# Herglotz's formalism, Eisenhart lift and Killing vectors

## TL;DR

This paper extends Eisenhart's lift to action-dependent Lagrangians, linking symmetries to conformal Killing vectors of the resulting metrics, and explores their equivalence and scaling invariance.

## Contribution

It introduces a generalized Eisenhart lift for action-dependent Lagrangians and analyzes the resulting conformal symmetries and metric equivalences.

## Key findings

- Symmetries lead to conformal Killing vectors of the extended metric.
- Time- and action-dependent descriptions can produce conformally equivalent metrics.
- Scaling invariance is naturally incorporated into the formalism.

## Abstract

The Eisenhart lift is extended to the case of dynamics described by action-dependent Lagrangians. The resulting Brinkmann metric depends on all coordinates. It is shown that the symmetries of the initial dynamics result in the existence of (conformal) Killing vectors of the Brinkmann metric. An example is given of equivalent time- and action-dependent descriptions which result in conformally equivalent metrics. It is also shown how the scaling invariance fits naturally into this scheme.

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Source: https://tomesphere.com/paper/2508.21588