Related papers: Poisson structures of some heavenly type dynamical…
This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants…
Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost…
The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian…
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…
We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint,…
We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of…
We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and…
Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible structure on $A$ that induces standard…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
This paper showed that Poisson brackets in quaternion variables can be obtained directly from canonical Poisson brackets on cotangent bundle of $SE(3)$ (or $SO(3)$) endowed by canonical symplectic geometry. Quaternion parameters in our case…
The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of…
We investigate conformal relative equilibria for Hamiltonian systems on exact Poisson manifolds equipped with scaling symmetries. By introducing conformally Poisson actions and conformal momentum maps, we characterize these equilibria…
The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are…
We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is that quadratic…
The concept of Lagrange structure allows one to systematically quantize the Lagrangian and non-Lagrangian dynamics within the path-integral approach. In this paper, I show that any Lagrange structure gives rise to a covariant Poisson…
We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and…
The Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a guiding-center Vlasov-Maxwell bracket. The bracket, which is shown to satisfy the Jacobi identity exactly, is…
We develop here a simple quantisation formalism that make use of Lie algebra properties of the Poisson bracket. When the brackets $\{H,\phi_i\}$ and $\{\phi_i,\phi_j\}$, where $H$ is the Hamiltonian and $\phi_i$ are primary and secondary…
The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real…
A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the subalgebras provides a complete set of integrals in involution with respect to the Sklyanin bracket. Using different representations of the…