The Dirac--Bergmann algorithm: Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance

The Dirac--Bergmann algorithm is a recipe for converting a singular Lagrangian system into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories---general relativity, electromagnetism, Yang Mills, string theory, etc. The Dirac--Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac--Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this work I analyze a finite--dimensional system that contains all of the major steps in the algorithm. The system includes primary and secondary constraints, first and second class constraints, restrictions on Lagrange multipliers, and both physical and gauge degrees of freedom. This relatively simple system provides a platform for discussing the Dirac conjecture, constructing Dirac brackets, and applying gauge conditions.