On the stability problem and a bifurcation phenomenon for a special class of equilibria of the free rigid body on $\mathfrak{so}(5)$

TL;DR

This paper investigates the stability and bifurcation phenomena of certain equilibria in the free rigid body dynamics on the Lie algebra rak{so}(5), identifying stability regions and the role of constants of motion.

Contribution

It introduces a set of non-canonical constants of motion and analyzes their impact on stability analysis using energy methods for rak{so}(5).

Findings

Established instability and stability regions for equilibria.

Presented constants of motion ensuring integrability.

Highlighted the importance of choosing appropriate constants in stability analysis.

Abstract

The stability of a special class of equilibria for the free rigid body on $\mathfrak{so}(5)$ is discussed. An instability region and two stability regions are established. The list of constants of motion which assure the complete integrability of the system is presented and used in the approach of stability with energy methods. The importance of appropriate choosing for the constants of motion in the energy methods is pointed out. A set of non-canonical constants of motion for the free rigid body on $\mathfrak{so}(5)$ and their role are analyzed.