A brief introduction to gauge-invariant dynamical systems

We describe gauge symmetry at the elementary level aiming at undergraduate and first-year graduate theoretical physics students. The formal definition of gauge invariance as a local symmetry of the action is given and further elaborated and interpreted in detail in two physical models, namely, a mechanical system and the prototypical gauge field theory of electrodynamics. The first model, consisting of two interacting particles with gauge symmetry, sets the background and paves the way for the canonical analysis of the electromagnetic field as a constrained system. The Dirac-Bergmann algorithm and the necessity of Dirac brackets for the canonical quantization of constrained systems are discussed within the two models context. The whole analysis is written in a didactical way trying to guide and motivate undergraduate students along an introduction to gauge symmetry. After going through this present brief introduction, the reader should be well prepared to boldly continue their journey through more advanced standard textbooks on the subject.