Eisenhart lift for higher derivative systems

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.Comment: V2: 12 pages, minor improvements, references added; the version to appear in PL

Eisenhart lift for higher derivative systems

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.Comment: V2: 12 pages, minor improvements, references added; the version to appear in PL

Hidden Symmetries of Dynamics in Classical and Quantum Physics

This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description of physical systems as varied as non-relativistic, relativistic, with or without gravity, classical or quantum, and are related to the existence of conserved quantities of the dynamics and integrability. In recent years their study has grown intensively, due to the discovery of non-trivial examples that apply to different types of theories and different numbers of dimensions. Applications encompass the study of integrable systems such as spinning tops, the Calogero model, systems described by the Lax equation, the physics of higher dimensional black holes, the Dirac equation, supergravity with and without fluxes, providing a tool to probe the dynamics of non-linear systems.Comment: 54 pages, review article. To be published in Rev. Mod. Phy

Eisenhart Lift of 22--Dimensional Mechanics

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.Comment: 20 page

Embedding nonrelativistic physics inside a gravitational wave

Gravitational waves with parallel rays are known to have remarkable properties: Their orbit space of null rays possesses the structure of a non-relativistic spacetime of codimension-one. Their geodesics are in one-to-one correspondence with dynamical trajectories of a non-relativistic system. Similarly, the null dimensional reduction of Klein-Gordon's equation on this class of gravitational waves leads to a Schroedinger equation on curved space. These properties are generalized to the class of gravitational waves with a null Killing vector field, of which we propose a new geometric definition, as conformally equivalent to the previous class and such that the Killing vector field is preserved. This definition is instrumental for performing this generalization, as well as various applications. In particular, results on geodesic completeness are extended in a similar way. Moreover, the classification of the subclass with constant scalar invariants is investigated.Comment: 56 pages, 9 figures, v3:Minor correction

Eisenhart lifts and symmetries of time-dependent systems

Certain dissipative systems, such as Caldirola and Kannai's damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with $n$ degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in $n+2$ dimensions, equipped with its covariantly constant null Killing vector field. Reparametrization of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola-Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schr\"odinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space. Time-dependent Lagrangians arise naturally also in cosmology and give rise to the phenomenon of Hubble friction. We provide an account of this for Friedmann-Lemaitre and Bianchi cosmologies and how it fits in with our previous discussion in the non-relativistic limit.Comment: 34 pages, no figures. Minor corrections, some references adde

Lyapunov exponents from geodesic spread in configuration space

The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold M_E = {q in R^N | V(q) < E} of a standard Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC equation can be used to study the degree of instability of the Hamiltonian flow. It is found that the solutions of the JLC equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute Lyapunov exponents. Therefore the instability exponents obtained through the JLC equation are in perfect quantitative agreement with usual Lyapunov exponents. This work completes a previous investigation that was limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure

Ricci-flat spacetimes admitting higher rank Killing tensors

Ricci-flat spacetimes of signature (2,q) with q=2,3,4 are constructed which admit irreducible Killing tensors of rank-3 or rank-4. The construction relies upon the Eisenhart lift applied to Drach's two-dimensional integrable systems which is followed by the oxidation with respect to free parameters. In four dimensions, some of our solutions are anti-self-dual.Comment: 12 page