Poincaré and the Three-Body Problem

Chenciner, Alain

doi:10.1007/978-3-0348-0834-7_2

Cited by 40 publications

(76 citation statements)

References 38 publications

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“…For example, in the planar isotropic harmonic oscillator with potential kr 2 /2 in plane polar coordinates, the gaussian curvature R = 16Ek/(2E −kr 2 ) 3 of the JM metric on configuration space is non-negative everywhere indicating stability. In the planar Kepler problem with Hamiltonian p 2 /2m − k/r , the gaussian curvature of the JM metric ds 2…”

Section: Trajectories As Geodesics Of the Jacobi-maupertuis Metricmentioning

confidence: 99%

“…Using Eq.10, we may express the Cartesian coordinates wi in terms of Hopf coordinates: 2w3 = r 2 cos 2η, 2w1 = r 2 sin(2η) cos(2ξ2) and 2w2 = r 2 sin(2η) sin(2ξ2). 2 Let f : (M, g) → (N, h) be a Riemannian submersion with local coordinates m i and n j . Let (r, m i ) and (r, n j ) be local coordinates on the cones C(M ) and C(N ) .…”

Section: Trajectories As Geodesics Of the Jacobi-maupertuis Metricmentioning

confidence: 99%

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Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

Krishnaswami

Senapati

2016

Journal of Mathematical Physics

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The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E . The geodesic flow on the full configuration space C 3 (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space C 2 and shape space R 3 (as well as S 3 and the shape sphere S 2 for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in 'Hopf' coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy 'pair of pants' JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on C 2 , R 3 and S 3 with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits and observe that the geodesic reformulation 'regularizes' pairwise and triple collisions on C 2 and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and zero energy, we find that the scalar curvature on C 2 is strictly negative though it could have either sign on R 3 . However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.

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Section: Trajectories As Geodesics Of the Jacobi-maupertuis Metricmentioning

confidence: 99%

Section: Trajectories As Geodesics Of the Jacobi-maupertuis Metricmentioning

confidence: 99%

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

Krishnaswami

Senapati

2016

Journal of Mathematical Physics

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“…This is achieved by making near-identity coordinate transformations to get rid of as many terms as possible from the equation. It was developed initially by Poincaré to integrate nonlinear systems [41,42]. The physical behavior should be invariant under analytic changes of coordinates, and the length (or time) parameter should stay the same, which the mathematical literature addresses by perturbative polynomial changes of coordinates (attempting removal of nth order nonlinearities in the flow by using nth order or lower terms in the change of variables).…”

Section: Normal Form Theorymentioning

confidence: 99%

Normal Form for Renormalization Groups

et al. 2019

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The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions. We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples. arXiv:1706.00137v3 [cond-mat.stat-mech]

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“…It is well known since the end of the 19th century that the problem of n point masses mutually interacting by the sole gravitational force is non-integrable for n ≥ 3 (see [9] for a detailed historical overview on this subject). Coming to more recent times, the birth of KAM theory in the mid-twentieth century led to new mathematical efforts in order to establish whether quasi-periodic motions persisted in the n-body problem for suitable perturbative parameters.…”

Section: Introductionmentioning

confidence: 99%

Sharp Nekhoroshev estimates for the three-body problem around periodic orbits

Barbieri

Niederman

2020

Journal of Differential Equations

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We construct a Nekhoroshev-like result of stability with sharp constants for the planar three body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted one.

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Poincaré and the Three-Body Problem

Cited by 40 publications

References 38 publications

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

Normal Form for Renormalization Groups

Sharp Nekhoroshev estimates for the three-body problem around periodic orbits

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