 181: 172:Semi-major axis 137: 127: 123: 93: 82: 76: 73: 66: 43: 39: 32: 17: 12: 11: 5: 8903: 8893: 8892: 8887: 8882: 8877: 8872: 8867: 8850: 8849: 8847: 8846: 8840: 8838: 8834: 8833: 8831: 8830: 8825: 8820: 8815: 8810: 8805: 8800: 8795: 8789: 8787: 8781: 8780: 8778: 8777: 8772: 8767: 8762: 8757: 8752: 8747: 8742: 8736: 8734: 8728: 8727: 8725: 8724: 8719: 8714: 8712:Backward Euler 8709: 8703: 8701: 8695: 8694: 8687: 8686: 8679: 8672: 8664: 8655: 8654: 8652: 8651: 8649:List of orbits 8640: 8637: 8636: 8634: 8633: 8628: 8623: 8618: 8613: 8608: 8603: 8601:Orbit equation 8598: 8590: 8585: 8580: 8575: 8570: 8565: 8560: 8555: 8550: 8545: 8539: 8537: 8529: 8528: 8526: 8525: 8520: 8515: 8510: 8505: 8500: 8495: 8490: 8485: 8480: 8475: 8473:Gravity assist 8470: 8468:Delta-v budget 8465: 8460: 8455: 8449: 8447: 8441: 8440: 8437: 8436: 8434: 8433: 8425: 8419: 8410: 8401: 8399:Orbital period 8391: 8389: 8385: 8384: 8382: 8381: 8379:True longitude 8372: 8370:Mean longitude 8363: 8354: 8337: 8327: 8325: 8321: 8320: 8318: 8317: 8308: 8299: 8290: 8280: 8278: 8274: 8273: 8271: 8270: 8257: 8248: 8239: 8229: 8227: 8225: 8224: 8221: 8217: 8211: 8205: 8204: 8201: 8200: 8198: 8197: 8196: 8195: 8187: 8186: 8185: 8180: 8175: 8174: 8173: 8160: 8155: 8154: 8153: 8148: 8143: 8138: 8130: 8129: 8128: 8126:Areostationary 8123: 8118: 8109: 8107: 8101: 8100: 8098: 8097: 8095:Very low Earth 8092: 8087: 8082: 8077: 8072: 8067: 8062: 8057: 8052: 8047: 8042: 8037: 8036: 8035: 8030: 8023:Geosynchronous 8019: 8017: 8011: 8010: 8008: 8007: 8005:Transfer orbit 8002: 8001: 8000: 7995: 7985: 7980: 7975: 7970: 7965: 7963:Lagrange point 7960: 7955: 7946: 7941: 7936: 7931: 7922: 7917: 7912: 7906: 7904: 7897: 7891: 7890: 7885:Gravitational 7882: 7881: 7874: 7867: 7859: 7852: 7851: 7839: 7827: 7806: 7805: 7800: 7796: 7790: 7785: 7780: 7775: 7770: 7754: 7749: 7737: 7732: 7727: 7717: 7715: 7714:External links 7712: 7710: 7709: 7700: 7670:-Body Problem" 7661: 7643:(2): 105–119. 7628: 7615:Gravitational 7610: 7601: 7592: 7583: 7574: 7525: 7496: 7487: 7474: 7461: 7452: 7441: 7438: 7437: 7399: 7397: 7390: 7384: 7381: 7379: 7378: 7360:(3): 411–468. 7347: 7285: 7260: 7254: 7237: 7198: 7162: 7135: 7110: 7104: 7091: 7061: 7030: 7024: 7007: 6976: 6970: 6949: 6943: 6930: 6901: 6872: 6866: 6858:Pergamon Press 6849: 6834: 6828: 6811: 6805: 6784: 6778: 6761: 6732: 6701:-body problem" 6692: 6686: 6670: 6652:(3): 167–179. 6641: 6606: 6571: 6536: 6530: 6513: 6507: 6490:10.1.1.15.9501 6472: 6425: 6419: 6398: 6392: 6379: 6373: 6357:Gravitational 6351: 6349: 6346: 6344: 6343: 6296: 6249: 6242: 6224: 6215: 6206: 6197: 6188: 6179: 6138: 6119:(2): 187–198. 6099: 6080:(3): 411–468. 6064: 6013: 5952: 5909: 5900: 5891: 5882: 5880:Chierchia 2010 5868: 5825: 5810:10.1086/108151 5776: 5767: 5761:See Moulton's 5754: 5745: 5730: 5728: 5727: 5724: 5721: 5702: 5692: 5679: 5677: 5676: 5673: 5670: 5636:center of mass 5627: 5617: 5593: 5568: 5559: 5557:Chenciner 2007 5545: 5536: 5527: 5518: 5469:Wang, Qiu Dong 5460: 5425:(3): 427–449, 5405: 5350: 5341: 5325: 5301:Florian Cajori 5291:, Book Three, 5280: 5270: 5251: 5231: 5223:Gravitational 5194: 5181: 5164: 5155: 5133: 5131: 5128: 5127: 5126: 5120: 5115: 5110: 5105: 5100: 5095: 5090: 5085: 5078: 5075: 4989:loss functions 4950:electrostatics 4934: 4933:-body problems 4927: 4886: 4883: 4882: 4881: 4864: 4848: 4798: 4778: 4694: 4691: 4650: 4646: 4642: 4637: 4632: 4626: 4621: 4616: 4611: 4606: 4600: 4591: 4587: 4581: 4577: 4573: 4565: 4562: 4559: 4556: 4553: 4550: 4547: 4543: 4539: 4534: 4530: 4512:, also called 4510:direct methods 