Systems of Hess-Appel'rot Type and Zhukovskii Property

TL;DR

This paper explores systems of Hess-Appel'rot type, focusing on their invariant relations, partial reductions, and the Zhukovskii property, illustrating their integrability and non-integrable dynamics with examples including the magnetic pendulum.

Contribution

It introduces the concept of the Zhukovskii property for Hess-Appel'rot systems and demonstrates its prevalence, along with classical and algebro-geometric integration methods for these systems.

Findings

Zhukovskii property is common in Hess-Appel'rot type systems

Partial reduction isolates integrable parts of the dynamics

Magnetic pendulum on Grassmannian is a partial reduction example

Abstract

We start with a review of a class of systems with invariant relations, so called {\it systems of Hess--Appel'rot type} that generalizes the classical Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the {\it Zhukovskii property}: these are Hamiltonian systems with invariant relations, such that partially reduced systems are completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rote type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type - the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian $Gr^+(4,2)$ has natural interpretation within Zhukovskii property and it is equivalent to a partial reduction of certain system of Hess-Appel'rot type. We perform a classical and an algebro-geometric integration of the system, as an example of an isoholomorphic system. The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems.