Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac-Bergmann Algorithm

Abstract

The Dirac-Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian 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 paper I analyze a finite-dimensional system that exhibits 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.

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References in corpus (3)

• The Dynamics of General Relativity
• Dynamical Structure of Irregular Constrained Systems
• Gauge Identities and the Dirac Conjecture

Cited by in corpus (10)

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• On the well-posed variational principle in degenerate point particle systems using embeddings of the symplectic manifold
• Geometry and proper time of a relativistic quantum clock
• Faddeev-Jackiw Approach to Classical Constrained Systems
• On Hamiltonian formulations of the Dirac system
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints