The commutativity principle and lagrangian symmetries

TL;DR

This paper explores how the commutativity of variation and time differentiation relates to symmetries in Lagrangian mechanics, unifying global and gauge symmetries and connecting Hamiltonian and Lagrangian frameworks.

Contribution

It introduces a unified approach to analyze both global and local symmetries using the commutativity principle, establishing their equivalence in Hamiltonian and Lagrangian formalisms.

Findings

Unified treatment of global and gauge symmetries.

Complete equivalence between Hamiltonian and Lagrangian formulations.

Application of Noether's theorem to both symmetry types.

Abstract

Using the commutativity of a general variation with the time differentiation we discuss both global and local (gauge) symmetries of a lagrangian from a unified point of view. The Noether considerations are thereby applicable for both cases. A complete equivalence between the hamiltonian and lagrangian formulations is established.