The use of sub-Riemannian geometry on the Heisenberg group H(n) provides a compact picture of symplectic geometry. Any Hamiltonian diffeomorphism on R lifts to a volume preserving bi-Lipschitz homeomorphisms of H(n), with the use of its generating function. Any curve of a flow of such homeomorphisms deviates from horizontality by the Hamiltonian of the flow. From the metric point of view this m...

We prove the following three results in Hamiltonian dynamics. • The Weinstein conjecture holds true for every displaceable hypersurface of contact type. • Every magnetic flow on a closed Riemannian manifold has contractible closed orbits for a dense set of small energies. • Every closed Lagrangian submanifold whose fundamental group injects and which admits a Riemannian metric without closed ge...

Aspects of noncanonical Hamiltonian field theory are reviewed. '·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular sys tem of thi s type is cons idered, namely reduced magnetohydrodynamics (RllHD) which was derived for tokamak modelling. The notion of a liePoisson bracket is reviewed; these are special P...

This paper aims at a historical and pedagogical presentation of some important contributions of the research on thermonuclear fusion by magnetic confinement to the study of Hamiltonian chaos. This chaos is defined with the help of Poincaré maps on a simple two-wave Hamiltonian system. A simple criterion for computing the transition to large scale chaos is introduced. A renormalization group app...

An N2 dimensional representation of the periodic Temperley-Lieb algebra TLL(x) is presented. It is also a representation of the cyclic group ZN . We choose x = 1 and define a Hamiltonian as a sum of the generators of the algebra acting in this representation. This Hamiltonian gives the time evolution operator of a stochastic process. In the finite-size scaling limit, the spectrum of the Hamilto...

Aspects of noncanonical Hamiltonian field theory are reviewed. f·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures. yet the equations are not in canonical form. A particular system of this type is considered. namely reduced magnetohydrodynamics (RflHD) which was derived for tokamak modelling. The notion of a LiePoisson bracket is reviewed; these are special Pois...

Aspects of noncanonical Hamiltonian field theory are reviewed. '·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular sys tem of thi s type is cons idered, namely reduced magnetohydrodynamics (RllHD) which was derived for tokamak modelling. The notion of a liePoisson bracket is reviewed; these are special P...

Symmetries play an important role in dynamical systems. They provide quantum numbers for the classification of states, determine selection rules and facilitate the calculation of matrix elements. An exact symmetry occurs when the Hamiltonian of the system commutes with all the generators (gi) of the symmetry-group, [H , gi ] = 0. In this case, all states have good symmetry and are labeled by th...

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a time...

The main result of this note is that every closed Hamiltonian S manifold is uniruled, i.e. it has a nonzero Gromov–Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of π1 of the Hamiltonian group in the small quantum homology of M as well as the blow up technique recently introduced by Hu, Li and Ruan. It applies more generally to manifolds that have...