4501: 4498: 4479:Main article: 4476: 4473: 4432: 4431: 4417: 4394: 4387: 4367: 4366: 4359: 4330: 4327: 4321: 4293: 4258: 4223: 4194: 4185: 4168: 4159: 4141: 4134: 4129: 4125: 4122: 4119: 4114: 4109: 4104: 4101: 4098: 4095: 4090: 4085: 4079: 4072: 4068: 4065: 4062: 4057: 4052: 4047: 4044: 4041: 4038: 4033: 4028: 4022: 4016: 4012: 4003: 3996: 3993: 3990: 3985: 3982: 3979: 3973: 3969: 3966: 3958: 3954: 3950: 3945: 3942: 3939: 3934: 3929: 3922: 3918: 3883: 3880: 3837: 3834: 3793:Main article: 3790: 3784: 3731: 3724: 3710:Main article: 3707: 3704: 3647: 3644: 3634: 3631: 3588: 3575: 3570: 3561: 3548:are where the 3545: 3541: 3499: 3498: 3483: 3472: 3443: 3424: 3379:Main article: 3376: 3373: 3361:Saturn's rings 3299: 3258: 3257: 3251: 3244: 3230: 3195: 3190: 3181: 3172: 3165: 3141: 3137: 3133: 3125: 3121: 3117: 3112: 3108: 3094: 3085: 3057: 3054: 3049: 3044: 3039: 3034: 3029: 3024: 3017: 3012: 3008: 3001: 2997: 2991: 2987: 2983: 2977: 2974: 2972: 2968: 2963: 2956: 2952: 2948: 2947: 2938: 2935: 2930: 2925: 2920: 2915: 2910: 2905: 2898: 2893: 2889: 2882: 2878: 2872: 2868: 2864: 2858: 2855: 2853: 2849: 2844: 2837: 2833: 2829: 2828: 2775:Main article: 2772: 2769: 2767: 2764: 2752: 2744: 2740: 2735: 2732: 2725: 2719: 2716: 2711: 2691:virial theorem 2617: 2614: 2611: 2608: 2605: 2602: 2594: 2590: 2586: 2581: 2576: 2572: 2497: 2492: 2487: 2482: 2477: 2470: 2466: 2460: 2455: 2452: 2449: 2445: 2441: 2436: 2431: 2426: 2421: 2416: 2409: 2405: 2399: 2394: 2391: 2388: 2384: 2380: 2377: 2339: 2320: 2264: 2259: 2254: 2249: 2244: 2239: 2232: 2227: 2224: 2221: 2217: 2213: 2209: 2187: 2178: 2169: 2160: 2151: 2141: 2116: 2112: 2106: 2101: 2098: 2095: 2091: 2083: 2078: 2071: 2067: 2061: 2056: 2053: 2050: 2046: 2039: 2035: 2024:center of mass 1955: 1947: 1943: 1939: 1933: 1928: 1923: 1918: 1913: 1904: 1899: 1896: 1893: 1889: 1885: 1882: 1872:kinetic energy 1855: 1852: 1849: 1846: 1843: 1819: 1811: 1806: 1801: 1796: 1793: 1787: 1784: 1778: 1775: 1768: 1763: 1758: 1746: 1741: 1736: 1731: 1728: 1722: 1716: 1713: 1706: 1701: 1696: 1658: 1644: 1636: 1618: 1612: 1606: 1601: 1596: 1591: 1586: 1580: 1573: 1569: 1563: 1559: 1555: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1525: 1521: 1518: 1515: 1502:self-potential 1477: 1472: 1467: 1462: 1459: 1453: 1450: 1443: 1438: 1432: 1427: 1422: 1417: 1412: 1406: 1399: 1393: 1388: 1383: 1378: 1373: 1367: 1361: 1357: 1351: 1347: 1343: 1335: 1328: 1325: 1322: 1317: 1314: 1311: 1305: 1301: 1293: 1289: 1285: 1278: 1273: 1266: 1262: 1253: 1249: 1238: 1219: 1203: 1192: 1179: 1170: 1144: 1137: 1132: 1126: 1121: 1116: 1111: 1106: 1100: 1093: 1087: 1082: 1077: 1072: 1067: 1061: 1055: 1051: 1045: 1041: 1037: 1031: 1025: 1019: 1014: 1009: 1004: 999: 993: 988: 982: 977: 972: 967: 962: 956: 950: 943: 938: 932: 927: 922: 917: 912: 906: 898: 894: 888: 884: 880: 874: 869: 866: 861: 845: 836: 810: 796: 782: 772: 747: 732: 729: 610:John Flamsteed 601: 598: 553:, and visible 518: 517: 515: 514: 507: 500: 492: 489: 488: 485: 484: 479: 473: 470: 469: 466: 465: 462: 461: 456: 454:Gravity assist 450: 447: 446: 443: 442: 439: 438: 433: 428: 423: 417: 414: 413: 410: 409: 402: 399: 398: 395: 394: 389: 381: 380: 372: 368: 367: 362: 361: 358: 357: 354: 353: 348: 343: 338: 332: 329: 328: 325: 324: 317: 314: 313: 310: 309: 304: 299: 294: 289: 287:Orbital period 284: 279: 274: 269: 263: 260: 259: 256: 255: 252: 251: 249:Decaying orbit 246: 241: 236: 228: 227: 221: 214: 212:Transfer orbit 210: 209: 208: 206:Elliptic orbit 203: 201:Circular orbit 197: 188: 187: 184: 183: 180: 179: 174: 169: 164: 159: 154: 149: 