Dynamics of generalized Poisson and Nambu–Poisson brackets

Solar System Research, 2001

The formulas for the Poisson bracket of a perturbed two-body problem and a perturbed planetary problem are found in different systems of Keplerian elements. As with canonical parametrization, the Poisson bracket is equal to a linear combination of partial brackets, but it contains coefficients depending on semimajor axis, eccentricity, and inclination. A simple relation between the Poisson brackets and matrices of coefficients of Lagrange-type equations determining the variations of osculating elements is derived. The Poisson bracket of D'Alembertian functions is proved to be a D'Alembertian one by itself.

Journal of Mathematical Physics, 2007

This paper presents a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. It is shown that one can define an almost aff-Poisson bracket on the constraint AV-bundle, which plays a prominent role in the description of nonholonomic dynamics. Moreover, these developments give a general description of nonholonomic systems and the unified treatment permits to study nonholonomic systems after or before reduction in the same framework. Also, it is not necessary to distinguish between linear or affine constraints and the methods are valid for explicitly time-dependent systems.

Mathematical proceedings of the Cambridge Philosophical Society, 1977

Starting from an arbitrary algebra in place of the algebra of smooth functions on a differentiable manifold M, we construct an algebra which generalizes the algebra of polynomial functions on T*M. We define a Poisson bracket on this algebra and show that it can be obtained from a natural one-form which generalizes the fundamental one-form on T*M.

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.

Journal of Mathematical Physics, 2008

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk . The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

Annals of Global Analysis and Geometry, 1994

We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frblicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.

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.

Russian Journal of Mathematical Physics, 2019

In the context of normal forms, we study a class of perturbed Hamiltonian systems on phase spaces with symmetry which arise from deformation of Poisson brackets. By combining the averaging method with some facts on the Poisson invariant cohomology, we derive various normalization criteria. In particular, we compute the invariant normal forms of first order for systems of adiabatic type on Poisson fibrations by using the technique of Hannay-Berry connections.

2002

Regular and Chaotic Dynamics, 2012

We discuss the nonholonomic Chaplygin and the Borisov-Mamaev-Fedorov systems, for which symplectic forms are different deformations of the square root from the corresponding invariant volume form. In both cases second Poisson bivectors are determined by Ltensors with non-zero torsion on the configurational space, in contrast with the well known Eisenhart-Benenti and Turiel constructions.