Herglotz's formalism, Eisenhart lift and Killing vectors

TL;DR

By extending Eisenhart's geometric lift to action-dependent dynamics via Herglotz's variational principle, the paper shows that the most general Brinkmann metric arises from lifting n-dimensional action-dependent systems; symmetries of the original dynamics correspond to (conformal) Killing vectors of the lifted metric, linking conservation laws to spacetime geometry. Time-dependent and action-dependent descriptions are shown to be conformally equivalent in the Brinkmann framework, with u no longer an affine parameter and charges becoming nonlocal. The work also clarifies how scaling symmetry fits into this scheme and sketches possible Hamiltonian formulations and Killing tensor extensions for future work.

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.