Gauge symmetry and partially Lagrangian systems

Abstract

We consider a classical field theory whose equations of motion follow from the least action principle, but the class of admissible trajectories is restricted by differential equations. The key element of the proposed construction is the complete gauge symmetry of these additional equations. The unfree variation of the trajectories reduces to the infinitesimal gauge symmetry transformation of the equations, restricting the trajectories. We explicitly derive the equations that follow from the requirement that this gauge variation of the action vanishes. The system of equations for conditional extrema is not a Lagrangian system as such, but it admits an equivalent Hamiltonian formulation with a non-canonical Poisson bracket. The bracket is degenerate, in general. Alternatively, the equations restricting the dynamics could be added to the action with Lagrange multipliers with unrestricted variation of the original variables. In this case, we would arrive at the Lagrangian equations for the original variables involving Lagrange multipliers and for Lagrange multipliers themselves. In general, these two methods are not equivalent because the multipliers can bring extra degrees of freedom compared to the case of equations derived by unfree variation of the action. We illustrate the general method with two examples. The first example is a particle in a central field with varying trajectories restricted by the equation of conservation of angular momentum. The phase space acquires one more dimension, and there is an extra conserved quantity $K$ which is responsible for the precession of trajectories. The second example is linearized gravity with the Einstein-Hilbert action, and the class of varying fields is restricted by the linearized Nordstr\"om equation. This conditional extrema problem is shown to lead to the linearized Cotton gravity equations.