On the geometry of generalized Chaplygin systems

Mathematical Proceedings of the Cambridge Philosophical Society

Volumn 132, Issue 2, 2002, Pages 323-351

  Cantrijn, Frans ^a   Cortés, Jorge ^b   De Leon, Manuel ^b   De Diego, David Martín ^b  

^a GHENT UNIVERSITY   (Belgium)

^b INSTITUTO DE MATEMÁTICAS Y FÍSICA FUNDAMENTAL   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036331982     PISSN: 03050041     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0305004101005179     Document Type: Conference Paper

References (45)

1. R. ABRAHAM and J. E. MARSDEN. Foundations of mechanics (Benjamin-Cuminings, 2nd edition, 1978).
2. V. I. ARXOLD. Dynamical systems III (Springer-Verlag, 1988).
3. L. BATES and R. CUSHMAN. What is a eompletely- integrable nonliolonomic dynamical system? Rep. Math. Phys. 44 (1999), 29-35.
4. L. BATES, H. GRAUMANN and C. MACDONNELL Examples of gauge conservation laws in nonliolonomic systems. Rep. Math. Phys. 37 (1990), 295-308.
5. L. BATES and J. ŚNIATYCKI. Nonliolonomic reduction. Rep. Math. Phys. 32 (1992), 99-115.
6. A. M. BLOCH and P. E. CROUCH. Newton's law and integrability of nonholonomic systems. SIAM J. Control Optim. 36 (1998), 2020-2039.
7. A. M. BLOCH, P. S. KRISHNAPRASAD, J. E. MARSDEN and R. M. MURRAY. Nonholonomic mechanical systems with symmetry. Arch. Rational Mech. Anal. 136 (1990), 21-99.
8. F. BULLO. Series expansions for the evolution of mechanical control systems. Preprint, 1999 (http://motion.csl. uiuc.edu/bullo/papers).
9. F. BULLO, N. E. LEONARD and A. D. LEWIS. Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups. IEEE Trans. Automat. Control 45 (2000), 1437-1454.
10. F. CANTRIJX, M. DE LEÓX, J. C. MARRERO and D. MARTÍN DE DIEGO. Reduction of nonholonomic mechanical systems with symmetries. Rep. Math. Phys. 42 (1998), 25-45.
11. F. CANTRIJN, M. DE LEÓN, J. C. MARRERO and D. MARTÍN DE DIEGO. Reduction of constrained systems with symmetries. J. Math. Phys. 40 (1999), 795-820.
12. F. CANTRIJN, M. DE LEÓN and D. MARTÍN DE DIEGO. On almost-Poisson structures in nonliolonomic mechanics. Nonlinearity 12 (1999), 721-737.
13. J. CORTÉS and M. DE LEÓN. Reduction and reconstruction of the dynamics of nonholonomic systems. J. Phys. A: Math. Gen. 32 (1999), 8615-8645.
14. J. CORTÉS and S. MARTÍNEZ. Configuration controllability of mechanical systems underactuated by one control. Preprint, 2000 (submitted to SIAM J. Control and Optim.).
15. R. CUSHMAN, D. KEMPPAINEN, J. ŚNIATYCKI and L. BATES. Geometry of nonholonomic constraints. Rep. Math. Phys. 36 (1995), 275-280.
16. V. DRAGOVIĆ, B. GAJIĆ B. JOVANOVIĆ. Generalizations of classical integrable nonliolonomic rigid body systems. J. Phys. A: Math. Gen. 31 (1998), 9801-9869.
17. B. JOVANOVIĆ. Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on SO(4). J. Phys. A:Math. Gen. 31 (1998), 1415-1422.
18. S. D. KELLY and R. M. MURRAY. Geometric phases and robotic locomotion. J. Robotic Systems 12(1995), 417-431.
19. S. KOBAYASHI and K. NOMIZU. Foundations of differential geometry. Interscience Tracts in Pure and Applied Mathematics (Intel-science Publishers, 1963).
20. J. KOILLER. Reduction of some classical non-holonomic systems with symmetry. Arch. Rational Mech. Anal. 118 (1992), 113-148.
