Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

Rañada, Manuel F.

doi:10.1063/1.4918611

Cited by 10 publications

(15 citation statements)

References 54 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, the study of systems with a position dependent mass is a matter highly studied in these last years but, in most of cases, these studies are related with the problem of the quantization (because the problem of order in the quantization of the kinetic term); the study presented in this paper is concerned with only the classical case and, although different, it has a close relation with the study presented in [9].…”

Section: Introductionmentioning

confidence: 93%

“…Second, quadratic superintegrability is a property very related with Hamilton-Jacobi (H-J) multiple separability (Schrödinger separability in the quantum case) and this property is also true for systems with a position dependent mass. This question (H-J separability approach to systems with a pdm) was studied in [9] (in this case the pdm depends on a parameter κ) and more recently in [10] (in this last case the pdm Hamiltonians studied were also related with those recently obtained through a differential Galois group analysis in [1]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems

Rañada

2016

Physics Letters A

Self Cite

View full text Add to dashboard Cite

The superintegrability of two-dimensional Hamiltonians with a position dependent mass (pdm) is studied (the kinetic term contains a factor m that depends of the radial coordinate). First, the properties of Killing vectors are studied and the associated Noether momenta are obtained. Then the existence of several families of superintegrable Hamiltonians is proved and the quadratic integrals of motion are explicitly obtained. These families include, as particular cases, some systems previously obtained making use of different approaches. We also relate the superintegrability of some of these pdm systems with the existence of complex functions endowed with interesting Poisson bracket properties. Finally the relation of these pdm Hamiltonians with the Euclidean Kepler problem and with the Euclidean harmonic oscillator is analyzed.

show abstract

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems

Rañada

2016

Physics Letters A

Self Cite

View full text Add to dashboard Cite

show abstract

“…The new geodesic dynamics determined by T r must be a deformation of the initial one provided by T r [84] in the sense that the PDM µ r , and so T r , will depend on a real parameter λ in such a way that the following properties must be satisfied: (iii) When taking the limit λ → 0 the PDM must satisfy µ r (λ) → 1, so that the dynamics of the original geodesic Hamiltonian T r is recovered.…”

Section: D Geodesic Hamiltonians T With a Position-dependent Massmentioning

confidence: 99%

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

Cariñena

Herranz

Rañada

2017

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…If −1 < σ < 1 then P 2 x increases monotonically from −∞ to +∞ so it vanishes for x = x * given by (17), and the geodesic is defined for x ∈ (x * , π). The relation (11), taking for initial conditions (x = x * , y = 0), we have (cos x * − η) cosh y = cos x − η which gives (16). On the following figure some geodesics are drawn: Figure 1: the special case σ = 0…”

Section: Remarksmentioning

confidence: 99%

“…More recently, further potentials were derived in [16], while in [8] with emphasis on the geodesics.…”

Section: Introductionmentioning

confidence: 99%

Global structure and geodesics for Koenigs superintegrable systems

Valent¹

2016

Regul. Chaot. Dyn.

View full text Add to dashboard Cite

Starting from the framework defined by Matveev and Shevchishin we derive the local and the global structure for the four types of super-integrable Koenigs metrics. These dynamical systems are always defined on non-compact manifolds, namely R 2 and H 2 . The study of their geodesic flows is made easier using their linear and quadratic integrals. Using Carter (or minimal) quantization we show that the formal superintegrability is preserved at the quantum level and in two cases, for which all of the geodesics are closed, it is even possible to compute the discrete spectrum of the quantum hamiltonian.

show abstract

Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

Cited by 10 publications

References 54 publications

Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems

Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems

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

Global structure and geodesics for Koenigs superintegrable systems

Contact Info

Product

Resources

About