Plugging the Gauge Fixing into the Lagrangian

TL;DR

This paper analyzes the effects of holonomic gauge fixing constraints on singular Lagrangians, showing that it removes gauge freedom and preserves second class constraints, with detailed application to electromagnetism.

Contribution

It provides a comprehensive analysis of how gauge fixing constraints alter the structure of singular Lagrangian systems, highlighting the loss of gauge freedom.

Findings

Reduced Lagrangian erases first class constraints

Retains second class constraints after gauge fixing

Reduced Lagrangian has no gauge freedom

Abstract

A complete analysis of the consequences of introducing a set of holonomic gauge fixing constraints (to fix the dynamics) into a singular Lagrangian is performed. It is shown in general that the dynamical system originated from the reduced Lagrangian erases all the information regarding the first class constraints of the original theory, but retains its second class. It is proved that even though the reduced Lagrangian can be singular, it never possesses any gauge freedom. As an application, the example of $n \cdot A = 0$ gauges in electromagnetism is treated in full detail.