Singular lagrangians and the Hamilton-Jacobi formalism in classical mechanics

TL;DR

This paper applies the Hamilton-Jacobi formalism to classical systems with constraints, comparing it with Dirac-Bergmann and Faddeev-Jackiw methods, and demonstrates its efficiency and broad applicability.

Contribution

It provides a detailed Hamilton-Jacobi analysis of constrained systems, including a comparison with other formalisms, highlighting its simplicity and computational advantages.

Findings

Hamilton-Jacobi formalism effectively handles complex constraints.

Comparison shows Hamilton-Jacobi is simpler and more efficient.

Analysis includes gauge systems and various constraint types.

Abstract

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.