Classical gauge theories as systems with constraints: a geometric point of view

TL;DR

This work provides a geometric Hamiltonian account of classical gauge theories by treating them as systems constrained to a submanifold of phase space, described by primary and, via the Dirac-Bergmann algorithm, subsequent secondary constraints. Gauge freedom is traced to the presence of first-class constraints and the nonuniqueness of Lagrange multipliers, with dynamics generated by the total Hamiltonian $H_T=H+\lambda^a\Phi_a$. The paper illustrates the framework with a simple constrained mechanical model and with electromagnetism, showing how primary and secondary constraints arise, how field functionals require functional Poisson brackets, and how a transverse decomposition isolates physical degrees of freedom while gauge fixing remains inconsequential to physical evolution. The approach clarifies the fundamental role of constraint surfaces and first-class constraints in gauge theories, connecting canonical structure to familiar gauge transformations such as $A^{\mu}\to A^{\mu}-\partial^{\mu}\chi$ and supporting a deeper, more geometric understanding of gauge invariance and fixing. Overall, the work offers a pedagogical, coordinate-free perspective on gauge theories that complements conventional Lagrangian treatments and standard references in constrained dynamics.

Abstract

In this paper, we briefly review the Hamiltonian formulation of classical systems that are constrained to submanifolds so that, within this context, the true meaning of classical gauge theories becomes clear. Please note that this paper is nothing more than a near-literal translation of Ref. [1], which we originally published in Brazilian Portuguese in 2018. Therefore, if you, the reader, find this paper useful enough to cite it in any of your works, we kindly ask that you (also) cite Ref. [1].