Dynamical systems and Poisson structures

Download

index.pdf

Date

2009-11-01

Author

Guerses, Metin
Guseinov, Gusein Sh
Zheltukhın, Kostyantyn

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

249
views

0
downloads

We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in R-3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al. [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that all dynamical systems in R-n are locally (n-1)-Hamiltonian. We give also an algorithm, similar to the case in R-3, to construct a rank two Poisson structure of dynamical systems in R-n. We give a classification of the dynamical systems with respect to the invariant functions of the vector field (X) over right arrow and show that all autonomous dynamical systems in R-n are super-integrable. (C) 2009 American Institute of Physics. [doi:10.1063/1.3257919]

Subject Keywords

Mathematical Physics, Statistical and Nonlinear Physics

URI

https://hdl.handle.net/11511/37377

Journal

JOURNAL OF MATHEMATICAL PHYSICS

DOI

https://doi.org/10.1063/1.3257919

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

Hamiltonian equations in R-3 Ay, Ahmet; GÜRSES, METİN; Zheltukhın, Kostyantyn (AIP Publishing, 2003-12-01) The Hamiltonian formulation of N=3 systems is considered in general. The most general solution of the Jacobi equation in R-3 is proposed. The form of the solution is shown to be valid also in the neighborhood of some irregular points. Compatible Poisson structures and corresponding bi-Hamiltonian systems are also discussed. Hamiltonian structures, the classification of irregular points and the corresponding reduced first order differential equations of several examples are given. (C) 2003 American Institute...
Quantum duality, unbounded operators, and inductive limits Dosi, Anar (AIP Publishing, 2010-06-01) In this paper, we investigate the inductive limits of quantum normed (or operator) spaces. This construction allows us to treat the space of all noncommutative continuous functions over a quantum domain as a quantum (or local operator) space of all matrix continuous linear operators equipped with G-quantum topology. In particular, we classify all quantizations of the polynormed topologies compatible with the given duality proposing a noncommutative Arens-Mackey theorem. Further, the inductive limits of oper...
SYMMETRIC SPACE PROPERTY AND AN INVERSE SCATTERING FORMULATION OF THE SAS EINSTEIN-MAXWELL FIELD-EQUATIONS ERIS, A; GURSES, M; Karasu, Atalay (AIP Publishing, 1984-01-01) We formulate stationary axially symmetric (SAS) Einstein–Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight‐dimensional flat manifold and formulate SAS Einstein–Maxwell fields as a σ‐model. We then give, in a coordinate free way, a Belinskii–Zakharov type of an inverse scattering transform technique for the field equations supplemented by a...
Time-dependent recursion operators and symmetries Gurses, M; Karasu, Atalay; Turhan, R (Informa UK Limited, 2002-05-01) The recursion operators and symmetries of nonautonomous, (1 + 1) dimensional integrable evolution equations are considered. It has been previously observed hat he symmetries of he integrable evolution equations obtained through heir recursion operators do not satisfy the symmetry equations. There have been several attempts to resolve his problem. It is shown that in the case of time-dependent evolution equations or time-dependent recursion operators associativity is lost. Due to this fact such recursion ope...
Hydrodynamic type integrable equations on a segment and a half-line Guerses, Metin; Habibullin, Ismagil; Zheltukhın, Kostyantyn (AIP Publishing, 2008-10-01) The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented. (C) 2008 American Institut...

Citation Formats

M. Guerses, G. S. Guseinov, and K. Zheltukhın, “Dynamical systems and Poisson structures,” JOURNAL OF MATHEMATICAL PHYSICS, pp. 0–0, 2009, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/37377.