Singular Lagrangians and the Hamilton-Jacobi Formalism in Classical Mechanics - INSPIRE

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Singular Lagrangians and the Hamilton-Jacobi Formalism in Classical Mechanics

Aug 28, 2024

41 pages

Published in:

Int.J.Theor.Phys. 64 (2025) 2, 36

Published: Feb 3, 2025

e-Print:

2408.15871 [gr-qc]

DOI:

10.1007/s10773-025-05887-w

View in:

ADS Abstract Service

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Abstract: (Springer)

This work conducts a Hamilton-Jacobi analysis of classical dynamical systems with internal constraints. We examine four systems, all previously analyzed by David Brown: three with familiar components (point masses, springs, rods, ropes, and pulleys) and one chosen specifically for its detailed illustration of the Dirac-Bergmann algorithm’s logical steps. Including this fourth system allows for a direct and insightful comparison with the Hamilton-Jacobi formalism, thereby deepening our understanding of both methods. To provide a thorough analysis, we classify the systems based on their constraints: non-involutive, involutive, and a combination of both. We then use generalized brackets to ensure the theory’s integrability, systematically remove non-involutive constraints, and derive the equations of motion. This approach effectively showcases the Hamilton-Jacobi method’s ability to handle complex constraint structures. Additionally, our study includes an analysis of a gauge system, highlighting the versatility and broad applicability of the Hamilton-Jacobi formalism. By comparing our results with those from the Dirac-Bergmann and Faddeev-Jackiw algorithms, we demonstrate that the Hamilton-Jacobi approach is simpler and more efficient in its mathematical operations and offers advantages in computational implementation.

Note:

41 pages, 5 figures

Hamilton-Jacobi formalism
Constrained system
Non-involutive system
Generalized brackets

References(49)

Figures(4)

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