Singular Lagrangians and the Dirac--Bergmann Algorithm in Classical Mechanics

Abstract

Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular. That is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption insures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definition of the conjugate momenta can be inverted for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive -- there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950's. After a brief review of the Dirac--Bergmann algorithm, several physical examples are constructed from familiar components: point masses connected by massless springs, rods, cords and pulleys. The algorithm is also used to develop an initial value formulation of systems with holonomic constraints.

arXiv comment28 pages, 7 figures

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• Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories
• Dynamical Structure of Irregular Constrained Systems
• A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism
• Léon Rosenfeld's general theory of constrained Hamiltonian dynamics
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Cited by in corpus (4)

• Partially Unitary Learning
• Singular Lagrangians and the Faddeev-Jackiw Formalism in Classical Mechanics
• Faddeev-Jackiw Approach to Classical Constrained Systems
• Introduction to Higher Order Classical Dynamics: Pais-Uhlenbeck Model and Coupled Oscillators