Geometry and Hamiltonian mechanics on discrete spaces

V. Talasila, J.J. Clemente Gallardo, Arjan van der Schaft

    Research output: Contribution to journalArticleAcademicpeer-review

    10 Citations (Scopus)
    13 Downloads (Pure)

    Abstract

    Numerical simulation is often crucial for analysing the behaviour of many complex systems which do not admit analytic solutions. To this end, one either converts a 'smooth' model into a discrete (in space and time) model, or models systems directly at a discrete level. The goal of this paper is to provide a discrete analogue of differential geometry, and to define on these discrete models a formal discrete Hamiltonian structure-in doing so we try to bring together various fundamental concepts from numerical analysis, differential geometry, algebraic geometry, simplicial homology and classical Hamiltonian mechanics. For example, the concept of a twisted derivation is borrowed from algebraic geometry for developing a discrete calculus. The theory is applied to a nonlinear pendulum and we compare the dynamics obtained through a discrete modelling approach with the dynamics obtained via the usual discretization procedures. Also an example of an energy-conserving algorithm on a simple harmonic oscillator is presented, and its effect on the Poisson structure is discussed.
    Original languageEnglish
    Pages (from-to)9705-9734
    Number of pages30
    JournalJournal of physics A: mathematical and general
    Volume37
    Issue number41
    DOIs
    Publication statusPublished - 2004

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