# A Study of the Approximate Singular Lagrangian -- Conditional Noether Symmetries and First Integrals

## TL;DR

This paper explores approximate and conditional Noether symmetries in reparametrization invariant Lagrangians, revealing how these symmetries relate to conformal Killing vectors and providing a geometric framework for their determination.

## Contribution

It introduces a geometric method to identify approximate and conditional Noether symmetries in constrained Lagrangians, including the lapse as a degree of freedom.

## Key findings

- Extra conditions lead to approximate and conditional symmetries.
- Noether symmetries correspond to conformal Killing vectors.
- Method applied to a pedagogical example.

## Abstract

The investigation of approximate symmetries of reparametrization invariant Lagrangians of n + 1 degrees of freedom and quadratic velocities is presented. We show that extra conditions emerge which give rise to approximate and conditional Noether symmetries of such constrained actions. The Noether symmetries are the simultaneous conformal Killing vectors of both the kinetic metric and the potential. In order to recover these conditional symmetry generators which would otherwise be lost in gauge fixing the lapse function entering the perturbative Lagrangian, one must consider the lapse among the degrees of freedom. We establish a geometric framework in full generality to determine the admitted Noether symmetries. Additionally, we obtain the corresponding first integrals (modulo a constraint equation). For completeness, we present a pedagogical application of our method.