Odd index of the amended potential implies linear instability

TL;DR

This paper establishes that for symmetric mechanical systems, an odd Morse index of the amended potential guarantees linear instability of the relative equilibrium, and also offers conditions for spectral instability.

Contribution

It introduces a criterion linking the oddness of the Morse index of the amended potential to linear instability, expanding understanding of stability in mechanical systems.

Findings

Odd Morse index implies linear instability.

Provides sufficient condition for spectral instability.

Enhances criteria for stability analysis in symmetric systems.

Abstract

For a relative equilibrium of a symmetric simple mechanical system, if the Morse index of the corresponding amended potential is odd, whether the nullity is zero or not, it is linearly unstable. We also provide a sufficient condition for spectral instability.