Variational Contact Symmetries of Constraint Lagrangians

TL;DR

This paper explores contact symmetries of re-parametrization invariant quadratic Lagrangians, revealing that integrals of motion relate to conformal Killing vectors and tensors, with gauge functions often unnecessary due to Hamiltonian constraints.

Contribution

It extends the analysis of contact symmetries to include linear velocity dependence and shows how integrals of motion connect to conformal Killing structures in the configuration space.

Findings

Integrals of motion are generated by conformal Killing vectors and tensors.

Re-parametrization freedom allows potential normalization, simplifying symmetry analysis.

Gauge functions are not essential in Noether's theorem due to Hamiltonian constraints.

Abstract

The investigation of contact symmetries of re--parametrization invariant Lagrangians of finite degrees of freedom and quadratic in the velocities is presented. The main concern of the paper is those symmetry generators which depend linearly in the velocities. A natural extension of the symmetry generator along the lapse function $N(t)$, with the appropriate extension of the dependence in $\dot{N}(t)$ of the gauge function, is assumed; this action yields new results. The central finding is that the integrals of motion are either linear or quadratic in velocities and are generated, respectively by the conformal Killing vector fields and the conformal Killing tensors of the configuration space metric deduced from the kinetic part of the Lagrangian (with appropriate conformal factors). The freedom of re--parametrization allows one to appropriately scale $N(t)$, so that the potential becomes constant; in this case the integrals of motion can be constructed from the Killing fields and Killing tensors of the scaled metric. A rather interesting result is the non--necessity of the gauge function in Noether's theorem due to the presence of the Hamiltonian constraint