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Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

In: Current and Future Directions in Applied Mathematics

Author

Listed:
  • Jerrold E. Marsden

    (California Institute of Technology, Control and Dynamical Systems 116-81)

  • Jeffrey M. Wendlandt

    (University of California at Berkeley, Mechanical Engineering)

Abstract

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators.

Suggested Citation

  • Jerrold E. Marsden & Jeffrey M. Wendlandt, 1997. "Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms," Springer Books, in: Mark Alber & Bei Hu & Joachim Rosenthal (ed.), Current and Future Directions in Applied Mathematics, pages 219-261, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2012-1_18
    DOI: 10.1007/978-1-4612-2012-1_18
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