Integrators for nonholonomic mechanical systems

Journal of Nonlinear Science

Volumn 16, Issue 4, 2006, Pages 283-328

  McLachlan, R ^a   Perlmutter, M ^a  

^a MASSEY UNIVERSITY   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84867946135     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-005-0698-1     Document Type: Review

References (30)

1. L. Bates and J. ́ Sniatycki, Nonholonomic reduction, Rep. Math. Phys. 32 (1993), no. 1, 99-115.
2. A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003.
3. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Ratl. Mech. Anal. 136 (1996), no. 1, 21-99.
4. A. M. Bloch, J. E. Marsden, and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc. 52 (2005), 324-333. (Pubitemid 41647479)
5. A. I. Bobenko and Y. B. Suris, Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top, Commun. Math. Phys. 204 (1998), 147-188.
6. A. I. Bobenko and Yu. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaŕe equations, and semidirect products, Lett. Math. Phys. 49 (1999), 79-93.
7. T. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A, to appear.
8. F. Cantrijn, J. Cort́es, M. de Léon, and D. Martin de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 2, 323-351. (Pubitemid 41652363)
9. H. Cendra, J. E. Marsden, and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems. Mathematics Unlimited-2001 and Beyond, 221-273, Springer, Berlin, 2001.
10. Chaplygin, S. A. On a ball's rolling on a horizontal plane, Regul. Chaotic Dynam. 7 (2002), no. 2, 131-148, translation of Mat. Sb. 24 (1903), 139-168.
11. J. Cort'es, Geometric, Control, and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics vol. 1793, Springer-Verlag, New York, 2002.
12. J. Cort'es, Energy-conserving nonholonomic integrators, Disc. Contin. Dynam. Syst., suppl., (2003), 189-199.
13. J. Cort'es and M. de Léon, Reduction and reconstruction of the dynamics of nonholonomic systems, J. Phys. A 32 (1999), 8615-8645.
14. J. Cort́es and S. Martinez, Non-holonomic integrators, Nonlinearity 14 (2001), no. 5, 1365-1392.
15. K. Ehlers, J.Koiller, R. Montgomery, and Pedro M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry, pp. 75-120, Progr. Math., 232, Birkḧauser, Boston, 2005.
16. J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Ratl. Mech. Anal. 118 (1992), no. 2, 113-148.
17. I. Kupka and W. Oliva, The non-holonomic mechanics, Special issue in celebration of Jack Hale's 70th birthday, J. Diff. Eq. 169 (2001), no. 1, 169-189.
18. B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge University Press, Cambridge, 2004.
19. M. de Léon, D. Martin de Diego, and A. Santamaria-Merino, Geometric integrators and nonholonomic mechanics, J. Math. Phys. 45 (2004), 1042-1062.
20. C.-M. Marle, Reduction of constrained mechanical systems and stability of relative equilibria, Commun. Math. Phys. 174 (1995), no. 2, 295-318. (Pubitemid 126006884)
21. C.-M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics (Calgary, AB, 1997), Rep. Math. Phys. 42 (1998), no. 1-2, 211-229. (Pubitemid 128181366)
22. C.-M. Marle, On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints. Classical and quantum integrability (Warsaw, 2001), 223-242, Banach Center Publ., 59, Polish Acad. Sci., Warsaw, 2003.
23. J. E. Marsden, S. Pekarsky, and S. Shkoller, Discrete Euler-Poincaŕe and Lie-Poisson Equations, Nonlinearity 12 (1999), no. 6, 1647-1662.
24. J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica 10 (2001), 1-159.
25. R. I. McLachlan and M. Perlmutter, Energy drift in reversible time integration, J. Phys. A 37, no. 45 (2004), L593-L598. (Pubitemid 39594040)
26. R. I. McLachlan and G. R.W. Quispel, Geometric integrators for ODEs, J. Phys. A, to appear.
27. J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Commun. Math. Phys. 139 (1991), no. 2, 217-243.
28. A. Pandolfi, C. Kane, J. E. Marsden, and M. Ortiz, Time-discretized variational formulation of nonsmooth frictional contact, Int. J. Num. Meth. Eng. 53 (2002), 1801-1829. (Pubitemid 34196027)
29. A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, in Dynamical Systems VII: Integrable Systems, Nonholonomic Dynamical Systems, pp. 1-81, Encyclopaedia of Mathematical Sciences 16, Springer-Verlag, Berlin, 1994.
30. A. Weinstein, Lagrangian mechanics and groupoids, Mechanics day (Waterloo, ON, 1992), Fields Inst. Commun. 7 (1996), 207-231.