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CONSERVED QUANTITIES ON MULTISYMPLECTIC MANIFOLDS

Part of: Symplectic geometry, contact geometry Infinite-dimensional Hamiltonian systems Hamiltonian and Lagrangian mechanics

Published online by Cambridge University Press:  26 December 2018

LEONID RYVKIN ,
TILMANN WURZBACHER  and
MARCO ZAMBON
LEONID RYVKIN
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany email leonid.ryvkin@rub.de
TILMANN WURZBACHER
Affiliation:
Institut Élie Cartan Lorraine, Université de Lorraine et C.N.R.S., Ile de Saulcy, 57045 Metz, France email tilmann.wurzbacher@univ-lorraine.fr
MARCO ZAMBON*
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium email marco.zambon@kuleuven.be
*
✉marco.zambon@kuleuven.be
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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. 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.

Keywords

conservation lawsmultisymplectic manifoldshomotopy co-momentum map

MSC classification

Primary: 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 53D05: Symplectic manifolds, general
Secondary: 70H33: Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 1 , February 2020 , pp. 120 - 144
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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