Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation variables

Tondo, Giorgio; Morosi, Carlo

doi:10.1016/s0034-4877(99)80167-0

Cited by 16 publications

(16 citation statements)

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“…While much progress has been made in this direction using Noether symmetries, see for example 6,7 , an explicit approach to quantising (3), allowing for example to display eigenfunctions in configuration space, is still lacking. As to comparison with earlier works, the following remarks are in order: the most general set of Hamiltonians described in this paper, displayed in (20), have been described in various papers within the context of bi-Hamiltonian and quasi-bi-Hamiltonian systems [8][9][10][11] . Results concerning their integrability and their amenability to treatment via separation of variables are obtained there, but the connection to the quite elementary approach presented here is not apparent to the author.…”

Section: Introductionmentioning

confidence: 99%

An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families

Leyvraz¹

2017

Journal of Mathematical Physics

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We discuss a general approach permitting the identification of a broad class of sets of Poisson-commuting Hamiltonians, which are integrable in the sense of Liouville. It is shown that all such Hamiltonians can be solved explicitly by a separation of variables Ansatz. The method leads in particular to a proof that the so-called "goldfish" Hamiltonian is maximally superintegrable, and leads to an elementary identification of a full set of integrals of motion. The Hamiltonians in involution with the "goldfish" Hamiltonian are also explicitly integrated. New integrable Hamiltonians are identified, among which some have the property of being isochronous, that is, that all their orbits have the same period. Finally, a peculiar structure is identified in the Poisson brackets between the elementary symmetric functions and the set of Hamiltonians commuting with the "goldfish" Hamiltonian: these can be expressed as products between elementary symmetric functions and Hamiltonians. The structure displays an invariance property with respect to one element, and has both a symmetry and a closure property. The meaning of this structure is not altogether clear to the author, but it turns out to be a powerful tool.

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