Continuous versus discrete structures II -- Discrete Hamiltonian systems and Helmholtz conditions

Preprint English OPEN
Cresson, Jacky; Pierret, Frédéric;
  • Subject: Mathematics - Dynamical Systems | Mathematics - Numerical Analysis
    acm: ComputingMethodologies_SIMULATIONANDMODELING

We define discrete Hamiltonian systems in the framework of discrete embeddings. An explicit comparison with previous attempts is given. We then solve the discrete Helmholtz's inverse problem for the discrete calculus of variation in the Hamiltonian setting. Several appl... View more
  • References (15)
    15 references, page 1 of 2

    [1] I.D. Albu and D. Opri¸s. Helmholtz type condition for mechanical integrators. Novi Sad J. Math., 29(3):11-21, 1999. XII Yugoslav Geometric Seminar (Novi Sad, 1998).

    [2] V. I. Arnold. Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979.

    [3] Z. Bartosiewicz, U. Kotta, E. Pawluszewicz, M. Wyrwas, Algebraic formalism of differntial oneforms for nonlinear control systems on time scales, Proc. Estonian Acad. Sci. Phys. Math., 2007, 56, 3, 264-282.

    [4] Bismut, J-M. 1981, M´ecanique al´eatoire, Lecture notes in mathematics (Springer-Verlag).

    [5] L. Bourdin, J. Cresson. Helmholtz's inverse problem of the discrete calculus of variations. Journal of Difference Equations and Applications, 19(9):1417-1436, 2013.

    [6] J. Cresson, F. Pierret, Discrete versus continuous structures I - Discrete embedding and differential equations, arXiv:1411.7117, 2014.

    [7] D. Cr˘aciun and D. Opri¸s. The Helmholtz conditions for the difference equations systems. Balkan J. Geom. Appl., 1(2):21-30, 1996.

    [8] S. Lall, M. West, Discrete variational Hamiltonian mechanics, J. Phys. A : Math. Gen. 39 (2006), 5509-5519.

    [9] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. Structurepreserving algorithms for ordinary differential equations.

    [10] P.E. Hydon, E.L. Mansfeld, A variational complex for difference equations, 44.p.

  • Metrics
    No metrics available
Share - Bookmark