On the Lyapunov instability in Lagrangian dynamics

TL;DR

This paper introduces a new Lyapunov instability criterion for Lagrangian dynamical systems, linking instability to solutions of a specific linear PDE, with additional conditions in magnetostatic fields.

Contribution

It presents a novel Lyapunov instability criterion for equilibrium points in Lagrangian mechanics, including cases with magnetic fields, based on PDE solutions.

Findings

New criterion for Lyapunov instability in Lagrangian systems

Additional instability conditions in magnetostatic fields

Link between PDE solutions and system stability

Abstract

In the context of mechanical Lagrangian dynamics, we prove a new Lyapunov instability criterion for a non strict local minimum equilibrium point of a smooth potential where the sufficient condition for instability is the existence of a smooth solution of a certain linear PDE derived from the mechanical Lagrangian governing the dynamics. In the presence of a magnetostatic field, we also give an additional sufficient condition for the motion of a charged particle to be Lyapunov unstable.