Related papers: The classification of nonsingular multidimensional…
A new four-dimensional family of skew-symmetric solutions of the Jacobi equations for Poisson structures is characterized. As a consequence, previously known types of Poisson structures found in a diversity of physical situations appear to…
We consider some issues of the representation in the Hamiltonian form of two problems of nonholonomic mechanics, namely, the Chaplygin's ball problem and the Veselova problem. We show that these systems can be written as generalized…
The gyrokinetic Vlasov-Maxwell equations are cast as an infinite-dimensional Hamiltonian system. The gyrokinetic Poisson bracket is remarkably simple and similar to the Morrison-Marsden-Weinstein bracket for the Vlasov-Maxwell equations. By…
We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which $G$ acts, then the…
A linear degenerate odd Poisson bracket (antibracket) realized solely on Grassmann variables is presented. It is revealed that this bracket has at once three nilpotent $\Delta$-like differential operators of the first, the second and the…
This paper develops the theory of affine Euler-Poincar\'e and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids,…
A method to construct Hamiltonian theories for systems of both ordinary and partial differential equations is presented. The knowledge of a Lagrangian is not at all necessary to achieve the result. The only ingredients required for the…
We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are…
The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known variational formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian formulation that involves a Poisson bracket…
We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient…
The interest in higher derivatives field theories has its origin mainly in their influence concerning the renormalization properties of physical models and to remove ultraviolet divergences. The noncommutative Podolsky theory is a…
A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the…
We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing…
This is a survey on the geometric classification of different varieties of algebras (nilpotent, nil-, associative, commutative associative, cyclic associative, Jordan, Kokoris, standard, noncommutative Jordan, commutative power-associative,…
A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the…
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables…
All real three dimensional Poisson-Lie groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie…
We construct a new local Poisson bracket compatible with the second unconstrained Adler-Gelfand-Dickey bracket. The resulting bihamiltonian structure admits a dispersionless limit and the leading term defines a logarithmic…
We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative…
Developing ideas based on combinatorial formulas for characteristic classes we introduce the algebra modeling secondary characteristic classes associated to $N$ connections. Certain elements of the algebra correspond to the ordinary and…