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.
Towards Relativistic Atomic Physics. I. The Rest-Frame Instant Form of Dynamics and a Canonical Transformation for a System of Charged Particles plus the Electro-Magnetic Field
Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the Non-Relativistic N-Body Problem
After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of { \it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian { \it left action}. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a { \it spin frame} for the N-body system. Then, the existence of appropriate non…
