Lagrange versus Symplectic Algorithm for Constrained Systems

Abstract

The systematization of the purely Lagrangean approach to constrained systems in the form of an algorithm involves the iterative construction of a generalized Hessian matrix W taking a rectangular form. This Hessian will exhibit as many left zero-modes as there are Lagrangean constraints in the theory. We apply this approach to a general Lagrangean in the first order formulation and show how the seemingly overdetermined set of equations is solved for the velocities by suitably extending W to a rectangular matrix. As a byproduct we thereby demonstrate the equivalence of the Lagrangean approach to the traditional Dirac-approach. By making use of this equivalence we show that a recently proposed symplectic algorithm does not necessarily reproduce the full constraint structure of the traditional Dirac algorithm.

20 pages, LaTeX

References in corpus (3)

• Symplectic embedding and Hamilton-Jacobi analysis of Proca model
• Constraint structure in modified Faddeev-Jackiw method
• Lagrangean Approach to Hamiltonian Gauge Symmetries and the Dirac Conjecture

Cited by in corpus (7)

• Faddeev-Jackiw quantization and the path integral
• Gauge Identities and the Dirac Conjecture
• The Difficulty of Symplectic Analysis with Second Class Systems
• An Extension of the Faddeev-Jackiw Technique to Fields in Curved Spacetimes
• Darboux coordinates for the Hamiltonian of first order Einstein-Cartan gravity
• Barcelos-Wotzasek symplectic algorithm for constrained systems revisited
• Symplectic algorithm for systems with second-class constraints