Geometric integrators and nonholonomic mechanics

Journal of Mathematical Physics

Volumn 45, Issue 3, 2004, Pages 1042-1064

  De Leon M ^a   Martin De Diego D ^a   Santamaria Merino A ^a  

^a INSTITUTO DE MATEMÁTICAS Y FÍSICA FUNDAMENTAL   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 12144288994     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1644325     Document Type: Article

References (52)

1. Arnold, V. I., Mathematical Methods of Classical Mechanics, Graduate Text in Mathematics 60 (Springer-Verlag, New York, 1978).
2. Baez, J. C., and Gilliam, J. W., "An algebraic approach to discrete mechanics," Lett. Math. Phys. 31, 205-212 (1994).
3. Bates, L., and Śniatycki, J., "Nonholonomic reduction," Rep. Math. Phys. 32, 99-115 (1992).
4. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., and Murray, R. M., "Nonholonomic mechanical systems with symmetry," Arch. Ration. Mech. Anal. 136, 21-99 (1996).
5. Bobenko, A. I., and Suris, Y. B., "Discrete Lagrangian reduction, discrete Euler-Poincare equations, and semidirect products," Lett. Math. Phys. 49, 79-93 (1999).
6. Bobenko, A. I., and Suris, Y. B., "Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top," Commun. Math. Phys. 204, 147-188 (1999).
7. Cantrijn, F., de León, M., Marrero, J. C., and Martín de Diego, D., "Reduction of nonholonomic mechanical systems with symmetries," Rep. Math. Phys. 42, 25-45 (1998).
8. Cadzow, J. A., "Discrete calculus of variations," Int. J. Control 11, 393-407 (1970).
9. Channell, P. J., and Scovel, C., "Symplectic integration of Hamiltonian Systems," Nonlinearity 3, 231-259 (1990).
10. Cortés, J., Geometric, Control and Numerical Aspects of Nonholonomic Systems (Lecture Notes in Mathematics (Springer-Verlag), Berlin, 2002), Vol. 1793.
11. Cortés, J., "Energy conserving nonholonomic integrators," Supplement Volume of Discrete Contin. Dyn. Syst., 189-199 (2003).
12. Cortés, J., and Martínez, S., "Nonholonomic integrators," Nonlinearity 14, 1365-1392 (2001).
13. Cushman, R., Kemppainen, D., Śniatycki, J., and Bates, L., "Geometry of nonholonomic constraints," Rep. Math. Phys. 36, 275-286 (1995).
14. Eden, R. J., "The Hamiltonian dynamics of nonholonomic systems," Proc. R. Soc. London, Ser. A 205, 564-583 (1951).
15. Gilliam, J. W., "Lagrangian and Symplectic techniques in discrete mechanics," Ph.D. thesis, Department of Mathematics, University of California Riverside, 1996.
16. González, O., "Mechanical systems subject to holonomic constraints: Differential-algebraic formulations and conservative integration," Physica D 132, 165-174 (1999).
17. Hairer, E., Lubich, C., and Wanner, G., Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics (Springer-Verlag, Berlin, 2002), Vol. 31.
18. Jaroszkiewicz, G., and Norton, K., "Principles of discrete time mechanics I: Particle systems," J. Phys. A 30, 3115-3144 (1997).
19. Jaroszkiewicz, G., and Norton, K., "Principles of discrete time mechanics II: Classical field theory," J. Phys. A 30, 3145-3163 (1997).
20. Jordan, B. W., and Polak, E., "Theory of a class of discrete optimal control systems," J. Electron. Control 17, 697-711 (1964).
21. Kane, C., Marsden, J. E., and Ortiz, M., "Symplectic energy-momentum integrators," J. Math. Phys. 40, 3353-3371 (1999).
22. Kane, C., Marsden, J. E., Ortiz, M., and West, M., "Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems," Int. J. Numer. Math. Eng. 49, 1295-1325 (2000).
23. Kijowski, J., and Tulczyjew, W. M., A Symplectic Framework for Field Theories, Lecture Notes in Physics (Springer-Verlag, Berlin, 1979), Vol. 107.
24. Koiller, J., "Reduction of some classical nonholonomic systems with symmetry," Arch. Ration. Mech. Anal. 118, 113-148 (1992).
25. Koon, W. S., and Marsden, J. E., "The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems," Rep. Math. Phys. 40, 21-62 (1997).
26. Lee, T. D., "Can time be a discrete dynamical variable?" Phys. Lett. 122B, 217-220 (1983).
