Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation

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We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation separates and the Riemann curvature tensor satisfies a certain equation (called the diagonal curvature condition). All orthogonal metrics of constant curvature satisfy the diagonal curvature condition. The metrics we find either correspond to a Benenti system or are warped product metrics where the induced metric on the base manifold corresponds to a Benenti system. Furthermore we show that most metrics we find are characterized by concircular tensors; these metrics, called Kalnins-Eisenhart-Miller (KEM) metrics, have an intrinsic characterization which can be used to obtain them. In conjunction with other results, we show that the metrics we found constitute all separable metrics for Riemannian spaces of constant curvature and de Sitter space.

“…. , x k ) are separable coordinates for (M, g) [RM14]. This observation motivates the following definition:…”

Section: Concircular Tensors and Kem Coordinatesmentioning

confidence: 99%

Section: Concircular Tensors and Kem Coordinatesmentioning

confidence: 99%

Section: Concircular Tensors and Kem Coordinatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature

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McLenaghan

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Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

Cariñena

Herranz

Rañada

2017

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian H = T + V into a geodesic Hamiltonian T with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians T r (r = a, b, c, d) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential U r leading to H r = T r + U r . Secondly, we study the superintegrability of the four Hamiltonians H r = H r /µ r , where µ r is a certain position-dependent mass, that enjoys the same separability as the original system H r . All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.It is well known that the harmonic (isotropic) oscillator and the Kepler-Coulomb (KC) problem are integrable systems admitting additional constants of motion (Demkov-Fradkin tensor [1,2] and Laplace-Runge-Lenz vector, respectively). Systems endowed with this property are called superintegrable. It is also known that if a system is separable (Hamilton-Jacobi (HJ) separable in the classical case or Schrödinger separable in the quantum case), then it is integrable with integrals of motion of at most second-order in momenta. Thus, if a system admits multiseparability (separability in several different systems of coordinates) then it is endowed with 'quadratic superintegrability' (superintegrability with linear or quadratic integrals of motion).Fris et al. studied in [3] the two-dimensional (2D) Euclidean systems admitting separability in more than one coordinate system and they obtained four families of potentials V r , r = a, b, c, d, possessing three functionally independent integrals of motion (they were mainly interested in the quantum 2D Schrödinger equation but their results also hold at the classical level). Then other authors studied similar problems on higher-dimensional Euclidean spaces [4]-[6], on 2D spaces with a pseudo-Euclidean metric (Drach potentials) [7]-[10], and on curved spaces [11]-[21] (see [22] for a recent review on superintegrability that includes a long list of references).The superintegrability property is related with different formalisms and it can be studied by making use of different approaches, that is, proving that all bounded classical trajectories are closed, HJ separability, action-angle variables formalism, exact solvability, degenerate quantum energy levels, complex functions whose Poisson bracket with the Hamiltonian are proportional to themselves, etc. In this paper, we relate superintegrability with a geometric formalism introduced many years ago by Eisenhart [23].The theory of general relativity states that the motion of a particle under the action of gravitational forces is described by a geodesic in the 4D Riemannian spacetime. The Eisenhart formalism (also known as Eisenhart lift) associates to a sys...

Equivalence problem for the orthogonal separable webs in 3-dimensional hyperbolic space

Cochran

McLenaghan

Smirnov

2017