Nonholonomic LR systems as generalized chaplygin systems with an invariant measure and flows on homogeneous spaces

Journal of Nonlinear Science

Volumn 14, Issue 4, 2004, Pages 341-381

  Fedorov, Y N ^a,b   Jovanovic B ^c  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^c MATHEMATICAL INSTITUTE   (Serbia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84867947272     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-004-0603-3     Document Type: Article

References (52)

1. Appel, P. 1901. Remarques d'orde analytique sur un nouvelle forme des equationes de la dynamique. J. Math. Pure Appl. 7, ser. 5, 5-12.
2. Arnold, V. I., Kozlov, V. V., Neishtadt, A. I. 1985. Mathematical aspects of classical and celestial mechanics. Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fundamental'nye Napravleniya, vol. 3, VINITI, Moscow. English trans.: Encyclopadia of Math. Sciences, vol. 3, Springer-Verlag, Berlin, 1989.
3. Baker, H. F. 1897. Abels Theorem and the Allied Theory Including the Theory of Theta Functions. Cambridge Univ. Press, Cambridge.
4. Bates, L., Cushman, R. 1999. What is a completely integrable nonholonomic dynamical system? Rep. Math. Phys. 44, no. 1-2, 29-35.
5. Bates, L., ́ Sniatycki, J. 1993. Nonholonomic reduction. Rep. Math. Phys. 37, 99-115.
6. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., Murray, R. M. 1996. Nonholonomic mechanical systems with symmetry. Arch. Rat. Mech. Anal. 136, 21-99.
7. Bloch, A. M., Brockett, R., Crouch, P. 1997. Double bracket equations and geodesic flows on symmetric spaces. Commun. Math. Phys. 187, 357-373. (Pubitemid 127354082)
8. Bogoyavlensky, O. I. 1986. Integrable cases of rigid body dynamics and integrable systems on the ellipsoid. Commun. Math. Phys. 103, 305-322.
9. Bolsinov, A. V., Kozlov, V. V., Fomenko, A. T. 1995. The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body. Usp. Mat. Nauk. 50, no. 3, 3-32; English trans.: Russian Math. Surveys 50 (1995), no. 3, 473-501.
10. Bolsinov, A. V., Jovanovíc, B. 2004. Complete involutive algebras of functions on cotangent bundles of homogeneous spaces. Math. Z. 246, no. 1-2, 213-236. (Pubitemid 38157185)
11. Bolsinov, A. V., Jovanovíc, B. 2003. Noncommutative integrability, moment map and geodesic flows. Annals Global Anal. Geometry 23, no. 4, 305-322.
12. Borisov, A. V., Mamaev, I. 2001. Chaplygin's ball rolling problem is Hamiltonian. (Russian) Mat. Zametki 70, no. 5, 793-795; English trans.: Math. Notes 70 (2001), no. 5-6, 720-723.
13. Borisov, A. V., Mamaev, I. S. 2002. Rolling of a rigid body on plane and sphere. Hierarchy of dynamics. Regul. Chaotic Dynam. 7, 177-200.
14. Brailov, A. V. 1986. Construction of complete integrable geodesic flows on compact symmetric spaces. Izv. Acad. Nauk SSSR, Ser. matem. 50, no. 2, 661-674, (Russian); English trans.: Math. USSR-Izv. 50 (1986), no. 4, 19-31.
15. Cantrijn, F., de Leon, M., Marrero, J. C., Martin de Diego, D. 1998. Reduction of nonholonomic mechanical systems with symmetries. Rep. Math. Phys., 42, 25-45. (Pubitemid 128181358)
16. Cantrijn, F., Cortes, J., de Leon, M., Martin de Diego, D. 2002.Onthe geometry of generalized Chaplygin systems. Math. Proc. Cambridge Philos. Soc. 132, no. 2, 323-351. (Pubitemid 41652363)
17. Chaplygin, S. A. 1903. On a rolling sphere on a horizontal plane. Mat. Sbornik 24 139-168 (Russian).
18. Chaplygin, S. A. 1911. On the theory of the motion of nonholonomic systems. Theorem on the reducing multiplier. Mat. Sbornik 28, no. 2, 303-314 (Russian).
19. Chaplygin, S. A. 1981. Selected works. Nauka, Moskva (Russian).
20. Clebsch, A., Gordan, P. 1866. Theorie der abelschen Funktionen. Teubner, Leipzig.
21. Dirac, P. A. 1950. On generalized Hamiltonian dynamics. Can. J. Math. 2, no. 2, 129-148.
22. Dragovíc, V. 2002. The Appell hypergeometric functions and classical separable mechanical systems. J. Phys. A: Math. Gen. 35, 2213-2221. (Pubitemid 37026667)
23. Dubrovin, B. A., Novikov, S. P., Fomenko, A. T. 1989. The Modern Geometry. Springer, New York.
