# Integrability and Dynamics of the *n*-Dimensional Symmetric Veselova Top

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## Abstract

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.

## Keywords

Nonholonomic dynamics Integrability Quasi-periodicity Symmetry Singular reduction## Mathematics Subject Classification

37J60 70E17 70E40 58D19 34A30## Notes

### Acknowledgements

LGN and JM are grateful to the hospitality of the Department of Mathematics Tullio Levi-Civita of the University of Padova, during its 2018 intensive period “Hamiltonian Systems”. LGN acknowledges the Alexander Von Humboldt Foundation for a Georg Forster Research Fellowship that funded a visit to TU Berlin where the last part of this work was completed. The authors are thankful to Božidar Jovanović for comments on an early draft of this paper and for sharing the recent preprint Gajić and Jovanović (2018) with us.

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