Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups. (English) Zbl 0799.58069

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.


37C80 Symmetries, equivariant dynamical systems (MSC2010)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems


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