24. Efimov, M. 1953. On the Chaplygin equations of nonholonomic mechanics and the method of reducing multiplier. Ph.D. Thesis. Institute of Mechanics RAS, Moscow (Russian).
25. Ehlers, K., Koiller, J., Rios, P., Nonholonomic Systems: Cartan's equivalence and Hamiltonization. Preprint http://www.esi.ac.at/Preprint-shadows/ esi1389.html.
26. Fedorov, Yu. 1989. On two integrable nonholonomic problems of classical dynamics. Vestn. Moskov. Univ. Ser. I, Mat. Mekh., no. 4, 38-41 (Russian).
27. Fedorov,Yu. 1999. Systems with an invariant measure on Lie groups. In Hamiltonian Systems with Three or More Degrees of Freedom. Ed. C. Simo. Nato ASI Series C. 533. Kluwer Academic Publishers, New York, 350-357.
28. Fedorov, Yu. N., Kozlov, V. V. 1995. Various aspects of n-dimensional rigid body dynamics. Amer. Math. Soc. Transl. Series 2, 168, 141-171.
29. Hermans, J. 1995. A symmetric sphere rolling on a surface. Nonlinearity 8, 493-515.
30. Iliev, I. 1985. On the conditions for the existence of the reducing Chaplygin factor. J. Appl. Math. Mech. 49, no. 3, 295-301.
31. Jacobi, K. 1884. Vorlesungen ̈uber Dynamik. Supplementband, Berlin.
32. Jovanovíc, B. 1999. Nonholonomic left and right flows on Lie groups. J. Phys. A: Math. Gen. 32, 8293-8302.
33. Jovanovíc, B. 2001. Geometry and integrability of Euler-Poincaŕe-Suslov equations. Nonlinearity 14, no. 6, 1555-1657.
34. Jovanovíc, B. 2003. Some multidimensional integrable cases of nonholonomic rigid body dynamics. Reg. Chaot. Dynam. 8, no. 1, 125-132.
35. Kn̈orrer, H. 1982. Geodesics on quadrics and a mechanical problem of C. Neumann. J. Reine Angew. Math. 334, 69-78 .
36. Koiller, J. 1992. Reduction of some classical nonholonomic systems with symmetry. Arch. Rat. Mech. 118, 113-148.
37. Koon, W. S., Marsden, J. E. 1997. The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems. Rep. Math. Phys. 40, 21-62. (Pubitemid 127163401)
38. Kozlov, V. V. 1985. On the integrability theory of equations of nonholonomic mechanics. Advances in Mechanics, 8, no. 3, 85-107 (Russian).
39. Kozlov, V. V. 1988. Invariant measures of the Euler-Poincaŕe equations on Lie algebras. Funkt. Anal. Prilozh. 22, 69-70 (Russian); English trans.: Funct. Anal. Appl. 22 (1988), no. 1, 58-59.
40. Moser, J. 1980. Geometry of quadric and spectral theory. In Chern Symposium 1979, Springer-Verlag, Berlin-Heidelberg-New York, 147-188.
41. Moshchuk, N. K. 1987. Reducing the equations of motion of certain nonholonomic Chaplygin systems to Lagrangian and Hamiltonian form. Prikl. Mat. Mekh. 51, no. 2, 223-229 (Russian); English trans.: J. Appl. Math. Mech. 51 (1987), no. 2, 172-177.
42. Neimark, J. I., Fufaev, N. A. 1972. Dynamics of nonholonomic systems. Trans. of Math. Mon. 33, AMS, Providence.
43. Neumann, C. 1859. De probleme quodam mechanico, quod ad primam integralium ultraellipticoram classem revocatum. J. Reine Angew. Math. 56.
44. Stanchenko, S. 1989. Nonholonomic Chaplygin systems. Prikl. Mat. Mekh. 53, no. 1, 16-23; English trans.: J. Appl. Math. Mech. 53 (1989), no. 1, 11-17.
45. Sumbatov, A. 2002. Nonholonomic systems. Regular & Chaotic Dynam. 7, no. 2, 221-238.
46. Thimm, A. 1981. Integrable geodesic flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 1, 495-517.
47. Veselov, A. P., Veselova, L. E. 1986. Flows on Lie groups with nonholonomic constraint and integrable non-Hamiltonian systems. Funkt. Anal. Prilozh. 20, no. 4, 65-66 (Russian); English trans.: Funct. Anal. Appl. 20 (1986), no. 4, 308-309.
48. Veselov, A. P., Veselova, L. E. 1988. Integrable nonholonomic systems on Lie groups. Mat. zametki 44, no. 5, 604-619 (Russian); English trans.: Math. Notes 44 (1988), no. 5.
49. Zenkov, D. V. 1995. The geometry of the Routh problem. J. Nonlin. Sci. 5, 503-519.
50. Zenkov, D. V., Bloch, A. M. 2000. Dynamics of the n-dimensional Suslov problem. J. Geom. Phys. 34, no. 2, 121-136.
51. Zenkov, D. V., Bloch, A. M. 2000. Dynamics of generalized Euler tops with constraints. Proc. Internat. Conference Dynam. Syst. & Diff. Eq., 398-405.
52. Zenkov, D. V., Bloch, A. M. 2003. Invariant measures of nonholonomic flows with internal degrees of freedom. Nonlinearity 16, 1793-1807. (Pubitemid 37316463)