Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton-Jacobi theory

摘要

Classical field theory is considered as a theory of unparametrized surfacesembedded in a configuration space, which accommodates, in a symmetric way,spacetime positions and field values. Dynamics is defined by a (Hamiltonian)constraint between multivector-valued generalized momenta, and points in theconfiguration space. Starting from a variational principle, we derive localequations of motion, that is, differential equations that determine classicalsurfaces and momenta. A local Hamilton-Jacobi equation applicable in the fieldtheory then follows readily. The general method is illustrated with threeexamples: non-relativistic Hamiltonian mechanics, De Donder-Weyl scalar fieldtheory, and string theory. Throughout, we use the mathematical formalism of geometric algebra andgeometric calculus, which allows to perform completely coordinate-freemanipulations.