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arXiv · 1308.6620

Collective symplectic integrators

Robert I McLachlan·Klas Modin·Olivier Verdier
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Abstract

We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge--Kutta methods. Their phase space is a symplectic vector space with a Hamiltonian action with momentum map $J$ whose range is the target Lie--Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by $J$. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on $\mathfrak{so}(3)^*$. The method specializes in the case that a sufficiently large symmetry group acts on the fibres of $J$, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented.

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Robert I McLachlan, Klas Modin, Olivier Verdier. 2013-08-29. Collective symplectic integrators. https://doi.org/10.1088/0951-7715%2F27%2F6%2F1525

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