Energy methods in the stability problem for the $\mathfrak{so}(4)$ free rigid body

TL;DR

This paper addresses the stability of the $rak{so}(4)$ free rigid body by constructing Lyapunov functions from Mishchenko's constants, providing a comprehensive solution to the stability problem.

Contribution

It introduces Lyapunov functions as linear combinations of Mishchenko's constants to analyze stability, completing the stability classification for the $rak{so}(4)$ rigid body.

Findings

Complete stability classification for $rak{so}(4)$ rigid body

Construction of Lyapunov functions from Mishchenko's constants

Verification of stability for all equilibrium cases

Abstract

For the $\mathfrak{so}(4)$ free rigid body the stability problem for the isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found we construct a Lyapunov function. These Lyapunov functions are linear combinations of Mishchenko's constants of motion.