Hamiltonian Dynamics of a Particle Interacting with a Wave Field

We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.     

Revealing the Escape Dynamics in a Hamiltonian System with Five Exits

The scope of this work is to reveal, by means of numerical methods, the escape process in a Hamiltonian system with five exits which describes the problem of rearrangement multichannel scattering. For determining the influence of the energy on the nature of the orbits we classify starting conditions of orbits in planes of two dimensions. All the complex basins of escape, associated with the five escape channels of the system, are illustrated by using color-coded diagrams. The distribution of time of the escape is correlated with the corresponding escape basins. The uncertainty (fractal) dimension along with the (boundary) basin entropy are computed for quantifying the degree of fractality of the dynamical system.     

Hamiltonian type Lie bialgebras

We first prove that,for any generalized Hamiltonian type Lie algebra H,the first co- homology group H~1(H,H(?)H) is trivial.We then show that all Lie bialgebra structures on H are triangular.     

Effective Computation of the Dynamics Around a Two-Dimensional Torus of a Hamiltonian System

The purpose of this paper is to give an explicit analysis of the nonlinear dynamics around a two-dimensional invariant torus of an analytic Hamiltonian system. The study is based on high-order normal form techniques and the computation of an approximated first integral around the torus. One of the main novel aspects of the current work is the implementation of the symplectic reducibility of the quasi-periodic time-dependent variational equations of the torus. We illustrate the techniques in a particular example that is a quasi-periodic perturbation of the well-known Restricted Three Body Problem. The results are useful for describing the neighborhood of the triangular points of the Sun-Jupiter system.     

Iterated Hamiltonian type systems and applications

We discuss, in arbitrary dimension, certain Hamiltonian type systems and prove existence, uniqueness and regularity properties, under the independence condition. We also investigate the critical case, define a class of generalized solutions and prove existence and basic properties. Relevant examples and counterexamples are also indicated. The applications concern representations of implicitly defined manifolds and their perturbations, motivated by differential systems involving unknown geometries.     

A model Hamiltonian of the BCS type for a Bose gas

Theoretical and Mathematical Physics -     

Conley type index applied to Hamiltonian inclusions

In the paper we develop the theory of a cohomological index of the Conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional Hilbert space. An application to the existence of periodic orbits to asymptotically linear Hamiltonian inclusions is presented.     

Real Hamiltonian forms of affine Toda field theories: Spectral aspects

Theoretical and Mathematical Physics - The paper is devoted to real Hamiltonian forms of $$2$$ -dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral...     

A Gutzwiller type formula for a reduced Hamiltonian within the framework of symmetry

For a classical Hamiltonian with a finite group of symmetries, we give semi-classical asymptotics in a neighbourhood of an energy E of a regularized spectral density of the quantum Hamiltonian restricted to symmetry subspaces of Peter–Weyl defined by irreducible characters of the group. If we suppose that the energy level ${\Sigma }_E$ is compact, non-critical, and that its periodic orbits are non-degenerate, we get a Gutzwiller type formula for the reduced Hamiltonian, whose oscillating part involves the symmetry properties of closed trajectories of ${\Sigma }_E$. To cite this article: R. Cassanas, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type

Let F be an algebraically closed field of prime characteristic p > 2,and g be a simple Lie superalgebra of special type or Hamiltonian type over F.We construct the simple g-modules with non-singular characters of height more than one,and some simple modules with singular characters of height more than five.Furthermore,for the case of special type Lie superalgebras,we also construct a class of simple modules with regular semisimple characters of height one.All those simple modules mentioned above are proved to be reduced Kac modules.     

Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems

In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative problems, thus providing the class of Line Integral Methods (LIMs). In this paper we derive a further extension, which we name Enhanced Line Integral Methods (ELIMs), more tailored for Hamiltonian problems, allowing for the conservation of multiple invariants of the continuous dynamical system. The analysis of the methods is fully carried out and some numerical tests are reported, in order to confirm the theoretical achievements.