144: 138: 133: 132: 129: 128: 121: 118: 117: 109: 108: 104: 103: 95: 94: 46: 44: 37: 15: 9: 6: 4: 3: 2: 8902: 8891: 8888: 8886: 8883: 8881: 8878: 8876: 8873: 8871: 8868: 8866: 8863: 8862: 8860: 8845: 8842: 8841: 8839: 8835: 8829: 8826: 8824: 8821: 8819: 8816: 8814: 8811: 8809: 8806: 8804: 8801: 8799: 8796: 8794: 8791: 8790: 8788: 8786: 8782: 8776: 8773: 8771: 8768: 8766: 8765:Heun's method 8763: 8761: 8758: 8756: 8753: 8751: 8748: 8746: 8743: 8741: 8738: 8737: 8735: 8733: 8729: 8723: 8720: 8718: 8715: 8713: 8710: 8708: 8705: 8704: 8702: 8700: 8696: 8692: 8685: 8680: 8678: 8673: 8671: 8666: 8665: 8662: 8650: 8642: 8641: 8638: 8632: 8629: 8627: 8624: 8622: 8619: 8617: 8614: 8612: 8609: 8607: 8604: 8602: 8599: 8597: 8596:-body problem 8595: 8591: 8589: 8586: 8584: 8581: 8579: 8576: 8574: 8571: 8569: 8566: 8564: 8561: 8559: 8556: 8554: 8551: 8549: 8546: 8544: 8541: 8540: 8538: 8536: 8530: 8524: 8521: 8519: 8516: 8514: 8511: 8509: 8506: 8504: 8501: 8499: 8498:Oberth effect 8496: 8494: 8491: 8489: 8486: 8484: 8481: 8479: 8476: 8474: 8471: 8469: 8466: 8464: 8461: 8459: 8456: 8454: 8451: 8450: 8448: 8446: 8442: 8432: 8424: 8420: 8418: 8417:Orbital speed 8411: 8409: 8402: 8400: 8393: 8392: 8390: 8386: 8380: 8373: 8371: 8364: 8362: 8355: 8353: 8338: 8336: 8329: 8328: 8326: 8322: 8316: 8309: 8307: 8300: 8298: 8291: 8289: 8282: 8281: 8279: 8275: 8269: 8258: 8256: 8249: 8247: 8240: 8238: 8231: 8230: 8228: 8222: 8219: 8218: 8215: 8212: 8210: 8206: 8194: 8191: 8190: 8188: 8184: 8181: 8179: 8176: 8172: 8171:Earth's orbit 8169: 8168: 8167: 8164: 8163: 8161: 8159: 8156: 8152: 8149: 8147: 8144: 8142: 8139: 8137: 8134: 8133: 8131: 8127: 8124: 8122: 8119: 8117: 8114: 8113: 8111: 8110: 8108: 8102: 8096: 8093: 8091: 8088: 8086: 8083: 8081: 8078: 8076: 8073: 8071: 8068: 8066: 8063: 8061: 8058: 8056: 8053: 8051: 8048: 8046: 8043: 8041: 8038: 8034: 8031: 8029: 8028:Geostationary 8026: 8025: 8024: 8021: 8020: 8018: 8016: 8012: 8006: 8003: 7999: 7996: 7994: 7991: 7990: 7989: 7986: 7984: 7981: 7979: 7976: 7974: 7971: 7969: 7966: 7964: 7961: 7959: 7956: 7954: 7950: 7947: 7945: 7942: 7940: 7937: 7935: 7932: 7930: 7926: 7923: 7921: 7918: 7916: 7913: 7911: 7908: 7907: 7905: 7901: 7898: 7896: 7892: 7888: 7880: 7875: 7873: 7868: 7866: 7861: 7860: 7857: 7850: 7840: 7838: 7833: 7828: 7826: 7821: 7816: 7815: 7812: 7804: 7801: 7799: 7797: 7794: 7791: 7789: 7786: 7784: 7781: 7779: 7776: 7774: 7771: 7769: 7765: 7762: 7755: 7753: 7750: 7748: 7744: 7741: 7738: 7736: 7733: 7731: 7728: 7726: 7722: 7719: 7718: 7706: 7701: 7696: 7691: 7687: 7683: 7679: 7675: 7671: 7662: 7658: 7654: 7650: 7646: 7642: 7638: 7629: 7626: 7620: 7611: 7607: 7602: 7598: 7593: 7589: 7584: 7580: 7575: 7571: 7567: 7559: 7555: 7547: 7543: 7535: 7531: 7526: 7522: 7518: 7514: 7510: 7506: 7502: 7501:Physics Today 7497: 7493: 7488: 7484: 7475: 7470: 7469: 7462: 7458: 7453: 7448: 7443: 7442: 7434: 7431: 7423: 7413: 7412:editing guide 7407: 7403: 7398: 7389: 7388: 7375: 7371: 7367: 7363: 7359: 7355: 7354: 7348: 7344: 7340: 7336: 7332: 7328: 7324: 7320: 7316: 7312: 7308: 7304: 7300: 7299: 7290: 7289:Wang, Qiudong 7286: 7282: 7278: 7274: 7270: 7261: 7257: 7251: 7247: 7243: 7238: 7233: 7228: 7224: 7220: 7216: 7212: 7208: 7199: 7195: 7191: 7187: 7183: 7179: 7175: 7168: 7163: 7158: 7153: 7149: 7145: 7141: 7136: 7132: 7128: 7124: 7120: 7116: 7111: 7107: 7101: 7097: 7092: 7082:on 2017-04-21 7078: 7074: 7067: 7062: 7059: 7055: 7049: 7045: 7041: 7040: 7035: 7034:Newton, Isaac 7031: 7027: 7021: 7017: 7013: 7008: 7003: 6998: 6994: 6990: 6986: 6977: 6973: 6967: 6963: 6959: 6958:-body