scholarly journals Poisson structures on (non)associative noncommutative algebras and integrable Kontsevich type Hamiltonian systems

2020 ◽  
Vol 3 (1) ◽  
pp. 001-006
Author(s):  
Hentosh Oksana E ◽  
Balinsky Alexander A ◽  
Prykarpatski Anatolij K
Keyword(s):  
Hamiltonian Systems ◽  
Poisson Structures ◽  
Noncommutative Algebras

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

2014 ◽  
Vol 6 (01) ◽  
pp. 87-106
Author(s):  
Xueyang Li ◽  
Aiguo Xiao ◽  
Dongling Wang
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Generating Function ◽  
Poisson Structure ◽  
Poisson Structures ◽  
Volterra Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

scholarly journals Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

2007 ◽  
Vol 365 (1858) ◽  
pp. 2333-2357 ◽  
Author(s):  
Boris Kolev
Keyword(s):  
Shallow Water ◽  
Lie Algebra ◽  
Hamiltonian Systems ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Vector Fields ◽  
Poisson Structures ◽  
Shallow Water Waves ◽  
Survey Article ◽  
Special Case

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie–Poisson structure and the second one is constant. These structures, called affine or modified Lie–Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

scholarly journals DYNAMICAL ASPECTS OF LIE-POISSON STRUCTURES

1993 ◽  
Vol 08 (31) ◽  
pp. 2973-2987 ◽  
Author(s):  
F. LIZZI ◽  
G. MARMO ◽  
G. SPARANO ◽  
P. VITALE
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Poisson Bracket ◽  
Lie Groups ◽  
Hamiltonian Systems ◽  
Quantum Groups ◽  
Equations Of Motion ◽  
Poisson Structures ◽  
Dimensional Phase Space ◽  
Deformation Procedure

Quantum groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU(2) and SU(1, 1), as submanifolds of a four-dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some Hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.

scholarly journals GEOMETRY OF DEFORMED BOSON ALGEBRAS

1996 ◽  
Vol 08 (07) ◽  
pp. 949-956
Author(s):  
P. CREHAN ◽  
T.G. HO
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Parameter Family ◽  
Poisson Structures ◽  
Nonlinear Transformations ◽  
Deformation Parameters ◽  
Special Cases ◽  
Non Linear ◽  
Deformed Algebras

We consider a phase space realisation of an infinite parameter family of deformations of the boson algebra in which the q-deformed algebras arise as special cases. The deformation parameters are identified with coefficients of non-linear terms in the normal forms expansion of a family of classical Hamiltonian systems. These deformations simply correspond to nonlinear transformations of the boson algebra. They are physically interesting, as the deformed commutators consistently quantise a class of noncanonical classical Poisson structures.

Deformations of Poisson structures on fibered manifolds and adiabatic slow–fast systems

2017 ◽  
Vol 14 (06) ◽  
pp. 1750086 ◽  
Author(s):  
Misael Avendaño-Camacho ◽  
Yury Vorobiev
Keyword(s):  
Perturbation Theory ◽  
Hamiltonian Systems ◽  
Deformation Theory ◽  
Normal Forms ◽  
Geometric Approach ◽  
Fiber Bundles ◽  
Poisson Structures ◽  
Averaging Theorem ◽  
Fibered Manifolds ◽  
Adiabatic Perturbation Theory

In the context of normal forms, we study a class of slow–fast Hamiltonian systems on general Poisson fiber bundles with symmetry. Our geometric approach is motivated by a link between the deformation theory for Poisson structures on fibered manifolds and the adiabatic perturbation theory. We present some normalization results which are based on the averaging theorem for horizontal 2-cocycles on Poisson fiber bundles.

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