Herglotz's formalism, Eisenhart lift and Killing vectors
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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