Quadratic Hamiltonians on non-symmetric Poisson structures (original) (raw)

Poisson structures for two nonholonomic systems with partially reduced symmetries

JGM, 2014

We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.

On some quadratic Hamilton-Poisson systems

We identify a class of quadratic Hamilton-Poisson systems on the three-dimensional Euclidean Lie-Poisson space. Specifically, we consider systems that are both homogeneous and for which the underlying quadratic form is positive semidefinite. Any such system is shown to be equivalent to one of four normal forms (of which two are parametrized families of systems). For the cases with non-trivial dynamics, the stability nature of the equilibrium states is fully investigated. Furthermore, we find explicit expressions for the integral curves (in terms of Jacobi elliptic functions).

DYNAMICAL ASPECTS OF LIE-POISSON STRUCTURES

Modern Physics Letters A, 1993

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 4-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 the 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.

Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Dynamical systems and Poisson structures

Journal of Mathematical Physics, 2009

We first consider the Hamiltonian formulation of n = 3 systems in general and show that all dynamical systems in R 3 are bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. We find the Poisson structures of a dynamical system recently given by Bender et al. Secondly, we show that all dynamical systems in R n are (n − 1)-Hamiltonian. We give also an algorithm, similar to the case in R 3 , to construct a rank two Poisson structure of dynamical systems in R n . We give a classification of the dynamical systems with respect to the invariant functions of the vector field X and show that all autonomous dynamical systems in R n are super-integrable. 0

Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics

The Journal of Geometric Mechanics, 2010

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.

Comments on the symmetry structure of bi-Hamiltonian systems

Czechoslovak Journal of Physics, 1982

The algebraic structure of exactly solvable equations is reviewed and results are reported which 1) establish that isospectral eigenvalue problems yield hereditary symmetries for bi-Hamiltonian equations and 2) show that if both an equation and its "m6dified" equation have known Hamiltonian formulatiens then their hereditary symmetries and bi-Hamiltonian formutations are readily obtained via their Miura transformation.

Symmetries of Hamiltonian systems with two degrees of freedom

1999

We classify the Lie point symmetry groups for an autonomous Hamiltonian system with two degrees of freedom. With the exception of the harmonic oscillator or a free particle where the dimension is 15, we obtain all dimensions between 1 and 7. For each system in the classification we examine integrability.© 1999 American Institute of Physics.