Canonical Transformations and Hamiltonian Evolutionary Systems

TL;DR

This paper establishes conditions for canonical transformations in Lagrangian field theories, showing how they affect Hamiltonian systems and conservation laws across different dimensions with illustrative examples.

Contribution

It provides a comprehensive characterization of canonical transformations in various dimensional settings of Hamiltonian field theories.

Findings

Derived necessary and sufficient conditions for canonical transformations.

Showed how transformations impact Hamiltonian systems and conservation laws.

Presented three illustrative examples.

Abstract

In many Lagrangian field theories one has a Poisson bracket defined on the space of local functionals. We find necessary and sufficient conditions for a transformation on the space of local functionals to be canonical in three different cases. These three cases depend on the specific dimensions of the vector bundle of the theory and the associated Hamiltonian differential operator. We also show how a canonical transformation transforms a Hamiltonian evolutionary system and its conservation laws. Finally we illustrate these ideas with three examples.