Multimomentum Hamiltonian Formalism in Field Theory. Geometric Supplementary

TL;DR

This paper reviews the geometric multi-momentum Hamiltonian formalism in classical field theory, emphasizing jet manifolds and connections, and aims to make these mathematical tools more accessible to physicists.

Contribution

It provides a comprehensive summary of jet manifolds and connections, facilitating the application of geometric Hamiltonian formalism in field theory for physicists.

Findings

Clarifies the role of jet manifolds in field theory

Summarizes the mathematical prerequisites for multi-momentum Hamiltonian formalism

Bridges the gap between advanced geometry and physical applications

Abstract

The well-known geometric approach to field theory is based on description of classical fields as sections of fibred manifolds, e.g. bundles with a structure group in gauge theory. In this approach, Lagrangian and Hamiltonian formalisms including the multiomentum Hamiltonian formalism are phrased in terms of jet manifolds. Then, configuration and phase spaces of fields are finite-dimensional. Though the jet manifolds have been widely used for theory of differential operators, the calculus of variations and differential geometry, this powerful mathematical methods remains almost unknown for physicists. This Supplementary to our previous article (hep-th/9403172) aims to summarize necessary requisites on jet manifolds and general connections.