V Talasila 1, J Clemente-Gallardo 2 and A J van der Schaft 1
1
Department of Applied Mathematics, University of Twente, 7500AE, Enschede, The Netherlands
2
Departamento de Matematica, Universidade de Coimbra, 3001-454 Coimbra, Portugal
talasilav@cs.utwente.nl jesus@mat.uc.pt a.j.vanderschaft@math.uc.utwente.nl
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V Talasila et al 2004 J. Phys. A: Math. Gen. 37 9705
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.
70Hxx Hamiltonian and Lagrangian mechanics (See also 37Jxx)
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
Issue 41 (15 October 2004)
Received 14 June 2004
Published 29 September 2004
V Talasila et al 2004 J. Phys. A: Math. Gen. 37 9705