Integrability and Dynamics of the n-Dimensional Symmetric Veselova Top

We consider the n-dimensional generalization 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, 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 quasiperiodic 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 quasiperiodic without the need of a time reparametrization.