Weak instability of Hamiltonian equilibria

TL;DR

This paper clarifies the concept of weak instability in Hamiltonian systems, illustrating that some equilibria can be unstable without nearby non-constant motions, and can even appear stable under linearization.

Contribution

It provides an expository analysis of weak instability in Hamiltonian equilibria, including examples that demonstrate this phenomenon and its distinction from classical stability.

Findings

Weak instability can occur without non-constant motions approaching the equilibrium.

A weakly unstable equilibrium can be linearly stable.

Examples illustrate the subtlety of stability concepts in Hamiltonian systems.

Abstract

This is an expository paper on Lyapunov stability of equilibria of autonomous Hamiltonian systems. Our aim is to clarify the concept of weak instability, namely instability without non-constant motions which have the equilibrium as limit point as time goes to minus infinity. This is done by means of some examples. In particular, we show that a weakly unstable equilibrium point can be stable for the linearized vector field.