The symplectic reduction of the linearized Hamiltonian systems at elliptic relative equilibria of four-body problem

TL;DR

This paper analyzes the linearized Hamiltonian systems at elliptic relative equilibria in the four-body problem, showing a symplectic reduction into simpler systems to aid stability analysis.

Contribution

It proves the symplectic reduction of the linearized Hamiltonian system at elliptic relative equilibria in the four-body problem, simplifying stability analysis.

Findings

The linearized system splits into two independent Hamiltonian systems.

One system corresponds to the Kepler 2-body problem.

The other is an implicit essential part of the linearized system.

Abstract

In this paper, we consider the elliptic relative equilibria of four-body problem. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic relative equilibria of $4$-bodies splits into two independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and the other system is the essential part of the linearized Hamiltonian system, which is given implicitly. The reduction can be applied to the stability problem of such elliptic relative equilibria of four-body problem.