SIGMA 3 (2007), 051, 12 pages      math.SG/0703665      doi:10.3842/SIGMA.2007.051
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Francesco Fassò and Andrea Giacobbe
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35131 Padova, Italy

Received November 20, 2006, in final form March 15, 2007; Published online March 22, 2007

Abstract
Bifibrations, in symplectic geometry called also dual pairs, play a relevant role in the theory of superintegrable Hamiltonian systems. We prove the existence of an analogous bifibrated geometry in dynamical systems with a symmetry group such that the reduced dynamics is periodic. The integrability of such systems has been proven by M. Field and J. Hermans with a reconstruction technique. We apply the result to the nonholonomic system of a ball rolling on a surface of revolution.

Key words: systems with symmetry; reconstruction; integrable systems; nonholonomic systems.

pdf (400 kb)   ps (322 kb)   tex (276 kb)

References

1. Blaom A.D., A geometric setting for Hamiltonian perturbation theory, Mem. Amer. Math. Soc. 153 (2001), 1-112.
2. Bogoyavlenskij O.I., Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys. 196 (1998), 19-51.
3. Bröcker T., tom Dieck T., Representations of compact Lie groups, Graduate Texts in Mathematics, Vol. 98, Springer-Verlag, New York, 1995.
4. Cushman R., Duistermaat J.J., Non-Hamiltonian monodromy, J. Differential Equations 172 (2001), 42-58.
5. Cushman R., Duistermaat H., Snyaticky J., Non-holonomic systems, in preparation.
6. Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.
7. Fassò F., Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87 (2005), 93-121.
8. Fassò F., Giacobbe A., Sansonetto N., Periodic flow, rank-two Poisson structures, and nonholonomic mechanics, Regular Chaotic Mech. 19 (2005), 267-284.
9. Fedorov Yu.N., Systems with an invariant measure on Lie groups, in Hamiltonian Systems with Three or More Degrees of Freedom (1995, S'Agarò), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 533, Kluwer, Dordrecht, 1999, 350-356.
10. Field M., Local structure of equivariant dynamics, in Singularity Theory and Its Applications, Part II (1988/1989, Coventry), Lecture Notes in Math., Vol. 1463, Springer, Berlin, 1991, 142-166.
11. Fomenko A.T., Trofimov V.V., Integrable systems on Lie algebras and symmetric spaces, Advanced Studies in Contemporary Mathematics, Vol. 2, Gordon and Breach Science Publishers, New York, 1988.
12. Hermans J., Rolling rigid bodies with and without symmetries, PhD Thesis, University of Utrecht, 1995.
13. Hermans J., A symmetric sphere rolling on a surface, Nonlinearity 8 (1995), 493-515.
14. Karasev M.V., Maslov V.P., Nonlinear Poisson brackets. Geometry and quantization, Translations of the AMS, Vol. 119, AMS, Providence, R.I., 1993.
15. Marsden J.E., Montgomery R., Ratiu T., Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc. 88 (1990), no. 436.
16. Meigniez G., Submersion, fibrations and bundles, Trans. Amer. Math. Soc. 354 (2002), 3771-3787.
17. Mischenko A.S., Fomenko A.T., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978), 113-121.
18. Routh E.J., Treatise on the dynamics of a system of rigid bodies (advanced part), Dover, New York, 1955.
19. Zenkov D.V., The geometry of the Routh problem, J. Nonlinear Sci. 5 (1995), 503-519.
20. Zung N.T., Torus action and integrable systems, in Topological Methods in the Theory of Integrable Systems, Editors A.V. Bolsinov, A.T. Fomenko and A.A. Oshemkov, Cambridge Scientific Publications, 2006, 289-328, math.DS/0407455.