Contact geometry of Hill's approximation in a spatial restricted four-body problem

TL;DR

This paper proves that Hill's approximation in the spatial restricted four-body problem exhibits contact geometry, enabling advanced mathematical tools to analyze the dynamics near Trojan asteroids.

Contribution

It extends the contact property known in three-body problems to the spatial restricted four-body problem, facilitating new analytical approaches.

Findings

Hill's four-body system has the contact property.

Enables use of holomorphic curve techniques and Floer theory.

Applicable in energy ranges where contact property holds.

Abstract

It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-Garc\'ia and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can "contact" the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.