Abstract
The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy–momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux–Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented.
Highlights
It is known since Eisenhart’s work on the geometrization of classical mechanics [1] that any dynamical system with d degrees of freedom qi, i = 1, . . . , d, which is governed by the Lagrangian L, can be embedded into the geodesic equations of the Brinkmann-type metric 2Ldt2 − dtdv, where t is the temporal variable and v is an extra coordinate
Second order Killing tensors are associated with separation of variables of the Hamilton–Jacobi equation, with Carter’s integration of the geodesic equations in the Kerr metric [4] being the prime example in General Relativity
While the geometric reformulation of Newtonian mechanics brings mostly aesthetic advantages, the construction of Brinkmann-type metrics with hidden symmetries is a source of new results
Summary
There are other examples of conformally flat spaces (but not constant curvature), possessing one Noether constant and a cubic integral (classified in [19] and further studied and generalised in [20,21]), which again cannot be represented as a cubic expression in the isometry algebra Being conformally flat, these spaces do have an abundant supply of conformal symmetries and in [22,23] a method was proposed for building quadratic and higher order invariants from appropriate polynomial expressions in conformal invariants. A physically viable metric, admitting a rank 2 Killing tensor, is constructed by uplifting a superintegrable model on S2, as well as other models with the additional functional freedom of being just Liouville integrable. Some final remarks are gathered in the concluding Sect. 6
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