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Collective symplectic integrators

Journal article
Authors Robert McLachlan
Klas Modin
Olivier Verdier
Published in Nonlinearity
Volume 27
Issue 6
Pages 1525-1542
ISSN 0951-7715
Publication year 2014
Published at Department of Mathematical Sciences, Mathematics
Pages 1525-1542
Language en
Subject categories Geometry, Numerical analysis


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 equipped 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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