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M. C. Gutzwiller, “Moon-Earth-Sun: The oldest three-body problem,” Rev. Mod. Phys. 70, 589 (1998).
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).
E. J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion (Macmillan, 1877).
J. Laskar, “Is the solar system stable?,” Prog. Math. Phys. 66, 239270 (2013).
R. Montgomery, “A new solution to the three-body problem,” Not. AMS 48(5), 471 (2001).
M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990), Vol. 1.
P. K. Newton, The N-Vortex Problem: Analytical Techniques (Springer-Verlag, New York, 2001).
V. Efimov, “Energy levels arising from resonant two-body forces in a three-body system,” Phys. Lett. B 33, 563 (1970).
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).
F. Calogero, “Solution of a three-body problem in one dimension,” J. Math. Phys. 10, 2191 (1969).
S. G. Rajeev, Advanced Mechanics: From Euler’s Determinism to Arnold’s Chaos (Oxford University Press, 2013).
C. Lanczos, The Variational Principles of Mechanics, 4th ed. (Dover, 1970), p. 139.
V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer Verlag, 1989), p. 245.
L. Casetti, M. Pettini, and E. G. D. Cohen, “Geometric approach to Hamiltonian dynamics and statistical mechanics,” Phys. Rep. 337, 237 (2000); e-print arXiv:cond-mat/9912092.
M. Pettini, in Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Springer, 2007), Chap. 3.
M. Cerruti-Sola, G. Ciraolo, R. Franzosi, and M. Pettini, “Riemannian geometry of Hamiltonian chaos: Hints for a general theory,” Phys. Rev. E 78, 046205 (2008).
M. Cerruti-Sola and M. Pettini, “Geometric description of chaos in two-degrees-of-freedom Hamiltonian systems,” Phys. Rev. E 53, 179 (1996).
K. Ramasubramanian and M. S. Sriram, “Global geometric indicator of chaos and Lyapunov exponents in Hamiltonian systems,” Phys. Rev. E 64, 046207 (2001).
R. Montgomery, “The Three-body problem and the shape sphere,” Am. Math. Mon. 122, 299321 (2015).
R. Montgomery, “Hyperbolic pants fit a three-body problem,” Ergodic Theory Dyn. Syst. 25, 921947 (2005).
R. Montgomery, “Infinitely many syzygies,” Arch. Ration. Mech. 164, 311 (2002).
R. Montgomery, “The zero angular momentum three-body problem: All but one solution has syzygies,” Ergodic Theory Dyn. Syst. 27, 1933 (2007).
A. Chenciner, “Three body problem,” Scholarpedia 2(10), 2111 (2007).
C. Moore, “Braids in classical dynamics,” Phys. Rev. Lett. 70, 3675 (1993).
R. Montgomery, “The N-body problem, the braid group, and action-minimizing periodic solutions,” Nonlinearity 11, 363 (1998).
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. 255264.
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.
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. 203230.
B. O’Neill, Semi-Riemannian Geometry (Academic Press, 1983).
The Hopf map S3S2 is often expressed in Cartesian coordinates. If defines the unit-S3C2 and defines a 2-sphere of radius 1/2 in R3, then and . Using Eq. (10), we may express the Cartesian coordinates wi in terms of Hopf coordinates,
Let f : (M, g) ↦ (N, h) be a Riemannian submersion with local coordinates mi and nj. Let (r, mi) and (r, nj) 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 ar + bimi in T(r,m)C(M). We will show that preserves its length. Now, if df(bimi) = cini, then . Since ∂r⊥∂mi, as f is a Riemannian submersion. Moreover since ∂r⊥∂ni. Thus is a Riemannian submersion.
R. Montgomery and C. Jackman, “No hyperbolic pants for the 4-body problem with strong potential,” Pac. J. Math. 280, 401 (2016).
S. G. Rajeev, “Geometry of the motion of ideal fluids and rigid bodies,” e-print arXiv:0906.0184 [math-ph] (2009).

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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 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 . The geodesic flow on the configuration space ℂ3 (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space ℂ2 and shape space ℝ3 (as well as 𝕊3 and the shape sphere 𝕊2 for the inverse-square potential when = 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 ℂ2, ℝ3, and 𝕊3 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 ℂ2 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 ℂ2 is strictly negative though it could have either sign on ℝ3. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.


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