About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels

Boualem, Hassan; Brouzet, Robert; Rakotondralambo, Joseph

doi:10.1016/j.difgeo.2008.04.008

Cited by 4 publications

(5 citation statements)

References 13 publications

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“…This concept was first introduced in [4] in the particular case of systems with two degrees of freedom and it was quickly extended in [23,24] for a higher-dimensional systems. Some recent papers considering properties of this particular class of systems are [1,2,3,4,5,6,8,14,15,23,24,25,36].…”

Section: Definitionmentioning

confidence: 99%

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

Cariñena

Rañada

2016

SIGMA

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Abstract. The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and in the second part a new quasi-bi-Hamiltonian structure is obtained by making use of the separability of the system in parabolic coordinates.

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Section: Definitionmentioning

confidence: 99%

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

Cariñena

Rañada

2016

SIGMA

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“…The additional assumption (BH) is that the corresponding recursion operator = ′ − N P P 1 has n functionally independent real eigenvalues, λ λ … , , n 1 . This formulation of the theorem is often mentioned in modern literature; see for instance [1,3,4,[13][14][15][16]. However, there is a trivial misprint, because the author omits one more necessary condition: that of the non-degeneracy (2.7) of the Hamiltonian function, which can be found in assumption 'ND' on page 5 in [8] and in the proof of the theorem.…”

Section: Discussionmentioning

confidence: 99%

“…, λ n . This formulation of the theorem is often mentioned in modern literature, see for instance [1,3,4,13,14,15,16]. Nevertheless, there is some trivial misprint because the author omits one more necessary condition of the non degeneracy (2.7) of the Hamiltonian function, which could be found in the assumption "ND" on the page 5 in [8] and in the proof of the theorem.…”

Section: Discussionmentioning

confidence: 99%

On bi-Hamiltonian formulation of the perturbed Kepler problem

Grigoryev¹,

Tsiganov²

2015

J. Phys. A: Math. Theor.

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The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which is against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, both the initial and perturbed Kepler systems are isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.

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“…Bi-Hamiltonian systems are systems endowed with interesting properties but, in general, it is quite difficult to find a bi-Hamiltonian formulation for a given Hamiltonian vector field and, for this reason, it is useful the concept of quasi-bi-Hamiltonian system. This concept was first introduced in [9,10] and then studied by other authors [11][12][13][14][15][16][17][18].…”

Section: A Geometric Introductionmentioning

confidence: 99%

Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems

Rañada¹

2017

J. Phys. A: Math. Theor.

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The existence of quasi-bi-Hamiltonian structures for the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems is studied. We first recall that the superintegrability of these two systems is related with the existence of certain complex functions endowed with interesting Poisson bracket properties, and then we prove the existence of several quasi-bi-Hamiltonian structures making use of these complex functions. This is done in two steps: first with complex 2-forms (wedge product of the differentials of the complex functions) and then with several real 2-forms. The properties of these geometric structures and the associated recursion operators are also analyzed. The paper can be considered as divided in two parts. In the first part a we study the TTW system (related to the harmonic oscillator) and in the second part we study the PW system (related to the Kepler problem).

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About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels

Cited by 4 publications

References 13 publications

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

On bi-Hamiltonian formulation of the perturbed Kepler problem

Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems

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