Linear instability of periodic orbits of free period Lagrangian systems

TL;DR

This paper establishes a criterion for determining the linear instability of periodic orbits in free period Lagrangian systems on Riemannian manifolds, based on spectral index parity.

Contribution

It introduces a new spectral index-based criterion for linear instability applicable to possibly degenerate periodic orbits with orbit cylinders.

Findings

Provides a sufficient condition for linear instability.

Connects spectral index parity to orbit stability.

Applicable to degenerate periodic orbits with orbit cylinders.

Abstract

In this paper we provide a sufficient condition for the linear instability of a periodic orbit for a free period Lagrangian system on a Riemannian manifold. The main result establish a general criterion for the linear instability of a maybe degenerate) periodic orbit admitting a orbit cylinder in terms to the parity of a suitable spectral index encoding the functional and symplectic property of the problem.