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An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families. (arXiv:1710.01750v1 [math-ph])
来源于:arXiv
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 {\em 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 stru 查看全文>>