On the stability problem for the $\mathfrak{so}(5)$ free rigid body

TL;DR

This paper investigates the stability of equilibria in the $rak{so}(5)$ free rigid body, identifying key integrals, classifying equilibria, and analyzing their stability properties, extending classical rigid body results to higher dimensions.

Contribution

It provides a detailed classification of equilibria for the $rak{so}(5)$ rigid body and analyzes their stability, introducing coordinate type Cartan subalgebras as analogues of classical axes.

Findings

Fifteen Weyl group orbits of equilibria identified

Stability and instability of these equilibria analyzed

Additional ten families of equilibria discovered

Abstract

In the general case of the $\mathfrak{so}(n)$ free rigid body we give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of $\mathfrak{so}(5)$ we prove that there are fifteen coordinate type Cartan subalgebras which on a regular adjoint orbit give fifteen Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body on $\mathfrak{so}(3)$. The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are ten other continuous families of equilibria.