Problem 6950: 6946: 6940: 6936: 6931: 6927: 6923: 6919: 6915: 6910: 6909: 6902: 6898: 6894: 6890: 6886: 6882: 6878: 6873: 6869: 6863: 6859: 6855: 6850: 6845: 6840: 6835: 6831: 6825: 6821: 6818:. Baltimore: 6817: 6812: 6808: 6802: 6798: 6785: 6781: 6775: 6771: 6768:. Cambridge: 6767: 6762: 6758: 6754: 6750: 6746: 6742: 6738: 6733: 6729: 6725: 6721: 6717: 6713: 6709: 6702: 6693: 6689: 6683: 6680: 6676: 6671: 6667: 6663: 6659: 6655: 6651: 6647: 6642: 6637: 6632: 6628: 6624: 6620: 6616: 6612: 6607: 6602: 6597: 6593: 6589: 6585: 6581: 6577: 6572: 6567: 6562: 6558: 6554: 6550: 6546: 6542: 6537: 6533: 6527: 6522: 6521: 6514: 6510: 6504: 6500: 6496: 6491: 6486: 6482: 6473: 6469: 6465: 6461: 6457: 6453: 6449: 6444: 6443:gr-qc/0108086 6439: 6435: 6431: 6426: 6422: 6416: 6412: 6407: 6406: 6399: 6395: 6389: 6385: 6380: 6376: 6370: 6366: 6363:. Cambridge: 6362: 6353: 6352: 6339: 6335: 6331: 6327: 6323: 6319: 6315: 6311: 6307: 6300: 6292: 6288: 6284: 6280: 6276: 6272: 6268: 6264: 6260: 6253: 6245: 6243:0-521-62186-0 6239: 6235: 6228: 6219: 6210: 6201: 6195:Blanchet 2001 6192: 6186:Alligood 1996 6183: 6174: 6169: 6165: 6161: 6158:(1): 80–111. 6157: 6153: 6149: 6142: 6134: 6130: 6126: 6122: 6118: 6114: 6110: 6103: 6095: 6091: 6087: 6083: 6079: 6075: 6068: 6060: 6056: 6052: 6048: 6044: 6040: 6036: 6032: 6028: 6024: 6017: 6009: 6005: 6001: 5997: 5993: 5989: 5985: 5981: 5976: 5971: 5967: 5963: 5956: 5948: 5944: 5940: 5936: 5932: 5928: 5924: 5920: 5913: 5907:Celletti 2008 5904: 5895: 5886: 5877: 5875: 5873: 5864: 5860: 5856: 5852: 5848: 5844: 5840: 5836: 5829: 5820: 5815: 5811: 5807: 5803: 5799: 5795: 5791: 5787: 5780: 5771: 5764: 5758: 5749: 5741: 5734: 5725: 5722: 5719: 5718: 5716: 5712: 5706: 5696: 5689: 5683: 5674: 5671: 5668: 5667: 5665: 5661: 5657: 5653: 5649: 5645: 5641: 5640:conic section 5637: 5631: 5621: 5612: 5610: 5608: 5606: 5604: 5602: 5600: 5598: 5578: 5572: 5563: 5554: 5552: 5550: 5540: 5531: 5522: 5514: 5510: 5506: 5502: 5498: 5494: 5490: 5486: 5482: 5478: 5474: 5470: 5464: 5456: 5452: 5448: 5444: 5440: 5436: 5432: 5428: 5424: 5420: 5416: 5409: 5401: 5397: 5393: 5389: 5385: 5381: 5377: 5373: 5369: 5365: 5361: 5354: 5345: 5338: 5332: 5330: 5322: 5318: 5314: 5310: 5306: 5302: 5298: 5294: 5290: 5284: 5274: 5266: 5262: 5255: 5245: 5244: 5235: 5228: 5220: 5212: 5208: 5207:Taylor series 5198: 5185: 5178: 5173: 5168: 5159: 5138: 5134: 5124: 5121: 5119: 5116: 5114: 5111: 5109: 5106: 5104: 5103:Natural units 5101: 5099: 5096: 5094: 5091: 5089: 5086: 5084: 5081: 5080: 5074: 5060: 5052: 5048: 5044: 5039: 5033: 5027: 5023: 5016: 5012: 5007: 5003: 4999: 4995: 4990: 4986: 4982: 4977: 4975: 4971: 4967: 4963: 4959: 4955: 4951: 4946: 4926: 4924: 4920: 4916: 4912: 4908: 4904: 4896: 4892: 4891:event horizon 4879: 4875: 4871: 4869: 4865: 4862: 4858: 4854: 4853: 4849: 4846: 4842: 4836: 4832: 4827: 4821: 4817: 4813: 4808: 4804: 4803: 4799: 4794: 4790: 4784: 4783: 4779: 4774: 4770: 4766: 4761: 4757: 4753: 4750: 4749: 4748: 4745: 4741: 4737: 4730: 4726: 4721: 4716: 4690: 4688: 4682: 4680: 4670: 4648: 4644: 4640: 4635: 4624: 4614: 4609: 4589: 4585: 4579: 4575: 4571: 4563: 4560: 4557: 4554: 4551: 4548: 4545: 4541: 4537: 4532: 4528: 4519: 4515: 4511: 4497: 4492:, in general 4489: 4482: 4472: 4470: 4466: 4460: 4453: 4448: 4444: 4438: 4427: 4423: 4418: 4413: 4409: 4404: 4403: 4402: 4393:-body problem 4386: 4379: 4373: 4364: 4360: 4357: 4356: 4355: 4351: 4344: 4337: 4326: 4320: 4310: 4301: 4296: 4292: 4289: 4275: 4266: 4261: 4257: 4254: 4240: 4231: 4226: 4222: 4219: 4205: 4193: 4188: 4184: 4181: 4167: 4162: 4158: 4152: 4139: 4132: 