Weak Moment Maps in Multisymplectic Geometry
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Date
2018-07-04
Authors
Herman, Jonathan
Journal Title
Journal ISSN
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Publisher
University of Waterloo
Abstract
We introduce the notion of a weak (homotopy) moment map associated to a Lie group
action on a multisymplectic manifold. We show that the existence/uniqueness theory governing
these maps is a direct generalization from symplectic geometry.
We use weak moment maps to extend Noether's theorem from Hamiltonian mechanics by
exhibiting a correspondence between multisymplectic conserved quantities and continuous
symmetries on a multi-Hamiltonian system. We nd that a weak moment map interacts
with this correspondence in a way analogous to the moment map in symplectic geometry.
We de ne a multisymplectic analog of the classical momentum and position functions
on the phase space of a physical system by introducing momentum and position forms. We
show that these di erential forms satisfy generalized Poisson bracket relations extending the
classical bracket relations from Hamiltonian mechanics. We also apply our theory to derive
some identities on manifolds with a torsion-free G2 structure.