Spectrum estimates of Hill's lunar problem

TL;DR

This paper studies the action spectrum of Hill's lunar problem by relating it to symplectic capacities and the rotating Kepler problem, providing estimates and existence results for periodic orbits with bounded action.

Contribution

It introduces a reinterpretation of spectral invariants as symplectic capacities and determines the action spectrum for the rotating Kepler problem, applying these to Hill's lunar problem.

Findings

Established bounds on the action spectrum of Hill's lunar problem.

Proved existence of a periodic orbit with action less than π for certain energies.

Connected spectral invariants with symplectic capacities in cotangent bundles.

Abstract

We investigate the action spectrum of Hill's lunar problem by observing inclusions between the Liouville domains enclosed by the regularized energy hypersurfaces of the rotating Kepler problem and Hill's lunar problem. In this paper, we reinterpret the spectral invariant corresponding to every nonzero homology class $\alpha \in H_*(\Lambda N)$ in the loop homology as a symplectic capacity $c_N(M, \alpha)$ for a fiberwise star-shaped domain $M$ in a cotangent bundle with canonical symplectic structure $(T^*N, \omega_{can}=d \lambda_{can})$. Also, we determine the action spectrum of the regularized rotating Kepler problem. As a result, we obtain estimates of the action spectrum of Hill's lunar problem. This will show that there exists a periodic orbit of Hill's lunar problem whose action is less than $\pi$ for any energy $-c < -{3^{4 \over 3} \over 2}$.