Geometry and Hamiltonian mechanics on discrete spaces


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

[img] PDF
Restricted to UT campus only
: Request a copy
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
Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:
Link to this item:
Official URL:
Export this item as:BibTeX
HTML Citation
Reference Manager


Repository Staff Only: item control page

Metis ID: 220484