# 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.

## References

- Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, Berlin (1988)Google Scholar
- Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal.
**136**, 21–99 (1996)MathSciNetCrossRefGoogle Scholar - Cushman, R., Duistermaat, J.J., Śniatycki, J.: Geometry of Nonholonomically Constrained Systems. Advanced Series in Nonlinear Dynamics, 26. World Scientific Publishing, Singapore (2010)zbMATHGoogle Scholar
- Chaplygin, S.A.: On the theory of the motion of nonholonomic systems. The reducing-multiplier theorem. Regul. Chaotic Dyn.
**13**, 369–376 (2008). [Translated from Matematicheskiǐ Sbornik (Russian) 28 (1911), by A. V. Getling]Google Scholar - Duistermaat, J.J.: Lie Groups. Springer, Berlin (2000)CrossRefGoogle Scholar
- Ehlers, K., Koiller, J., Montgomery, R., Rios, P.M.: Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization. In: The Breath of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 75–120. Birkhäuser Boston, Boston, MA (2005)Google Scholar
- Fassò, F., García-Naranjo, L.C., Giacobbe, A.: Quasiperiodicity in relative quasiperiodic tori. Nonlinearity
**28**, 4281–4301 (2015)MathSciNetCrossRefGoogle Scholar - Fassò, F., Giacobbe, A.: Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system. SIGMA Symmetry Integr. Geom. Methods Appl.
**3**, 051 (2007)MathSciNetzbMATHGoogle Scholar - Fassò, F., Giacobbe, A., Sansonetto, N.: Periodic flows, rank-two Poisson structures, and nonholonomic mechanics. Regul. Chaotic Dyn.
**10**, 267–284 (2005)MathSciNetCrossRefGoogle Scholar - Fassò, F., Sansonetto, N.: Conservation of ‘moving’ energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces. J. Nonlinear Sci.
**26**, 519–544 (2016)MathSciNetCrossRefGoogle Scholar - Fedorov, Y.N., Jovanović, B.: Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces. J. Nonlinear Sci.
**14**, 341–381 (2004)MathSciNetCrossRefGoogle Scholar - Fedorov, Y.N., Jovanović, B.: Hamiltonization of the generalized Veselova LR system. Regul. Chaotic Dyn.
**14**, 495–505 (2009)MathSciNetCrossRefGoogle Scholar - Fedorov, Y.N., Kozlov, V.V.: Various aspects of \(n\)-dimensional rigid body dynamics. Am. Math. Soc. Transl. (2)
**168**, 141–171 (1995)MathSciNetzbMATHGoogle Scholar - Field, M.J.: Equivariant dynamical systems. Trans. Am. Math. Soc.
**259**, 185–205 (1980)MathSciNetCrossRefGoogle Scholar - Field, M.J.: Dynamics and Symmetry. Imperial College Press, London (2007)CrossRefGoogle Scholar
- Gajić, B., Jovanović, B.: Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere. arXiv:1805.10610 (2018)
- García-Naranjo, L.C., Montaldi, J.: Gauge momenta as Casimir functions of nonholonomic systems. Arch. Ration. Mech. Anal.
**228**, 563–602 (2018)MathSciNetCrossRefGoogle Scholar - Hermans, J.: A symmetric sphere rolling on a surface. Nonlinearity
**8**, 493–515 (1995)MathSciNetCrossRefGoogle Scholar - Izosimov, A.: Stability of relative equilibria of multidimensional rigid body. Nonlinearity
**27**, 1419–1443 (2014)MathSciNetCrossRefGoogle Scholar - Jovanović, B.: LR and L + R systems. J. Phys. A
**42**, 225202 (2009)MathSciNetCrossRefGoogle Scholar - Jovanović, B.: Hamiltonization and integrability of the Chaplygin sphere in \(\mathbb{R}^n\). J. Nonlinear Sci.
**20**, 569–593 (2010)MathSciNetCrossRefGoogle Scholar - Jovanović, B.: Rolling balls over spheres in \(\mathbb{R}^n\). Nonlinearity
**31**, 4006–4030 (2018)MathSciNetCrossRefGoogle Scholar - Koiller, J.: Reduction of some classical nonholonomic systems with symmetry. Arch. Ration. Mech. Anal.
**118**, 113–148 (1992)MathSciNetCrossRefGoogle Scholar - Ratiu, T.S.: The motion of the free n-dimensional rigid body. Indiana Univ. Math. J.
**29**, 609–629 (1980)MathSciNetCrossRefGoogle Scholar - Veselova, L.E.: New cases of the integrability of the equations of motion of a rigid body in the presence of a nonholonomic constraint. Geometry, differential equations and mechanics (Russian) (Moscow, 1985), pp. 64–68, Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow (1986)Google Scholar
- Veselov, A.P., Veselova, L.E.: Integrable nonholonomic systems on Lie groups. Mat. Notes
**44**(5–6), 810–819 (1988). [Russian original in Mat. Zametki**44**(5), 604–619 (1988).]MathSciNetCrossRefGoogle Scholar