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arXiv · 1610.05592

Conserved quantities on multisymplectic manifolds

Leonid Ryvkin·Tilmann Wurzbacher·Marco Zambon
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Abstract

Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved under transgression to spaces of maps into M. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. We show that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.

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BibTeXRIS

Leonid Ryvkin, Tilmann Wurzbacher, Marco Zambon. 2016-10-18. Conserved quantities on multisymplectic manifolds. https://doi.org/10.1017/s1446788718000381

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