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.

[1]  B. Jovanović.  Rolling balls over spheres in Rn , 2018, 1804.03697.

[2]  J. Koiller.  Reduction of some classical non-holonomic systems with symmetry , 1992 .

[3]  Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System , 2007, math/0703665.

[4]  S. A. Chaplygin.  On the theory of motion of nonholonomic systems. The reducing-multiplier theorem , 2008 .

[5]  Jeff Moehlis,et al.  Equivariant dynamical systems , 2007, Scholarpedia.

[6]  A. Veselov,et al.  Integrable nonholonomic systems on Lie groups , 1988 .

[7]  Yuri N. Fedorov.  2 1 2 A ug 2 00 3 Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces ∗ , 2003 .

[8]  P. Krishnaprasad,et al.  Nonholonomic mechanical systems with symmetry , 1996 .

[9]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[10]  Jair Koiller,et al.  Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization , 2005 .

[11]  Michael Field,et al.  Dynamics and Symmetry , 2007 .

[12]  Yuri N. Fedorov,et al.  Hamiltonization of the generalized Veselova LR system , 2009 .

[13]  Tudor S. Ratiu,et al.  The motion of the free $n$-dimensional rigid body , 1980 .

[14]  B. Jovanović.  Rolling balls over spheres in $ \newcommand{\m}{\mathfrak m} {\mathbb{R}^n}$ , 2018, Nonlinearity.

[15]  PERIODIC FLOWS, RANK-TWO POISSON STRUCTURES, AND NONHOLONOMIC MECHANICS , 2005 .

[16]  Yuri N. Fedorov,et al.  Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces , 2003, J. Nonlinear Sci..

[17]  B. Gajic,et al.  Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere , 2018, Nonlinearity.

[18]  Gauge Momenta as Casimir Functions of Nonholonomic Systems , 2016, 1610.05618.

[19]  Quasi-periodicity in relative quasi-periodic tori , 2014, 1411.7976.

[20]  Francesco Fassò,et al.  Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces , 2015, J. Nonlinear Sci..

[21]  Michael Field.  Equivariant dynamical systems , 1980 .

[22]  Bozidar Jovanovic,et al.  Hamiltonization and Integrability of the Chaplygin Sphere in ℝn , 2009, J. Nonlinear Sci..

[23]  A. Izosimov.  Stability of relative equilibria of multidimensional rigid body , 2012, 1203.3985.

[24]  J Hermans,et al.  A symmetric sphere rolling on a surface , 1995 .

[25]  B. Jovanović.  LR and L+R systems , 2009, 0902.1656.