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Discrete mechanics and variational integrators. (English) Zbl 1123.37327
Summary: This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge-Kutta schemes are presented.

MSC:
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
70-08 Computational methods for problems pertaining to mechanics of particles and systems
70H05 Hamilton’s equations
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