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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@article{Ryvkin-Conserved-quantities-on-2020,
 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.},
 author = {Ryvkin, Leonid and Wurzbacher, Tilmann and Zambon, Marco},
 doi = {10.1017/S1446788718000381},
 eprint = {1610.05592},
 fjournal = {Journal of the Australian Mathematical Society},
 issn = {1446-7887},
 journal = {J. Aust. Math. Soc.},
 keywords = {37J39,37J06,53D20,53D05},
 language = {English},
 number = {1},
 pages = {120--144},
 title = {Conserved quantities on multisymplectic manifolds},
 volume = {108},
 year = {2020},
 zbl = {1440.37064},
 zbmath = {7154935}
}