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

### Abstract

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.

Published by AIP Publishing.

Received 26 June 2016
Accepted 20 September 2016
Published online 07 October 2016

Acknowledgments:
We thank K. G. Arun, A. Lakshminarayan, R. Montgomery, S. G. Rajeev, and A. Thyagaraja for useful discussions and references. This work was supported in part by the Infosys Foundation and a Ramanujan grant of the Department of Science & Technology, Government of India.

Article outline:

I. INTRODUCTION
II. TRAJECTORIES AS GEODESICS OF THE JACOBI-MAUPERTUIS METRIC
III. PLANAR THREE-BODY PROBLEM WITH INVERSE-SQUARE POTENTIAL
A. Jacobi-Maupertuis metric on configuration space and Hopf coordinates
B. Quotient JM metrics on shape space, the three-sphere, and the shape sphere
C. JM metric in the near-collision limit and its completeness
1. Geometry near pairwise collisions
2. Geometry on ℝ^{3} and ℂ^{2} near triple collisions
D. Scalar curvature for equal masses and zero energy
E. Sectional curvature for three equal masses
F. Stability tensor and linear stability of geodesics
IV. PLANAR THREE-BODY PROBLEM WITH NEWTONIAN POTENTIAL
A. JM metric and its curvature on configuration and shape space
B. Near-collision geometry and “geodesic incompleteness”

/content/aip/journal/jmp/57/10/10.1063/1.4964340

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A. Chenciner, “Poincaré and the three-body problem,” in Henri Poincaré, 1912-2012, Poincaré Seminar 2012, edited byB. Duplantier and V. Rivasseau (Springer, Basel, 2015).

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E. J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion (Macmillan, 1877).

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R. Montgomery, “A new solution to the three-body problem,” Not. AMS 48(5), 471 (2001).

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M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990), Vol. 1.

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P. K. Newton, The N-Vortex Problem: Analytical Techniques (Springer-Verlag, New York, 2001).

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T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nagerl, and R. Grimm, “Evidence for Efimov quantum states in an ultracold gas of caesium atoms,” Nature 440, 315 (2006).

http://dx.doi.org/10.1038/nature04626
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S. G. Rajeev, Advanced Mechanics: From Euler’s Determinism to Arnold’s Chaos (Oxford University Press, 2013).

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C. Lanczos, The Variational Principles of Mechanics, 4th ed. (Dover, 1970), p. 139.

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V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer Verlag, 1989), p. 245.

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M. Pettini, in Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Springer, 2007), Chap. 3.

26.

A. Chenciner, “Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry,” in Proceedings of the ICM (World Scientific, Beijing, 2002), Vol. 3, pp. 255–264.

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D. K. Yeomans, “Exposition of Sundman’s regularization of the three-body problem,” NASA Goddard Space Flight Center Technical Report NASA-TM-X-55636, X-640-66-481, 1966.

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A. Celletti, “Basics of regularization theory,” in Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems, edited by B. A. Steves, A. J. Maciejewski, and M. Hendry (Springer, Netherlands, 2006), pp. 203–230.

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B. O’Neill, Semi-Riemannian Geometry (Academic Press, 1983).

30.

The Hopf map S^{3} → S^{2} is often expressed in Cartesian coordinates. If defines the unit-S^{3} ⊂ C^{2} and defines a 2-sphere of radius 1/2 in R^{3}, then and . Using Eq. (10), we may express the Cartesian coordinates w_{i} in terms of Hopf coordinates,

31.

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). Then defines a submersion from C(M) to C(N). Consider a horizontal vector a∂_{r} + b_{i}∂_{mi} in T_{(r,m)}C(M). We will show that preserves its length. Now, if df(b_{i}∂_{mi}) = c_{i}∂_{ni}, then . Since ∂_{r}⊥∂_{mi}, as f is a Riemannian submersion. Moreover since ∂_{r}⊥∂_{ni}. Thus is a Riemannian submersion.

33.

S. G. Rajeev, “

Geometry of the motion of ideal fluids and rigid bodies,” e-print

arXiv:0906.0184 [math-ph] (

2009).

http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/10/10.1063/1.4964340

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2016-10-07

2016-10-26

### Abstract

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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