Who's Afraid of the Hill Boundary?

TL;DR

This paper investigates the behavior of geodesics in the Jacobi-Maupertuis metric near the Hill boundary, revealing conjugate points and the structure of the conjugate locus through analysis of a simplified physics model.

Contribution

It proves the existence of conjugate points near the Hill boundary and characterizes the conjugate locus as a tangent hypersurface, using a reduction to a constant force model.

Findings

Geodesics near the Hill boundary contain conjugate points close to the boundary.

The conjugate locus near the boundary is a hypersurface tangent to it.

Analysis of a constant force model explains geodesic behavior near the boundary.

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.