Geometry and Hamiltonian mechanics on discrete spaces

Talasila, V. and Clemente-Gallardo, J. and Schaft van der, A.J. (2004) Geometry and Hamiltonian mechanics on discrete spaces. Journal of Physics A: Mathematical and General, 37 (41). pp. 9705-9734. ISSN 1751-8113

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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.
Item Type:	Article
Copyright:	© 2004 Institute of Physics
Faculty:	Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:	Mathematical Systems and Control Theory (MSCT)
Link to this item:	http://purl.utwente.nl/publications/69155
Official URL:	http://dx.doi.org/10.1088/0305-4470/37/41/008
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