Oğul Esen

A hierarchy of Poisson brackets in non-equilibrium thermodynamics

Abstract Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed and what additional features we also gain by the construction. In particular, the Poisson brackets governing reversible evolution in one-particle kinetic theory, kinetic theory of binary mixtures, binary fluid mixtures, classical irreversible thermodynamics and classical hydrodynamics are derived from Liouville equation. Although the construction is quite natural, a few examples where it does not work are included (e.g. the BBGKY hierarchy). Finally, a new infinite grand-canonical hierarchy of Poisson brackets is proposed, which leads to Poisson brackets expressing non-local phenomena such as turbulent motion or evolution of polymeric fluids. Eventually, Lie–Poisson structures standing behind some of the brackets are identified.

Bi-Hamiltonian Structures of 3D Chaotic Dynamical Systems

We study Hamiltonian structures of dynamical systems with three degrees of freedom which are known for their chaotic properties, namely Lu, modified Lu, Chen, T and Qi systems. We show that all these flows admit bi-Hamiltonian structures depending on the values of their parameters.

Superintegrable cases of four-dimensional dynamical systems

Degenerate tri-Hamiltonian structures of the Shivamoggi and generalized Raychaudhuri equations are exhibited. For certain specific values of the parameters, it is shown that hyperchaotic Lü and Qi systems are superintegrable and admit tri-Hamiltonian structures.

Hamiltonian dynamics on matched pairs

It is shown that the cotangent bundle of a matched pair Lie group is itself a matched pair Lie group. The trivialization of the cotangent bundle of a matched pair Lie group are presented. On the trivialized space, the canonical symplectic two-form and canonical Poisson bracket are explicitly written. Various symplectic and Poisson reductions are perfomed. The Lie-Poisson bracket is derived. As an example, Lie-Poisson equations on

A Hamilton–Jacobi theory for implicit differential systems

\mathfrak{sl}(2,\mathbb{C})^\ast

On time-dependent Hamiltonian realizations of planar and nonplanar systems

are obtained.

On integrals, Hamiltonian and metriplectic formulations of polynomial systems in 3D

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of

Lagrangian dynamics on matched pairs

TT^*Q

On Geometry of Schmidt Legendre Transformation

generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton--Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.

A hierarchy of Poisson brackets

Abstract In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems in 2 D . We generalize the cosymplectic structures to time-dependent Nambu–Poisson Hamiltonian systems and corresponding Jacobi’s last multiplier for 3 D systems. We illustrate our constructions with various examples.