A partial reciprocal of Dirichlet Lagrange Theorem detected by Jets

TL;DR

This paper investigates the stability of certain equilibrium points in conservative Hamiltonian systems using jet analysis, demonstrating instability and the existence of asymptotic trajectories under specific conditions.

Contribution

It introduces a novel approach to identify instability in Hamiltonian systems when equilibrium is not a potential energy minimum, extending Krasovskii's results.

Findings

Equilibrium points not at potential minima are generally unstable.

Existence of asymptotic trajectories to these equilibrium points.

Application of jet analysis to stability in Hamiltonian systems.

Abstract

We study the stability of an equilibrium point in a conservative Hamiltonian system in the case that equilibrium is not a minimum of the potential energy and this fact is shown by a jet of this function. Thanks to a modification of a result of Krasovskii, we prove that for a large class of systems under these conditions equilibrium is unstable and there is an asymptotic trajectory to that point.