Gauge invariance in simple mechanical systems

TL;DR

This paper explores the Hamiltonian formulation of simple gauge-invariant mechanical systems, illustrating key concepts like constraints and gauge symmetries through a model of four masses connected by rods, aiding understanding of classical and quantum field theories.

Contribution

It provides a geometric derivation of the Hamiltonian formulation for a specific gauge-invariant system, clarifying fundamental concepts relevant to advanced physics education.

Findings

Derived Hamiltonian formulation for the model

Clarified the role of constraints and gauge symmetries

Discussed gauge fixing and reduced phase space

Abstract

This article discusses and explains the Hamiltonian formulation for a class of simple gauge invariant mechanical systems consisting of point masses and idealized rods. The study of these models may be helpful to advanced undergraduate or graduate students in theoretical physics to understand, in a familiar context, some concepts relevant to the study of classical and quantum field theories. We use a geometric approach to derive the Hamiltonian formulation for the model considered in the paper: four equal masses connected by six ideal rods. We obtain and discuss the meaning of several important elements, in particular, the constraints and the Hamiltonian vector fields that define the dynamics of the system, the constraint manifold, gauge symmetries, gauge orbits, gauge fixing, and the reduced phase space.