4127: 4120: 4112: 4102: 4096: 4088: 4077: 4070: 4063: 4055: 4045: 4039: 4031: 4020: 4014: 4010: 4001: 3994: 3991: 3988: 3983: 3980: 3977: 3971: 3967: 3964: 3956: 3952: 3948: 3940: 3932: 3920: 3916: 3904: 3899: 3897: 3896:Taylor series 3879: 3877: 3873: 3869: 3859: 3849: 3847: 3843: 3833: 3829: 3822: 3817: 3813: 3807: 3802: 3796: 3783: 3781: 3777: 3773: 3767: 3760: 3751: 3747: 3742: 3734: 3730: 3723: 3719: 3713: 3703: 3700: 3699:quasiperiodic 3696: 3693:proved using 3692: 3686: 3684: 3680: 3676: 3670: 3665: 3661: 3653: 3643: 3641: 3640:astrodynamics 3630: 3628: 3624: 3621: 3620:deterministic 3617: 3612: 3610: 3606: 3602: 3598: 3593: 3587: 3574:is Jupiter. L 3569: 3560: 3555: 3551: 3539: 3530: 3526: 3524: 3520: 3516: 3508: 3503: 3496: 3492: 3488: 3484: 3481: 3477: 3473: 3468: 3462: 3458: 3453: 3448: 3444: 3441: 3437: 3433: 3429: 3425: 3421: 3417: 3413: 3412: 3411: 3407: 3400: 3390: 3382: 3372: 3370: 3366: 3362: 3358: 3354: 3350: 3346: 3342: 3333: 3328: 3323: 3321: 3316: 3314: 3310: 3306: 3298: 3292: 3287: 3280: 3272: 3264: 3260:The equation 3250: 3243: 3239: 3235: 3231: 3223: 3215: 3212: 3204:acceleration 3203: 3196: 3189: 3180: 3171: 3164: 3160: 3156: 3155: 3154: 3135: 3123: 3119: 3115: 3110: 3106: 3093: 3084: 3078: 3047: 3037: 3032: 3015: 3010: 3006: 2999: 2995: 2989: 2985: 2981: 2975: 2973: 2966: 2954: 2950: 2928: 2918: 2913: 2896: 2891: 2887: 2880: 2876: 2870: 2866: 2862: 2856: 2854: 2847: 2835: 2831: 2818: 2812: 2806: 2800: 2796: 2795:Kepler's Laws 2792: 2788: 2784: 2778: 2766:Special cases 2763: 2750: 2742: 2738: 2733: 2730: 2723: 2709: 2692: 2686: 2666: 2654: 2646: 2643: 2633: 2628: 2615: 2612: 2609: 2606: 2603: 2600: 2592: 2588: 2584: 2579: 2574: 2570: 2558: 2549: 2541: 2518: 2513: 2495: 2485: 2468: 2464: 2458: 2453: 2450: 2447: 2443: 2439: 2434: 2424: 2419: 2407: 2403: 2397: 2392: 2389: 2386: 2382: 2378: 2375: 2363: 2358: 2354: 2347: 2342: 2338: 2335: 2328: 2323: 2319: 2314: 2310: 2297: 2288: 2283: 2278: 2277:cross product 2262: 2257: 2247: 2242: 2230: 2225: 2222: 2219: 2215: 2211: 2198: 2194: 2186: 2177: 2168: 2159: 2150: 2146: 2140: 2136: 2114: 2110: 2104: 2099: 2096: 2093: 2089: 2081: 2069: 2065: 2059: 2054: 2051: 2048: 2044: 2037: 2025: 2021: 2017: 2008: 2005: 1998: 1992: 1988: 1982: 1977: 1966: 1953: 1945: 1941: 1937: 1931: 1921: 1902: 1897: 1894: 1891: 1887: 1883: 1880: 1873: 1853: 1850: 1847: 1844: 1841: 1833: 1817: 1809: 1794: 1785: 1782: 1776: 1773: 1766: 1756: 1744: 1729: 1720: 1714: 1711: 1704: 1694: 1679: 1670: 1661: 1657: 1654: 1647: 1639: 1635: 1629: 1616: 1604: 1594: 1589: 1571: 1567: 1561: 1557: 1553: 1545: 1542: 1539: 1536: 1533: 1530: 1527: 1523: 1519: 1516: 1513: 1505: 1503: 1493: 1475: 1460: 1451: 1448: 1441: 1430: 1420: 1415: 1397: 1391: 1381: 1376: 1365: 1359: 1355: 1349: 1345: 1341: 1333: 1326: 1323: 1320: 1315: 1312: 1309: 1303: 1299: 1291: 1287: 1283: 1276: 1264: 1260: 1251: 1247: 1237: 1235: 1226: 1224: 1218: 1212: 1206: 1202: 1195: 1191: 1182: 1178: 1173: 1169: 1162: 1142: 1135: 1124: 1114: 1109: 1091: 1085: 1075: 1070: 1059: 1053: 1049: 1043: 1039: 1035: 1029: 1017: 1007: 1002: 986: 980: 970: 965: 954: 948: 941: 930: 920: 915: 896: 892: 886: 882: 878: 872: 867: 864: 848: 839: 831: 822: 813: 809: 806: 799: 791: 785: 781: 775: 763: 758: 754: 750: 743:point masses 728: 726: 720: 713: 708: 704: 700: 694: 692: 685: 683: 680:, advised by 679: 670: 664: 659: 657: 651: 646: 644: 636: 635: 625: 613: 611: 607: 597: 595: 592: 588: 582: 577: 574: 572: 560: 556: 552: 548: 544: 540: 536: 532: 531:-body problem 525: 513: 508: 506: 501: 499: 494: 493: 491: 490: 483: 480: 478: 