On the Lagrange-Dirichlet converse in dimension three

TL;DR

This paper proves that in three-dimensional real analytic mechanical systems, most non-strict local minima of the potential are Lyapunov unstable equilibria, highlighting instability in typical cases.

Contribution

It establishes that in dimension three, a generic set of non-strict local minima are Lyapunov unstable, extending the understanding of stability in such systems.

Findings

Most non-strict local minima are Lyapunov unstable in dimension three.

Open and dense subset of potential minima are unstable.

Results apply to real analytic potentials in three-dimensional systems.

Abstract

Consider a mechanical system with a real analytic potential. We prove that in dimension three, there is an open and dense subset of the set of non strict local minimums of the potential such that every one of its points is a Lyapunov unstable equilibrium point.