Lagrangian Noether symmetries as canonical transformations

TL;DR

This paper demonstrates that all infinitesimal Noether symmetries in a time-independent Lagrangian system can be represented as canonical transformations in an extended phase space, especially useful for singular Lagrangians.

Contribution

It proves that any Noether symmetry can be realized as a canonical transformation in an enlarged phase space, extending the Hamiltonian formalism to singular Lagrangian systems.

Findings

All infinitesimal Noether symmetries can be expressed as canonical transformations.

The method applies to singular Lagrangians and rederives Noether identities.

An example illustrates the theoretical results.

Abstract

We prove that, given a time-independent Lagrangian defined in the first tangent bundle of configuration space, every infinitesimal Noether symmetry that is defined in the $n$-tangent bundle and is not vanishing on-shell, can be written as a canonical symmetry in an enlarged phase space, up to constraints that vanish on-shell. The proof is performed by the implementation of a change of variables from the the $n$-tangent bundle of the Lagrangian theory to an extension of the Hamiltonian formalism which is particularly suited for the case when the Lagrangian is singular. This result proves the assertion that any Noether symmetry can be canonically realized in an enlarged phase space. Then we work out the regular case as a particular application of this ideas and rederive the Noether identities in this framework. Finally we present an example to illustrate our results.