Brouwer’s satellite solution redux

Brouwer’s solution to the artificial satellite problem is revisited. We show that the complete Hamiltonian reduction is rather achieved in the plain Poincaré’s style, through a single canonical transformation, than using a sequence of partial reductions based on von Zeipel’s alternative for dealing with perturbed degenerate Hamiltonian systems. Beyond the theoretical interest of the new approach as regards the complete reduction of perturbed Keplerian motion, we also show that a solution based on a single set of corrections may yield computational benefits in the implementation of an analytic orbit propagator.

Topological obstacles to the realizability of integrable Hamiltonian systems by billiards

We introduce the following classes of integrable billiards: elementary billiards, topological, books, with potential, magnetic field, geodesic billiards. These classes are used to test the A.T. Fomenko conjecture about the realizability up to Liouville equivalence by billiards of integrable non-degenerate Hamiltonian systems with two degrees of freedom. In the class of book billiards found topological obstacles to realizability.

LOCALIZING TRANSPORT BARRIERS IN DEGENERATE 3/2 D.O.F. HAMILTONIAN SYSTEMS WITH APPLICATIONS TO MAGNETIC CONFINEMENT FUSION

The existence of transport barriers in 3/2 degrees of freedom degenerate Hamiltonian systems is studied using the associated stroboscopic maps. For small enough amplitudes of the perturbations, a transport barrier (formed by infinitely many invariant rotational circles) is proven to exist and to form in the degenerate annulus. These results are applied to Hamiltonian models which describe some magnetic configurations in tokamak plasmas.

Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.