On variations in discrete mechanics and field theory

Journal of Mathematical Physics

Volumn 44, Issue 12, 2003, Pages 5978-6004

  Guo, Han Ying ^a,b   Wu, Ke ^c  

^a CCAST WORLD LABORATORY   (China)

^b INSTITUTE OF THEORETICAL PHYSICS   (China)

^c CAPITAL NORMAL UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345149792     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1621058     Document Type: Article

References (34)

1. Abraham, R., and Marsden, J. E., Foundation of Mechanics, 2nd ed. (Addison-Wesley, New York, 1978).
2. Anderson, I. M., "The variational bicomplex," (1989) appeared as a technical report on http://www.math.usu.edu/tild;fg.mp (2001).
3. Anderson, I. M., "Introduction to the variational bicomplex," in Mathematical Aspects of Classical Field Theory, edited by M. Gotay, J. Marsden, and V. Moncrief [Contemp. Math. 132, 51-73 (1992)].
4. Anderson, I. M., "Introduction to the variational bicomplex," in Mathematical Aspects of Classical Field Theory, edited by M. Gotay, J. Marsden, and V. Moncrief [Contemp. Math. 132, 51-73 (1992)].
5. Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Graduate texts in Math. 60 (1978) (Springer-Verlag, New York, 1989).
6. Bridges, T. J., "Multisymplectic structures and wave propagation," Math. Proc. Cambridge Philos. Soc. 121, 147-190 (1997).
7. Chen, J. B., Guo, H. Y., and Wu, K., "Total variation and variational symplectic energy-momentum integrators," hep-th/0109178.
8. Chen, J. B., Guo, H. Y., and Wu, K., "Total variation in Hamiltonian formalism and symplectic-energy integrators," J. Math. Phys. 44, 1688-1702 (2003); hep-th/0111185.
9. Courant, R., and Hubert, D., Methods of Mathematical Physics (Interscience, New York, 1953).
10. Dittrich, W., and Reuter, M., Classical and Quantum Dynamics (Springer-Verlag, New York, 1994).
11. Feng, K., "On difference schemes and symplectic geometry," in Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations - Computation of Partial Differential Equations, edited by Feng Keng (Science Press, Beijing, 1985). Selected works of Feng Keng II (1995) and the references therein.
12. Ge, Z., and Marsden, J. E., "Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory," Phys. Lett. A 133, 134-139 (1988).
13. Guo, H. Y., Li, Y. Q., and Wu, K., "On symplectic and multisymplectic structures and their discrete versions in Lagrangian formalism," Commun. Theor. Phys. 35, 703-710 (2001).
14. Guo, H. Y., Li, Y. Q., Wu, K., and Wang, S. K., "Symplectic, multisymplectic structures and Euler-Lagrange cohomology," hep-th/0104140.
15. Guo, H. Y., Li, Y. Q., Wu, K., and Wang, S. K., "Difference discrete variational principle, Euler-Lagrange cohomology and symplectic, multisymplectic structures I: Difference discrete variational principle," Commun. Theor. Phys. 37, 1-10 (2002); "II: Euler-Lagrange cohomology, " ibid, 37, 129-138 (2002); "III: Application to symplectic and multisymplectic algorithms," ibid. 37, 257-264 (2002). See also hep-th/0106001.
16. Guo, H. Y., Li, Y. Q., Wu, K., and Wang, S. K., "Difference discrete variational principle, Euler-Lagrange cohomology and symplectic, multisymplectic structures I: Difference discrete variational principle," Commun. Theor. Phys. 37, 1-10 (2002); "II: Euler-Lagrange cohomology, " ibid, 37, 129-138 (2002); "III: Application to symplectic and multisymplectic algorithms," ibid. 37, 257-264 (2002). See also hep-th/0106001.
17. Guo, H. Y., Li, Y. Q., Wu, K., and Wang, S. K., "Difference discrete variational principle, Euler-Lagrange cohomology and symplectic, multisymplectic structures I: Difference discrete variational principle," Commun. Theor. Phys. 37, 1-10 (2002); "II: Euler-Lagrange cohomology, " ibid, 37, 129-138 (2002); "III: Application to symplectic and multisymplectic algorithms," ibid. 37, 257-264 (2002). See also hep-th/0106001.
18. Guo, H. Y., Pan, J. Z., Wu, K., and Zhou, B., "The Euler-Lagrange cohomology groups on symplectic manifolds," physics/0304074.
19. Guo, H. Y., Wu, K., Wang, S. H., Wang, S. K., and Wei, G. M., "Noncommutative differential calculus approach to symplectic algorithm on regular lattice," Commun. Theor. Phys. 34, 307-318 (2000).
20. Guo, H. Y., Wu, K., and Zhang, W., "Noncommutative differential calculus on Abelian groups and its applications," Commun. Theor. Phys. 34, 245-250 (2000).
21. Kane, C., Marsden, J. E., and Ortiz, M., "Symplectic-energy-momentum preserving variational integrators," J. Math. Phys. 40, 3353-3371 (1999).
22. Lee, T. D., "Can time be a discrete dynamical variable?" Phys. Lett. 122B, 217-220 (1983).
23. Lee, T. D., "Difference equations and conservation laws," J. Stat. Phys. 46, 843-860 (1987).
24. Lee, T. D., "Discrete mechanics," Lectures given at the International School of Subnuclear Physics, Erice, 1983.
25. Lew, A., Marsden, J. E., Ortiz, M., and West, M., "Asynchronous variational integrators," Arch. Rat. Mech. Anal. (2003).
26. Marsden, J. E., and Shkoller, S., "Multisymplectic geometry, covariant Hamiltonian and water waves," Math. Proc. Cambridge Philos. Soc. 125, 535-575 (1999).
27. Marsden, J. E., Patrick, G. W., and Shkoller, S., "Multisymplectic geometry, variational integrators, and nonlinear PDEs," Commun. Math. Phys. 199, 351-395 (1998).
28. McDuff, D., and Salamon, D., Introduction to Symplectic Topology (Clarendon, Oxford, 1998).
29. 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).
30. Olver, P. J., Applications of Lie Groups to Differential Eequations, 2nd ed. GTM 107 (Springer-Verlag, New York, 1993).
31. Ruth, R. D., "A canonical integration technique," IEEE Trans. Nucl. Sci. 30, 1669-2671 (1983).
32. Sanz-Serna, J. M., and Calvo, M. P., Numerical Hamiltonian Problems (Chapman and Hall, London, 1994), and references therein.
33. Veselov, A. P., "Integrable discrete-time systems and difference operators," Funkc. Anal. Priloz. 22, 1-13 (1988).
34. Zhou, B., Guo, H. Y., and Wu, K., "General volume-preserving mechanical systems," Lett. Math. Phys. (to be published); see also math-ph/0208033.