Trace formula for linear Hamiltonian systems with its applications to elliptic Lagrangian solutions

TL;DR

This paper develops trace formulas for linear Hamiltonian and Sturm-Liouville systems, linking the monodromy matrix to eigenvalues of the action functional's Hessian, and applies these to stability analysis of elliptic Lagrangian solutions in the three-body problem.

Contribution

It introduces new trace formulas connecting monodromy matrices with eigenvalues, enabling stability and non-degeneracy analysis of periodic orbits in Hamiltonian systems.

Findings

Derived trace formulas for Hamiltonian and Sturm-Liouville systems.

Provided stability criteria for symmetric periodic orbits.

Estimated stable and hyperbolic regions for elliptic Lagrangian solutions.

Abstract

In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm-Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem. It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagranian solutions.