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Published online by Cambridge University Press:  26 December 2018

Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany email
Institut Élie Cartan Lorraine, Université de Lorraine et C.N.R.S., Ile de Saulcy, 57045 Metz, France email
KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium email


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. Our main result is 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.

Research Article
© 2018 Australian Mathematical Publishing Association Inc. 

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