  With the covariant formulation in hand from the first paper of this series
(physics/9801019), we begin in this second paper to study the canonical (or
``instantaneous'') formulation of classical field theories. The canonical
formluation works with fields defined as time-evolving cross sections of
bundles over a Cauchy surface, rather than as sections of bundles over
spacetime as in the covariant formulation. In Chapter 5 we begin to relate
these approaches to classical field theory; in particular, we show how
covariant multisymplectic geometry induces the instantaneous symplectic
geometry of cotangent bundles of sections of fields over a Cauchy surface. In
Chapter 6, we proceed to consider field dynamics. A crucial feature of our
discussion here is the degeneracy of the Lagrangian functionals for the field
theories of interest. As a consequence of this degeneracy, we have constraints
on the choice of initial data, and gauge freedom in the evolution of the
fields. Chapter 6 considers the role of initial value constraints and gauge
transformations in Hamiltonian field dynamics. In Chapter 7, we then describe
how covariant momentum maps defined on the multiphase space induce
"energy-momentum maps'' on the instantaneous phase spaces. We show that for a
group action which leaves the Cauchy surface invariant, this energy-momentum
map coincides with the usual notion of a momentum map. We also show, when the
gauge group"includes'' the spacetime diffeomorphism group, that one of the
components of the energy-momentum map corresponding to spacetime
diffeomorphisms can be identified (up to sign) with the Hamiltonian for the
theory.
