A frame bundle generalization of multisymplectic field theories

TL;DR

This paper introduces a new geometric framework based on a principal bundle that generalizes multisymplectic geometry for classical field theories, enabling vector-valued observables and addressing key limitations of existing models.

Contribution

It develops a vertically adapted linear frame bundle that extends multisymplectic geometry, incorporating vector-valued observables and resolving fundamental issues in previous models.

Findings

Defines a generalized symplectic structure on the vertically adapted frame bundle

Establishes a Poisson bracket for vector-valued field observables

Connects the new geometry to existing multivelocity and multiphase spaces

Abstract

This paper presents a generalization of symplectic geometry to 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 a generalized symplectic structure from the full frame bundle. The geometric structure of the vertically adapted frame bundle admits vector-valued field observables and produces vector-valued Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We show that the linear and affine multivelocity spaces and multiphase spaces for geometric field theories are associated to the vertically adapted frame bundle. In addition, the new geometry not only generalizes both the linear and the affine models of multisymplectic geometry but also resolves fundamental problems found in both multisymplectic models.