475: 474: 468: 467: 460: 459:Oberth effect 457: 455: 452: 451: 445: 444: 437: 434: 432: 429: 427: 424: 422: 419: 418: 412: 411: 407: 401: 400: 393: 390: 388: 385: 384: 378: 374: 373: 371: 365: 364:N-body orbits 360: 359: 352: 349: 347: 346:Perturbations 344: 342: 339: 337: 334: 333: 327: 326: 322: 316: 315: 308: 305: 303: 300: 298: 295: 293: 290: 288: 285: 283: 280: 278: 275: 273: 270: 268: 265: 264: 258: 257: 250: 247: 245: 242: 240: 237: 235: 232: 231: 225: 222: 220: 216: 215: 213: 207: 204: 202: 199: 198: 192: 186: 185: 178: 175: 173: 170: 168: 167:Orbital nodes 165: 163: 160: 158: 155: 153: 150: 148: 145: 143: 140: 139: 136: 131: 130: 126: 120: 119: 115: 111: 110: 107:Astrodynamics 106: 105: 101: 100: 91: 88: 80: 70: 65: 63: 57: 53: 52: 45: 36: 35: 30: 26: 22: 8707:Euler method 8611:Perturbation 8593: 8592: 8568:Ground track 8478:Gravity turn 8429: 8422: 8415: 8406: 8397: 8377: 8368: 8359: 8352:True anomaly 8350: 8335:Mean anomaly 8333: 8313: 8304: 8295: 8286: 8266: 8253: 8244: 8237:Eccentricity 8235: 8193:Lunar cycler 8166:Heliocentric 8106:other points 8055:Medium Earth 7953:Non-inclined 7725:Scholarpedia 7704: 7680:(1): 27–43. 7677: 7673: 7640: 7636: 7614: 7605: 7596: 7590:. CRC Press. 7587: 7578: 7569: 7565: 7557: 7553: 7545: 7541: 7533: 7529: 7504: 7500: 7491: 7482: 7467: 7456: 7446: 7426: 7417: 7405: 7357: 7351: 7305:(1): 73–88. 7302: 7296: 7272: 7241: 7214: 7211:Scholarpedia 7210: 7177: 7173: 7147: 7143: 7114: 7095: 7084:. Retrieved 7077:the original 7072: 7038: 7014:. New York: 7011: 6992: 6988: 6960:. New York: 6953: 6934: 6907: 6879:. New York: 6876: 6853: 6815: 6792: 6765: 6740: 6736: 6714:(3): 66–70. 6711: 6707: 6677:. New York: 6674: 6649: 6645: 6618: 6615:Scholarpedia 6614: 6586:(10): 2111. 6583: 6580:Scholarpedia 6579: 6548: 6545:Scholarpedia 6544: 6519: 6480: 6433: 6429: 6409:. New York: 6404: 6383: 6356: 6313: 6309: 6299: 6269:(1): 51–73. 6266: 6262: 6252: 6233: 6227: 6218: 6209: 6200: 6191: 6182: 6155: 6151: 6141: 6116: 6112: 6102: 6077: 6073: 6067: 6029:(1): 73–88. 6026: 6022: 6016: 5975:math/0011268 5965: 5961: 5955: 5922: 5918: 5912: 5903: 5894: 5885: 5838: 5834: 5828: 5793: 5790:NASA TND-501 5789: 5779: 5770: 5762: 5757: 5748: 5742:. MIT Press. 5739: 5733: 5714: 5710: 5705: 5695: 5687: 5682: 5642:which has a 5630: 5625:mathematics. 5620: 5584:. Retrieved 5571: 5562: 5539: 5530: 5521: 5483:(1): 73–88, 5480: 5476: 5472: 5463: 5422: 5418: 5414: 5408: 5370:(1): 43–57, 5367: 5363: 5359: 5353: 5344: 5336: 5320: 5316: 5315:, 1970; and 5312: 5308: 5296: 5292: 5288: 5283: 5273: 5264: 5254: 5242: 5234: 5222: 5197: 5184: 5177:steady-state 5176: 5172:Quasi-steady 5171: 5167: 5158: 5137: 5098:Lunar theory 5040: 5031: 5025: 5021: 5014: 5010: 4978: 4965: 4947: 4936: 4888: 4866: 4860: 4856: 4850: 4834: 4830: 4819: 4815: 4811: 4800: 4792: 4788: 4780: 4772: 4768: 4764: 4754:, such as a 4751: 4746: 4739: 4735: 4728: 4724: 4714: 4696: 4683: 4668: 4513: 4509: 4503: 4487: 4484: 4458: 4451: 4436: 4433: 4425: 4421: 4411: 4407: 4396: 4378:Qiudong Wang 4371: 4368: 4349: 4342: 4335: 4332: 4318: 4308: 4299: 4294: 4290: 4287: 4273: 4264: 4259: 4255: 4252: 4238: 4229: 4224: 4220: 4217: 4203: 4191: 4186: 4182: 4179: 4165: 4160: 4156: 4153: 3900: 3885: 3857: 3850: 3839: 3827: 3820: 3816:figure eight 3805: 3800: 3798: 3771: 3765: 3758: 3749: 3745: 3740: 3732: 3728: 3721: 3715: 3687: 3668: 3651: 3649: 3636: 3623:chaos theory 3613: 3594: 3585: 3567: 3558: 3537: 3535: 3512: 3494: 3490: 3466: 3460: 3456: 3454:, for which 3451: 3419: 3416:Isaac Newton 3405: 3398: 3391: 3384: 3364: 3356: 3353:Robert Hooke 