1 results
CONSERVED QUANTITIES ON MULTISYMPLECTIC MANIFOLDS
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 26 December 2018, pp. 120-144
- Print publication:
- February 2020
-
- Article
- Export citation
-
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.
