Evolutionary Laws, Initial Conditions, and Gauge Fixing in Constrained Systems

TL;DR

This paper presents a detailed method for removing nonphysical degrees of freedom in constrained systems, ensuring consistency between Lagrangian and Hamiltonian formulations, with applications to reparameterization invariant theories and cosmological models.

Contribution

It introduces a novel Lagrangian approach for fixing ambiguities and determining initial data in constrained systems, and proves the equivalence of degrees of freedom in both formulations.

Findings

Final degrees of freedom agree in Lagrangian and Hamiltonian formulations.

One constraint in reparameterization invariant theories must be explicitly time dependent.

Illustrated with spacetime trajectories and cosmological models.

Abstract

We describe in detail how to eliminate nonphysical degrees of freedom in the Lagrangian and Hamiltonian formulations of a constrained system. Two important and distinct steps in our method are the fixing of ambiguities in the dynamics and the determination of inequivalent initial data. The Lagrangian discussion is novel, and a proof is given that the final number of degrees of freedom in the two formulations agrees. We give applications to reparameterization invariant theories, where we prove that one of the constraints must be explicitly time dependent. We illustrate our procedure with the examples of trajectories in spacetime and with spatially homogeneous cosmological models. Finally, we comment briefly on Dirac's extended Hamiltonian technique.