A frame bundle generalization of multisymplectic geometries

Abstract

We present a generalized symplectic geometry on a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits an “ n-symplectic” structure from the full frame bundle. This geometric structure admits vector-valued field observables and produces Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We show that both the linear and the affine models of multisymplectic geometry are obtained by mapping the vertically adapted frame bundle to associated fiber bundles. In addition, the new geometry resolves a fundamental problem that arises in each model.