Variational symmetries as the existence of ignorable coordinates

TL;DR

This paper demonstrates that variational symmetries in a Lagrangian system are equivalent to the existence of coordinates where one coordinate is ignorable, simplifying the analysis of conserved quantities.

Contribution

It establishes a direct link between variational symmetries and the existence of ignorable coordinates in the extended configuration space.

Findings

Variational symmetry implies an ignorable coordinate exists.

Existence of an ignorable coordinate indicates a variational symmetry.

Provides a coordinate transformation approach for symmetry analysis.

Abstract

It is shown that given a Lagrangian for a system with a finite number of degrees of freedom, the existence of a variational symmetry is equivalent to the existence of coordinates in the extended configuration space such that one of the coordinates is ignorable.