21. W. S. KOON and J. E. MARSDEN. The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Rep. Math. Phys. 40 (1997), 21-62.
22. W. S. KOON and J. E. MARSDEN. Poisson reduction of nonholonomic mechanical systems with symmetry. Rep. Math. Phys. 42 (1998), 101-134.
23. V. V. KOZLOV. Invariant measures of the Euler-Poincaré equations on Lie algebras. Funkt. Anal. Prilozh. 22 (1988), 69-70 [in Russian]
24. Funct. Anal. Appl. 22 (1988) [Engl. transl.].
25. O. KRUPKOVA. Mechanical systems with nonholonomic constraints. J. Math. Phys. 38 (1997), 5098-5126.
26. M. DE LEÓX and P. R. RODRIGUES. Methods of differential geometry in analytical mechanics. Math. Ser., no. 152 (North-Holland, 1989).
27. M. DE LEÓX, J. C. MARUERO and D. MARTÍX DE DIEGO. Mechanical systems with non-linear constraints. Int. J. Theor. Phys. 36 (1997), 973-989.
28. M. DE LEÓDECON and D. MARTÍNE DIECO. On the geometry of non-holonomic Lagrangian systems. J. Math. Phys. 37 (1996), 3389-3414.
29. A. D. LEWIS. Affine connections and distributions with applications to nonholonomic mechanics. Rep. Math. Phys. 42 (1998), 135-164.
30. A. D. LEWIS and R. M. MURRAY. Configuration controllability of simple mechanical control systems. SI AM J. Control Optim. 35 (1997), 766-790.
31. C.-M. MARLE. Reduction of Constrained mechanical systems and stability of relative equilibria. Commun. Math. Phys. 174 (1995), 295-318.
32. J. E. MARSDEN, R. MONTGOMERY and T. RATIU. Reduction, symmetry, and phases in mechanics. Mem. Amer. Math. Soc., vol. 436 (AMS, 1990).
33. E. MASSA and E. PAOAXI. A new look at classical mechanics of constrained systems. Ann. Inst. H. Poincaré (Phys. Théor.) 66 (1997), 1-36.
34. J. NEIMARK and N. FUFAEV. Dynamics of nonholonomic systems. Transactions of Mathematical Monographs, vol. 33 (AMS, 1972).
35. J. OSTROWSKI. Geometric perspectives on the mechanics and control of undulatory locomotion. PhD thesis (California Institute of Technology, 1995).
36. A. SHAPERE and F. WILCZEK. Geometry of self-propulsion at low Reynolds number. J. Fluid. Mech. 198 (1989), 557-585.
37. S. V. STANCHENKO. Nonholonomic Chaplygin systems. Prikl. Mat. Mehk. 53 (1989), 16-23 [in Russian].
38. D. SAUXDERS, F. CAXTRIJX and W. SARLET. Regularity aspects and Hamiltonization of nonholonomic systems. J. Phys. A:Math. Gen. 32 (1999), 6869-6890.
39. J. Z. SYXOE. Geodesies in nonholonomic geometry. Math. Annalen 99 (1928), 738-751.
40. A. J. VAX DER SCHAFF and B. M. MASCIIKE. On the Hamiltonian formulation of nonholonomic mechanical systems. Hep. Math. Phys. 34 (1994), 225-233.
41. A. M. VERSHIK and L. D. FADEEV. Lagrangian mechanics in invariant form. Sel. Math. Sov. 1 (1981), 339-350.
42. A. P. VESELOV and L. E. VESELOYA. Flows on Lie groups with nonholonomic constraint and integrable nonhamiltonian systems. Funkt. Anal. Prilozh. 20 (1986), 65-66 [in Russian]
43. Funct. Anal. Appl. 20 (1986), 308-309 [Engl. transl.].
44. L. S. WANG and P. S. KRISHNAPRASAD. Gyroscopic control and stabilization. J. Nonlinear Sci. 2 (1992), 367-415.
45. R. YANG. Nonholonomic Geometry, mechanics and control. PhD thesis (Systems Research Institute, University of Maryland, 1992).