27. Lee, T. D., "Difference equations and conservation laws," J. Stat. Phys. 46, 843-860 (1987).
28. de León, M., Marrero, J. C., and Martín de Diego, D., "Mechanical systems with nonlinear constraints," Int. J. Theor. Phys. 36, 973-989 (1997).
29. de León, M., and Martín de Diego, D., "Solving nonholonomic Lagrangian dynamics in terms of almost product structures," Extr. Math. 11, 325-347 (1996).
30. de León, M., and Martín de Diego, D., "On the geometry of non-holonomic Lagrangian systems," J. Math. Phys. 37, 3389-3414 (1996).
31. de León, M., and Martín de Diego, D., "Variational integrators and time-dependent Lagrangian systems," Rep. Math. Phys. 49, 183-192 (2002).
32. de León, M., Martín de Diego, D., and Santamaría, A., "Geometric numerical integration of nonholonomic systems and optimal control problems," 2nd IFAC Workshop on lagrangian and Hamiltonian Methods for Nonlinear Control, Seville 2003, pp. 163-168.
33. de León, M., Martín de Diego, D., and Santamaría, A., "Discrete optimal control theory and symplectic integrators" (preprint IMAFF-CSIC, 2002).
34. de León, M., and Rodrigues, P. R., Methods of Differential Geometry in Analytical Mechanics, Math. Ser. 152 (North-Holland, Amsterdam, 1989).
35. McLachlan, R. I., and Scovel, C., "Equivariant constrained symplectic integration," J. Nonlinear Sci. 5, 233-256 (1995).
36. Maeda, S., "Canonical structure and symmetries for discrete systems," Math. Japonica 25, 405-420 (1980).
37. Maeda, S., "Extension of discrete Noether theorem," Math. Japonica 26, 85-90 (1981).
38. Marie, Ch. M., "Reduction of constrained mechanical systems and stability of relative equilibria," Commun. Math. Phys. 174, 295-318 (1995).
39. Marsden, J. E., Park City Lectures on Mechanics, Dynamics and Symmetry, in Symplectic Geometry and Topology, edited by Y. Eliashberg and L. Traynor (American Mathematical Society, Providence, 1999), Vol. 7 of LAS/Park City Math. Ser., pp. 335-430.
40. Marsden, J. E., Patrick, G. W., and Shkoller, S., "Multisymplectic geometry, variational integrators, and nonlinear PDEs," Commun. Math. Phys. 199, 351-395 (1998).
41. Marsden, J. E., Pekarsky, S., and Shkoller, S., "Discrete Euler-Poincaré and Lie-Poisson equations," Nonlinearity 12, 1647-1662 (1999).
42. Marsden, J. E., Pekarsky, S., and Shkoller, S., "Symmetry reduction of discrete Lagrangian mechanics on Lie groups," J. Geom. Phys. 36, 140-151 (1999).
43. Marsden, J. E., and West, M., "Discrete mechanics and variational integrators," Acta Numerica 10, 357-514 (2001).
44. Moser, J., and Veselov, A. P., "Discrete versions of some classical integrable systems and factorization of matrix polynomials," Commun. Math. Phys. 139, 217-243 (1991).
45. Neimark, J., and Fufaev, N., Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1972), Vol. 33.
46. Norton, K., and Jaroszkiewicz, G., "Principles of discrete time mechanics, III: Quantum field theory," J. Phys. A 31, 977-1000 (1998).
47. Pandolfl, A., Kane, C., Marsden, J. E., and Ortiz, M., "Time-discretized variational formulation of nonsmooth factional contact," Int. J. Numer. Methods Eng. 53, 1801-1829 (2002).
48. Sanz-Serna, J. M., and Calvo, M. P., Numerical Hamiltonian Problems (Chapman and Hall, London, 1994).
49. Vershik, A. M., and Faddeev, L. D., "Differential geometry and Lagrangian mechanics with constraints," Sov. Phys. Dokl. 17, 34-36 (1972).
50. Veselov, A. P., "Integrable discrete-time systems and difference operators," Funct. Anal. Appl. 22, 83-93 (1988).
51. Wendlandt, J. M., and Marsden, J. E., "Mechanical integrators derived from a discrete variational principle," Physica D 106, 223-246 (1997).
52. Wendlandt, J. M., and Marsden, J. E., Mechanical Systems with Symmetry, Variational Principles and Integration Algorithms, Current and Future Directions in Applied Mathematics, edited by M. Alber, B. Hu, and J. Rosenthal (Birkhauser, New York, 1997), pp. 219-261.