Classical gauge principle -- From field theories to classical mechanics

TL;DR

This paper explores how the gauge principle, traditionally used in field theories, can be applied to classical mechanics models with finite degrees of freedom, emphasizing local invariance and classification of Lagrangians.

Contribution

It introduces a formal method to implement the gauge principle in classical mechanics and classifies Lagrangians based on their local invariance properties.

Findings

Gauge invariance can be applied to finite-dimensional classical systems.

Lagrangians are classified into equivalence classes by local structure.

Modern applications of gauge invariance in classical mechanics are discussed.

Abstract

In this paper we discuss how the gauge principle can be applied to classical-mechanics models with finite degrees of freedom. The local invariance of a model is understood as its invariance under the action of a matrix Lie group of transformations parametrized by arbitrary functions. It is formally presented how this property can be introduced in such systems, followed by modern applications. Furthermore, Lagrangians describing classical-mechanics systems with local invariance are separated in equivalence classes according to their local structures.