Symplectic-energy-momentum preserving variational integrators. (English) Zbl 0983.70014

This paper, a continuation of the article by J. E. Marsden, S. Pekarsky and S. Shkoller [Nonlinearity 12, No. 6, 1647-1662 (1999; Zbl 0978.37045)], presents an integration method for special mechanical systems. These methods are variational, they have adaptative time steps, they are symplectic, energy preserving, and they also preserve momentum maps for systems invariant under a symmetry group. The authors derive theoretical constraints which limit the possibilities of constant time stepping algoritms to be symplectic and energy and momentum preserving.
Additionally, the authors give a review of variational integrators, examining the accuracy of solutions, dissipation, constraints, symmetry and reduction. The authors also examine the energy conservation and symplectic conservation in the continuous case before giving the variational algorithm. Numerical examples and an extensive bibliography conclude the paper.
For other constrained optimization methods, see C. D. Rahn and C. D. Mote jun. [J. Dyn. Syst. Meas. Control 118, No 2, 309-314 (1996; Zbl 0877.70019)] and O. Gonzalez [Physica D 132, No. 1-2, 165-174 (1999; Zbl 0942.70002)].


70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70-08 Computational methods for problems pertaining to mechanics of particles and systems
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
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