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.
Highlights
In classical mechanics several methods are known which provide a consistent geometric description of a second order dynamical system
Given a dynamical system with n degrees of freedom, the former operates with a Riemannian metric on an n–dimensional manifold, while the latter yields a Brinkmann–type metric [4] of Lorentzian signature in an (n + 2)–dimensional spacetime which is of interest in the general relativistic context
In [11, 12] Ricci–flat spacetimes of the ultrahyperbolic signature which support higher rank Killing tensors or possess maximally superintegrable geodesic flows have been built along similar lines
Summary
In classical mechanics several methods are known which provide a consistent geometric description of a second order dynamical system. An alternative method, which operates with a larger set of extra degrees of freedom, is proposed and shown to yield a consistent geometric description for a particular class of potentials which are the sum of homogeneous functions with arbitrary coefficients (coupling constants). Geometric properties of such metric are discussed in detail. A simple canonical transformation applied to Ostrogradsky’s Hamiltonian It makes the conventional Eisenhart lift feasible, provided the potential depends on the variable and its derivatives of even order only.
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