Conserving algorithms for the dynamics of Hamiltonian systems on lie groups |
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Authors: | D Lewis J C Simo |
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Institution: | (1) Board of Mathematics, University of California Santa Cruz, 95064 Santa Cruz, CA, USA;(2) Division of Applied Mechanics, Stanford University, 94305 Stanford, CA, USA |
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Abstract: | Summary Three conservation laws are associated with the dynamics of Hamiltonian systems with symmetry: The total energy, the momentum
map associated with the symmetry group, and the symplectic structure are invariant under the flow. Discrete time approximations
of Hamiltonian flows typically do not share these properties unless specifically designed to do so. We develop explicit conservation
conditions for a general class of algorithms on Lie groups. For the rigid body these conditions lead to a single-step algorithm
that exactly preserves the energy, spatial momentum, and symplectic form. For homogeneous nonlinear elasticity, we find algorithms
that conserve angular momentum and either the energy or the symplectic form. |
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Keywords: | Hamilton's equations symmetry groups canonical transformations rigid bodies homogeneous elasticity |
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