Heavy rigid body with a gyroscope in $\mathbb R^n$

TL;DR

This paper extends classical multidimensional rigid body models by adding gyroscopes, providing Lax representations and proving their integrability, along with geometric insights into their motion.

Contribution

It introduces integrable multidimensional rigid body systems with gyroscopes, offering polynomial Lax pairs and geometric interpretations, advancing the understanding of such systems.

Findings

Constructed Lax representations for new systems

Proved Liouville integrability of extended models

Presented geometric motion representations

Abstract

Starting from the following multidimensional integrable generalizations of the heavy rigid body systems: the Euler top, the Lagrange top, the Lagrange bitop, and the totally symmetric case, we add to each of them a gyroscope. For each of the newly constructed systems, we provide a polynomial matrix Lax representation and prove Liouville integrability. We also present Zhukovskiy's geometric representation of motion of the Euler top with a gyroscope.