Integrability and dynamics of the n-dimensional symmetric Veselova top

TL;DR

This paper extends the Veselova top to n dimensions, deriving equations, invariant measures, and identifying conditions for integrability and quasi-periodic dynamics in symmetric cases.

Contribution

It generalizes the Veselova problem to higher dimensions, providing new insights into its integrability and invariant structures.

Findings

Invariant measure formula derived

Existence of steady rotation solutions shown

Phase space foliated by invariant tori with quasi-periodic dynamics

Abstract

We consider the the n-dimensional generalisation of the nonholonomic Veselova problem. We derive the reduced equations of motion in terms of the mass tensor of the body and determine some general properties of the dynamics. In particular we give a closed formula for the invariant measure, we indicate the existence of steady rotation solutions, and obtain some results on their stability. We then focus our attention on bodies whose mass tensor has a specific type of symmetry. We show that the phase space is foliated by invariant tori that carry quasi-periodic dynamics in the natural time variable. Our results enlarge the known cases of integrability of the multi-dimensional Veselova top. Moreover, they show that in some previously known instances of integrability, the flow is quasi-periodic without the need of a time reparametrisation.