Geometry and Hamiltonian mechanics on discrete spaces

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

    Research output: Contribution to journalArticleAcademicpeer-review

    9 Citations (Scopus)

    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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