Integrability and dynamics of the n-dimensional symmetric Veselova top

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.

37 pp

References in corpus (4)

• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
• Gauge momenta as Casimir functions of nonholonomic systems
• Rolling balls over spheres in R^n
• Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

Cited by in corpus (5)

• Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere
• Gyroscopic Chaplygin systems and integrable magnetic flows on spheres
• Demchenko's nonholonomic case of gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis
• Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere
• Integrability of the $n$-dimensional axially symmetric Chaplygin sphere