3338: 3325: 3319: 3317: 3296: 3293: 3285: 3278: 3270: 3262: 3259: 3248: 3241: 3237: 3233: 3221: 3213: 3210: 3201: 3187: 3185:relative to 3178: 3169: 3162: 3158: 3091: 3082: 3079: 2810: 2807: 2780: 2684: 2664: 2652: 2644: 2641: 2631: 2629: 2559:states that 2556: 2547: 2539: 2516: 2514:is given by 2511: 2359: 2352: 2345: 2340: 2336: 2333: 2326: 2321: 2317: 2298: 2281: 2184: 2175: 2166: 2157: 2148: 2144: 2138: 2134: 2009: 2003: 1996: 1986: 1979:first-order 1975: 1967: 1668: 1659: 1655: 1652: 1642: 1637: 1633: 1630: 1506: 1501: 1495: 1239: 1227: 1216: 1204: 1200: 1193: 1189: 1180: 1176: 1171: 1167: 850:is given by 843: 834: 820: 811: 807: 804: 794: 783: 779: 770: 756: 752: 745: 734: 725:Qiudong Wang 718: 711: 703:chaos theory 696: 690: 687: 671: 660: 653: 648: 642: 632: 626: 614: 606:Isaac Newton 603: 590: 584: 579: 575: 527: 521: 363: 244:Radial orbit 195:eccentricity 177:True anomaly 162:Mean anomaly 152:Eccentricity 83: 74: 67:Please help 62:Template:Sfn 59: 48: 8573:Hill sphere 8408:Mean motion 8288:Inclination 8277:Orientation 8178:Mars cycler 8116:Areocentric 7988:Synchronous 7825:Mathematics 7217:(5): 3930. 7150:: 105–179. 6621:(9): 2123. 6551:(9): 4079. 6316:: 235–258. 5615:Trenti 2008 5073:particles. 4839:time using 4824:time using 4693:Many bodies 4463:by Gerver. 4281:results in 3601:Hill sphere 3403:. The case 3349:Tycho Brahe 3307:, and this 2555:. Then the 755:= 1, 2, …, 377:Halo orbits 341:Hill sphere 157:Inclination 71:if you can. 8859:Categories 8513:Rendezvous 8209:Parameters 8045:High Earth 8015:Geocentric 7968:Osculating 7925:Elliptical 7507:(12): 55. 7420:March 2017 7404:" section 7086:2014-03-28 6918:B0000CLA7B 6889:B0006AVKQW 6856:. London: 6348:References 6213:Board 1999 5968:(3): 881. 5889:Féjoz 2004 4981:statistics 4895:black hole 4868:Mean field 4500:Few bodies 4475:Simulation 4376:bodies by 3746:homothetic 3695:KAM theory 3605:Roche lobe 3305:barycenter 2791:barycenter 1830:where the 591:restricted 421:Mass ratio 336:Barycenter 77:March 2017 8558:Ephemeris 8535:mechanics 8445:Maneuvers 8388:Variation 8151:Libration 8146:Lissajous 8050:Low Earth 8040:Graveyard 7939:Horseshoe 7849:Astronomy 7343:118132097 7327:0923-2958 7281:848738761 7194:951409281 7186:1908/4228 7123:Doubleday 7048:915353069 6926:802752879 6839:CiteSeerX 6757:123461135 6728:119728316 6485:CiteSeerX 6468:119101016 6338:0021-9991 6291:0021-9991 6113:Exp. Math 6074:Ann. Math 6059:118132097 6051:0923-2958 5863:213600592 5660:hyperbola 5513:118132097 5455:120617936 5400:120358878 5289:Principia 5047:vorticity 5032:dual tree 4841:multigrid 4645:ε 4615:− 4561:≤ 4549:≤ 4542:∑ 4533:ε 4316:which at 4103:− 4046:− 3992:≠ 3972:∑ 3509:behaviour 3485:In 1917, 3445:In 1772, 3432:collinear 3426:In 1767, 3420:Principia 3414:In 1687, 3365:Principia 3116:η 3107:α 3038:− 2919:− 2610:− 2444:∑ 2425:⋅ 2383:∑ 2248:× 2216:∑ 2090:∑ 2045:∑ 1888:∑ 1800:∂ 1792:∂ 1786:− 1735:∂ 1727:∂ 1595:− 1543:≤ 1531:≤ 1524:∑ 1520:− 1466:∂ 1458:∂ 1452:− 1421:− 1382:− 1324:≠ 1304:∑ 1115:− 1076:− 1008:− 971:− 949:⋅ 921:− 643:Principia 634:Principia 261:Equations 189:Types of 8324:Position 7949:Inclined 7920:Circular 7764:Archived 7743:Archived 7036:(1687). 6995:: I–VI. 6222:Ram 2010 6133:23816314 6008:10024592 5947:10053934 5656:parabola 5586:25 March 5339:article. 