Natural coordinates for a class of Benenti systems

Błaszak, Maciej; Sergyeyev, Artur

doi:10.1016/j.physleta.2007.01.001

Cited by 27 publications

(46 citation statements)

References 13 publications

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“…Remark 3.4). Consider now the point transformation from (q, p)-coordinates (2.11), (3.1) to non-orthogonal coordinates (r, s) such that r i are given by [7] q 1 = r 1 ,…”

Section: A Class Of F Lat and Constant Curvature Stäckel Systemsmentioning

confidence: 99%

Classical and Quantum Superintegrability of Stäckel Systems

Błaszak¹,

Marciniak²

2017

SIGMA

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Abstract. In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

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“…Remark 3.4). Consider now the point transformation from (q, p)-coordinates (2.11), (3.1) to non-orthogonal coordinates (r, s) such that r i are given by [7] q 1 = r 1 ,…”

Section: A Class Of F Lat and Constant Curvature Stäckel Systemsmentioning

confidence: 99%

Classical and Quantum Superintegrability of Stäckel Systems

Błaszak¹,

Marciniak²

2017

SIGMA

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“…We will now turn to the main question of this article: how to relate two Stäckel systems by a single Stäckel transform and in such a way that their solutions are related by a reciprocal transform? As we mentioned above, The Hamiltonians H i defined by (21) or by (22) do not depend on any additional parameters α i so in order to perform a Stäckel transform on (21) we have to embed it into a parameter-dependent system. Of course, there is infinitely many ways of embedding of our Stäckel system into an n-parameter system but the choice below is natural in the sense that the corresponding Stäckel transform transforms a Stäckel system into a new Stäckel system.…”

Section: Stäckel Equivalence Of Stäckel Systemsmentioning

confidence: 99%

“…and as we remember it is coordinate free. The passage to the flat coordinates (x 1 , x 2 ) is given by the point transformation [21]…”

Section: Examplesmentioning

confidence: 99%

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On Reciprocal Equivalence of Stäckel Systems

Błaszak

Marciniak

2012

Stud Appl Math

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In this paper we ivestigate Stäckel transforms between di¤erent classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

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“…Our construction encompasses two known cases: Jacobi elliptic coordinates (introduced in [3] and fully described in [4]) and Jacobi parabolic coordinates and also one of the less known cases considered recently by Blaszak and Sergyeyev in [5] (but with no degeneration of coordinate systems).…”

Section: Introductionmentioning

confidence: 99%