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Guo, H. Y.; Li, Y. Q.; Wu, K.; Wang, S. K. (2001)
Languages: English
Types: Preprint
Subjects: Mathematics - Symplectic Geometry, Mathematical Physics, Mathematics - Numerical Analysis

Classified by OpenAIRE into

arxiv: Mathematics::Symplectic Geometry
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of this variational principle, we get the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory. We also explore the difference discrete versions for the Euler-Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both Lagrangian and Hamiltonian formalism. In terms of the difference discrete Euler-Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure preserving properties can always be established not only in the solution spaces of the discrete Euler-Lagrange/canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler-Lagrange cohomological conditions are satisfied. We also apply the difference discrete variational principle and cohomological approach directly to the symplectic and multisymplectic algorithms.
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    • [18] H.Y. Guo, K. Wu and W. Zhang, Noncommutative Differential Calculus on Abelian Groups and Its Applications, Comm. Theor. Phys. 34 (2000) 245-250.
    • [19] H.Y.Guo, K. Wu, S.H. Wang and G.M. Wei, Discrete Symplectic Algorithm on Regular Lattice, Talk given by H.Y. Guo at the CCAST-WL workshop on Computational Methods and Their Applications in Physics and Mechanics, March, 1999, the CCAST-WL workshop on Genetic Algorithm and Its applications, April 5-9, 1999, the CCAST-WL workshop on Integrable System, May 3-7, 1999. CCAST-WL workshop series, 104, 167-192.
    • [20] A. Connes, Noncommutative Geometry, Academic Press, INC. 1994.
    • [21] S. Reich, Multisymplectic Runge-Kutta Collocation Methods for Hamiltonian Wave Equations, J. Comput. Phys. 157 (2000), 473-499.
    • [22] K. Feng, H.M. Wu, M.Z. Qin and D.L. Wang, Construction of Canonical Difference Schemes for Hamiltonian Formalism via Generating Functions, J. Comp. Math. 7 (1989) 71-96.
    • [23] A.P. Veselov, Integrable Discrete-time Systems and Difference Operators, Funkts. Anal. Prilozhen, 22 (1988) 1-13.
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