The Bohlin variant of the Eisenhart lift embeds Lagrangian systems into timelike geodesics of conformally flat (d+2)-dimensional metrics and yields novel examples of such metrics admitting higher-rank Killing tensors.
Eisenhart lift for higher derivative systems
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abstract
The Eisenhart lift provides an elegant geometric description of a dynamical system of second order in terms of null geodesics of the Brinkmann-type metric. In this work, we attempt to generalize the Eisenhart method so as to encompass higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian. A consistent geometric description seems feasible only for a particular class of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.
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The Bohlin variant of the Eisenhart lift
The Bohlin variant of the Eisenhart lift embeds Lagrangian systems into timelike geodesics of conformally flat (d+2)-dimensional metrics and yields novel examples of such metrics admitting higher-rank Killing tensors.