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Singular Lagrangians and the Dirac–Bergmann algorithm in classical mechanics.
- Source :
- American Journal of Physics; Mar2023, Vol. 91 Issue 3, p214-224, 11p
- Publication Year :
- 2023
-
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 ensures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definitions of the conjugate momenta can be inverted to solve 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 1950s. After a brief review of the Dirac–Bergmann algorithm, several examples are presented using familiar components: point masses connected by massless springs, rods, cords, and pulleys. [ABSTRACT FROM AUTHOR]
- Subjects :
- DYNAMICAL systems
CLASSICAL mechanics
ALGORITHMS
LAGRANGE equations
Subjects
Details
- Language :
- English
- ISSN :
- 00029505
- Volume :
- 91
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- American Journal of Physics
- Publication Type :
- Academic Journal
- Accession number :
- 161960003
- Full Text :
- https://doi.org/10.1119/5.0107540