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Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem
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The Hopf map S3 → S2 is often expressed in Cartesian coordinates. If defines the unit-S3 ⊂ C2 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 a∂r + bi∂mi in T(r,m)C(M). We will show that preserves its length. Now, if df(bi∂mi) = ci∂ni, then . Since ∂r⊥∂mi, as f is a Riemannian submersion. Moreover since ∂r⊥∂ni. Thus is a Riemannian submersion.
S. G. Rajeev
, “Geometry of the motion of ideal fluids and rigid bodies
,” e-print arXiv:0906.0184
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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 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 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 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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