  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 find
that a weak moment map interacts with this correspondence in a way analogous to
the moment map in symplectic geometry.
  We define 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 differential 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 closed $G_2$ structure.
