Lagrange versus Symplectic Algorithm for Constrained Systems

TL;DR

This paper compares Lagrangian and symplectic algorithms for constrained systems, demonstrating the equivalence of the Lagrangian approach to Dirac's method and highlighting limitations of the symplectic algorithm in capturing the full constraint structure.

Contribution

It systematizes the Lagrangian approach into an algorithm and compares it with the symplectic method, revealing their differences in handling constraints.

Findings

Lagrangian approach is equivalent to Dirac's method for constrained systems.

The symplectic algorithm may not reproduce the full constraint structure.

A generalized Hessian matrix W is constructed for the Lagrangian system.

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.