On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian
نویسندگان
چکیده
منابع مشابه
Bi–Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation variables
We discuss from a bi-Hamiltonian point of view the Hamilton–Jacobi separability of a few dynamical systems. They are shown to admit, in their natural phase space, a quasi–bi– Hamiltonian formulation of Pfaffian type. This property allows us to straightforwardly recover a set of separation variables for the corresponding Hamilton–Jacobi equation.
متن کاملv 1 1 0 N ov 1 99 8 ON A CLASS OF DYNAMICAL SYSTEMS BOTH QUASI - BI - HAMILTONIAN AND BI - HAMILTONIAN
It is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero in [1]) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained trough a non canonical map whose form is directly suggested by the associated Nijenhuis tensor.
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We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.
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We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degenera...
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In 1975, one of the present authors 1 showed how to obtain the quantized levels of the nonlinear Schrodinger equation using the action-angle variables (canonical coordinates) of the AKNS scattering data. The symplectic form used to effect the reduction to canonical coordinates was based on the standard Hamiltonian structure for the nonlinear Schrooinger equation. The method used was a nonlinear...
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ژورنال
عنوان ژورنال: Physics Letters A
سال: 1998
ISSN: 0375-9601
DOI: 10.1016/s0375-9601(98)00543-x