scholarly journals Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z-graded graphs

2021 ◽  
Vol 62 (4) ◽  
pp. 042204
Author(s):  
Gamal Mograby ◽  
Maxim Derevyagin ◽  
Gerald V. Dunne ◽  
Alexander Teplyaev
Keyword(s):  
Hamiltonian Systems ◽  
Lax Pairs ◽  
Toda Lattices

Finite-Dimensional Hamiltonian Systems from Li Spectral Problem by Symmetry Constraints

2007 ◽  
Vol 62 (7-8) ◽  
pp. 399-405
Author(s):  
Lin Luo ◽  
Engui Fan
Keyword(s):  
Spectral Problem ◽  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Lax Pairs ◽  
Curvature Equation ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Symmetry Constraints

A hierarchy associated with the Li spectral problem is derived with the help of the zero curvature equation. It is shown that the hierarchy possesses bi-Hamiltonian structure and is integrable in the Liouville sense. Moreover, the mono- and binary-nonlinearization theory can be successfully applied in the spectral problem. Under the Bargmann symmetry constraints, Lax pairs and adjoint Lax pairs are nonlineared into finite-dimensional Hamiltonian systems (FDHS) in the Liouville sense. New involutive solutions for the Li hierarchy are obtained.

scholarly journals Lax pairs for linear Hamiltonian systems

2019 ◽  
Vol 60 (4) ◽  
pp. 760-776 ◽  
Author(s):  
A. B. Zheglov ◽  
D. V. Osipov
Keyword(s):  
Hamiltonian Systems ◽  
Lax Pairs ◽  
Linear Hamiltonian Systems

scholarly journals Lax Pairs for Linear Hamiltonian Systems

2019 ◽  
Vol 60 (4) ◽  
pp. 592-604 ◽  
Author(s):  
A. B. Zheglov ◽  
D. V. Osipov
Keyword(s):  
Hamiltonian Systems ◽  
Lax Pairs ◽  
Linear Hamiltonian Systems

scholarly journals Bargmann Type Systems for the Generalization of Toda Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Fang Li ◽  
Liping Lu
Keyword(s):  
Hamiltonian Systems ◽  
Toda Lattice ◽  
Type Systems ◽  
Integrals Of Motion ◽  
Lattice Equation ◽  
Finite Dimensional ◽  
Symplectic Map ◽  
Representation Of Solutions ◽  
Toda Lattice Equation ◽  
Toda Lattices

Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete hierarchy of a generalization of the Toda lattice equation is proposed, which leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Finally, the representation of solutions for a lattice equation in the discrete hierarchy is obtained.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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