Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

Valent, Galliano

doi:10.1134/s1560354717040013

Cited by 12 publications

(10 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another open problem is whether Lorentzian metrics admitting third or higher rank Killing tensors, linked to the work in [19,20] give rise to physically meaningful solutions.…”

Section: Resultsmentioning

confidence: 99%

“…Specifically, there are the metrics found by Koenigs [15], which are described and analysed in [16,17]. There are other examples of conformally flat spaces (but not constant curvature), possessing one Noether constant and a cubic integral (classified in [18] and further studied and generalised in [19,20]), 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 [21,22] a method was proposed for building quadratic and higher order invariants from appropriate polynomial expressions in conformal invariants.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Eisenhart lift of 2-dimensional mechanics

Fordy

Galajinsky

2019

Eur. Phys. J. C

View full text Add to dashboard Cite

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.

show abstract

“…Another open problem is whether Lorentzian metrics admitting third or higher rank Killing tensors, linked to the work in [19,20] give rise to physically meaningful solutions.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Eisenhart lift of 2-dimensional mechanics

Fordy

Galajinsky

2019

Eur. Phys. J. C

View full text Add to dashboard Cite

show abstract

“…Some third order integrals were considered in [2], where one of the cases of [10] was constructed, but even in 2 degrees of freedom this was complicated. This general problem is discussed in [12], for systems in 2 degrees of freedom, with one Killing vector.…”

Section: Specific Cases Listed In Section 53mentioning

confidence: 99%

Superintegrable systems on 3 dimensional conformally flat spaces

Fordy

Huang

2020

Journal of Geometry and Physics

View full text Add to dashboard Cite

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system. 1 2 n(n + 1) = 6 Killing vectors (when n = 3).The conformal algebra of this 3D metric has dimension 1 2 (n + 1)(n + 2) = 10 (when n = 3). A convenient

show abstract

“…Superintegrable systems with one linear and one cubic integral were described in [69]; the methods can be generalized for the case when one integral is linear and the second of arbitrary degree; e.g. [70]. Problem 3.10.…”

Section: (C) Superintegrable Systemsmentioning

confidence: 99%

Open problems, questions and challenges in finite- dimensional integrable systems

Bolsinov

Matveev

Miranda

et al. 2018

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

show abstract

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

Cited by 12 publications

References 14 publications

Eisenhart lift of 2-dimensional mechanics

Eisenhart lift of 2-dimensional mechanics

Superintegrable systems on 3 dimensional conformally flat spaces

Open problems, questions and challenges in finite- dimensional integrable systems

Contact Info

Product

Resources

About