### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 101, 11 pages      arXiv:1407.7163      https://doi.org/10.3842/SIGMA.2014.101

### Who's Afraid of the Hill Boundary?

Richard Montgomery
Math Dept. UC Santa Cruz, Santa Cruz, CA 95064, USA

Received August 25, 2014, in final form October 28, 2014; Published online November 02, 2014

Abstract
The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove the conjugate locus of any point near enough to the boundary is a hypersurface tangent to the boundary. Our method of proof is to reduce analysis of geodesics near the boundary to that of solutions to Newton's equations in the simplest model case: a constant force. This model case is equivalent to the beginning physics problem of throwing balls upward from a fixed point at fixed speeds and describing the resulting arcs, see Fig. 2.

Key words: Jacobi-Maupertuis metric; conjugate points.

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