The integral manifolds of the three body problem

McCord, Christopher; Meyer, Kenneth R.; Wang, Quidong

doi:10.1090/memo/0628

Cited by 24 publications

(45 citation statements)

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“…A standard mathematical procedure [Whittaker, 1937] allows solving such ODEs by quadratures if independent integrals of motion exist. Therefore, we shall now determine the number of integrals of motion for the general three-body problem [McCord et al, 1998]. …”

Section: From Poincaré To the Present Timementioning

confidence: 99%

The three-body problem

Musielak

Quarles

2014

Rep. Prog. Phys.

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Abstract. The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.

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Section: From Poincaré To the Present Timementioning

confidence: 99%

The three-body problem

Musielak

Quarles

2014

Rep. Prog. Phys.

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“…1 In the spatial 3-body problem, it is β 3 that must be non-zero. But an inspection of the tables in [19] or [17] shows that β 3 = 0 for all energy levels.…”

Section: Geodesic Flows In the N -Body Problemmentioning

confidence: 99%

“…So it is on the reduced manifolds that we might look for geodesic structures. Combining the topological and homological results of [6], [7], [16], [19] with the results of this paper, we can determine whether or not the N -body flow on the reduced integral manifolds is a geodesic flow in the following cases: (i) the planar N -body problem for all N , c and h, (ii) the spatial 3-body problem for non-zero c and all h, (iii) the spatial N -body problem with hc 2 positive. The two remaining cases are the spatial N -body problem with N ≥ 4, negative energy and non-zero angular momentum, and the spatial N -body problem with zero angular momentum.…”

Section: Geodesic Flows In the N -Body Problemmentioning

confidence: 99%

“…For the 3-body problem, the cohomology groups are computed in [19]. 1 In the spatial 3-body problem, it is β 3 that must be non-zero.…”

Section: Geodesic Flows In the N -Body Problemmentioning

confidence: 99%

“…In a series of papers [16], [17], [18], [19] we have computed the homology of the integral manifolds of the various versions of the three-body and N -body problem. In § 4, we apply our geodesic criteria to the reduced integral manifolds.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Are Hamiltonian flows geodesic flows?

McCord

Meyer

Offin

2002

Trans. Amer. Math. Soc.

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Abstract. When a Hamiltonian system has a "Kinetic + Potential" structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N -body problem. We show that the flow of the reduced planar N -body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.

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Syzygies and the Integral Manifolds of the Spatial N-Body Problem

McCord

2004

New Advances in Celestial Mechanics and Hamiltonian Systems

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The integral manifolds of the three body problem

Cited by 24 publications

References 0 publications

The three-body problem

The three-body problem

Are Hamiltonian flows geodesic flows?

Syzygies and the Integral Manifolds of the Spatial N-Body Problem

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