Trace formula for linear Hamiltonian systems with its applications to 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.

52 pages, 5 figures

References in corpus (2)

• Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory
• Hill's formula

Cited by in corpus (9)

• Collision index and stability of elliptic relative equilibria in planar n-body problem
• Trace estimation of a family of periodic Sturm-Liouville operators with application to Robe's restricted three-body problem
• Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem
• An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian $n$-body Problem
• Eigenvalue problem of Sturm-Liouville systems with separated boundary conditions
• Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases
• Hill-type formula for Hamiltonian system with Lagrangian boundary conditions
• Rotating Quasi-periodic Solutions of Second Order Hamiltonian Systems with Sub-quadratic Potential
• Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs