Quadratic and cubic invariants in classical mechanics

Fokas, A. S.; Lagerstrom, P. A.

doi:10.1016/0022-247x(80)90132-8

Cited by 59 publications

(51 citation statements)

References 3 publications

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“…The special cases with frequency ratios 1:2 143,364 and 1:3 141 have also been considered. While at the classical level it is clear that the su(N) or sp(2N,R) algebras can be used for the description of the N-dimensional anisotropic oscillator, the situation at the quantum level, even in the two-dimensional case, is not as simple.…”

Section: Other Approachesmentioning

confidence: 99%

Quantum groups and their applications in nuclear physics

Bonatsos

Daskaloyannis

1999

Progress in Particle and Nuclear Physics

198

203

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Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the su q (2) rotator model and its extensions, the construction of deformed exactly soluble models (u(3)⊃so (3) model, Interacting Boson Model, Moszkowski model), the 3-dimensional q-deformed harmonic oscillator and its relation to the nuclear shell model, the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to the structure of molecules and of atomic clusters, as well as an outlook are also given.

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Section: Other Approachesmentioning

confidence: 99%

Quantum groups and their applications in nuclear physics

Bonatsos

Daskaloyannis

1999

Progress in Particle and Nuclear Physics

198

203

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“…For third order integrals this is no longer the case (as was pointed out by Hietarinta in [16]). For integrable systems with third or higher order integrals in classical mechanics, see also [7,11,12,15,20,22,25] 2 Conditions for the existence of a third order invariant in classical mechanics…”

Section: Introductionmentioning

confidence: 99%

Superintegrability with third-order integrals in quantum and classical mechanics

Gravel

Winternitz

2002

Journal of Mathematical Physics

119

225

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“…We suppose that all these systems may be embedded into the family of the Stäckel systems [12], either into the subset of the generalized Stäckel systems [14] or these systems may be related to the Toda lattices and the Calogero-Moser systems [4,14]. As an example the last case (l) (3.10) of the Drach systems and the Fokas-Lagerstrom [3] model belong to the generalized Stäckel systems [14]. The complete classification will be presented in the forthcoming publication.…”

Section: Resultsmentioning

confidence: 99%

“…An application of the direct method [3,5,4] or the Jacobi method [6] does not allows us to obtain this system (4.9).…”

Section: Other Degenerate Systems On the Plane With A Qubic Integral mentioning

confidence: 99%

The Drach superintegrable systems

Tsiganov

2000

J. Phys. A: Math. Gen.

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Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated Stäckel problems with quadratic integrals of motion. For the superintegrable Stäckel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.

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Quadratic and cubic invariants in classical mechanics

Cited by 59 publications

References 3 publications

Quantum groups and their applications in nuclear physics

Quantum groups and their applications in nuclear physics

Superintegrability with third-order integrals in quantum and classical mechanics

The Drach superintegrable systems

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