5077:See also 4954:proteins 4631:‖ 4599:‖ 4443:Painlevé 3842:isometry 3776:topology 3770:gives a 3727:(0), …, 3616:Poincaré 3447:Lagrange 3202:Eulerian 2510:and the 2491:‖ 2476:‖ 2350:for any 2299:Because 2155:, where 1927:‖ 1912:‖ 1680:for the 1611:‖ 1579:‖ 1437:‖ 1405:‖ 1185:‖ 1165:‖ 1131:‖ 1099:‖ 1024:‖ 992:‖ 937:‖ 905:‖ 699:Poincaré 49:require 8885:Gravity 8823:Yoshida 8533:Orbital 8503:Phasing 8463:Delta-v 8268:Apsides 8262:, 8060:Molniya 7978:Parking 7915:Capture 7903:General 7837:Physics 7811:Portals 7682:Bibcode 7657:2323910 7509:Bibcode 7449:. AIAA. 7374:2946572 7335:1117788 7307:Bibcode 7219:Bibcode 6897:1219303 6654:Bibcode 6623:Bibcode 6588:Bibcode 6553:Bibcode 6448:Bibcode 6318:Bibcode 6271:Bibcode 6160:Bibcode 6094:2946572 6031:Bibcode 6000:2661357 5980:Bibcode 5927:Bibcode 5843:Bibcode 5798:Bibcode 5796:: 347. 5652:ellipse 5505:1117788 5485:Bibcode 5447:1225892 5427:Bibcode 5392:0538663 5372:Bibcode 4870:methods 4712:⁠ 4700:⁠ 4679:chaotic 4469:measure 4313:⁠ 4284:⁠ 4278:⁠ 4249:⁠ 4243:⁠ 4214:⁠ 4208:⁠ 4176:⁠ 3660:Jupiter 3654:is the 3609:chaotic 3536:In the 3519:Jupiter 3507:chaotic 3283:⁠ 3267:⁠ 3226:⁠ 3207:⁠ 3200:is the 2787:gravity 2681:⁠ 2669:⁠ 2657:⁠ 2638:⁠ 2552:⁠ 2536:⁠ 2533:⁠ 2521:⁠ 1983:, with 1870:is the 1673:⁠ 1649:⁠ 1504:energy 1500:is the 1159:is the 825:⁠ 801:⁠ 689:series 600:History 551:planets 524:physics 51:cleanup 8890:Orbits 8837:Theory 8189:Other 8090:Tundra 7958:Kepler 7934:Escape 7887:orbits 7655: 7572:(3–4). 7400:This " 7372: 7341: 7333: 7325: 7279: 7252: 7192: 7131:689289 7129: 7102: 7056: 7046: 7022: 6968: 6941: 6924: 6916: 6895: 6887: 6864: 6841: 6826: 6803: 6776: 6755: 6726: 6684: 6528: 6505: 6487: 6466: 6417: 6390: 6371: 6336: 6289: 6240: 6131: 6092: 6057: 6049: 6006: 5998: 5945: 5861: 5648:circle 5511: 5503: 5453: 5445: 5398: 5390: 5004:, and 4960:, the 4929:Other 4673:, the 4671:> 2 4490:> 2 4471:zero. 4439:> 3 4374:> 3 4338:> 3 3872:Galois 3664:Saturn 3550:Trojan 3469:> 0 3430:found 3345:Kepler 3153:Where 2512:virial 2364:of an 2355:> 0 1496:where 1232:-body 1155:where 760:in an 721:> 3 581:times. 526:, the 8431:Epoch 8220:Shape 8158:Lunar 8112:Mars 8104:About 8075:Polar 7895:Types 7653:JSTOR 7370:JSTOR 7339:S2CID 7170:(PDF) 7080:(PDF) 7069:(PDF) 6981:"The 6753:S2CID 6724:S2CID 6704:(PDF) 6464:S2CID 6438:arXiv 6129:S2CID 6090:JSTOR 6055:S2CID 6004:S2CID 5996:JSTOR 5970:arXiv 5859:S2CID 5644:focus 5580:(PDF) 5509:S2CID 5451:S2CID 5396:S2CID 5130:Notes 4893:of a 4445:(see 4383:[0,∞) 4363:below 3544:and L 3428:Euler 2315:: if 555:stars 142:Apsis 8223:Size 8162:Sun 8141:Halo 7993:semi 7570:1971 7560:(2). 7558:1969 7548:(3). 7546:1968 7536:(3). 7534:1968 7323:ISSN 7277:OCLC 7250:ISBN 7190:OCLC 7127:OCLC 7100:ISBN 7073:NIPS 7054:ISBN 7044:OCLC 7020:ISBN 6966:ISBN 6939:ISBN 6922:OCLC 6914:ASIN 6893:OCLC 6885:ASIN 6862:ISBN 6824:ISBN 6801:ISBN 6774:ISBN 6682:ISBN 6526:ISBN 6503:ISBN 6415:ISBN 6388:ISBN 6369:ISBN 6334:ISSN 6287:ISSN 6238:ISBN 6047:ISSN 5943:PMID 5588:2014 5287:See 5258:See 4983:and 4855:and 4818:log 4771:log 4555:< 4347:and 4340:(or 4173:and 3870:and 3868:Abel 3801:same 3650:The 3597:mass 2667:⟩ = 2360:The 2307:are 2303:and 2182:and 1866:and 1537:< 1223:norm 1213:the 1198:and 1163:and 735:The 709:for 637:the 585:The 547:Moon 27:and 7998:sub 7910:Box 7723:at 7690:doi 7645:doi 7517:doi 7362:doi 7358:135 7315:doi 7227:doi 7182:hdl 7178:III 7152:doi 6997:doi 6745:doi 6716:doi 6662:doi 6650:244 6631:doi 6596:doi 6561:doi 6495:doi 6456:doi 6326:doi 6314:245 6279:doi 6267:225 6168:doi 6121:doi 6082:doi 6078:135 6039:doi 5988:doi 5966:152 5935:doi 5851:doi 5814:hdl 5806:doi 5658:or 5493:doi 5435:doi 5380:doi 4996:in 4979:In 4828:or 4722:of 4461:= 4 4454:= 5 4441:by 4352:= 0 4345:= 3 4154:As 3898:". 3860:≥ 3 3830:≥ 3 3823:= 3 3808:= 3 3778:of 3768:= 0 3761:= 1 3736:(0) 3671:− 1 3556:); 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