Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body

TL;DR

This paper completely characterizes the stability of all generic equilibria in the Lie-Poisson dynamics of the $rak{so}(4)$ rigid body, revealing new types of equilibria and their stability properties.

Contribution

It identifies and analyzes the stability of all generic equilibria for the $rak{so}(4)$ rigid body, including coordinate type Cartan subalgebras and additional equilibria on curves.

Findings

Three coordinate type Cartan subalgebras yield three Weyl group orbits of equilibria.

Stability of all generic equilibria is fully determined.

Existence of additional equilibria forming curves.

Abstract

The stability for all generic equilibria of the Lie-Poisson dynamics of the $\mathfrak{so}(4)$ rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of $\mathfrak{so}(n)$ are equilibrium points for the rigid body dynamics. In the case of $\mathfrak{so}(4)$ there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit give three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in $\mathfrak{so}(3)$. In addition to these coordinate type Cartan equilibria there are others that come in curves.