Bruno Cordani

The Arnold web and the Arnold diffusion arise when an integrable Hamiltonian system is slightly perturbed: the first concerns the peculiar topology characterizing the set of the resonance lines in phase space, the latter the extremaly slow motion (if any) along these lines. While Arnold has proved the possibility of diffusion, it is still unknown if the phenomenon is generic in realistic physical systems. The system we consider is the Hydrogen atom (or Kepler problem) subject to the combined action of a constant electric and magnetic field, which is known as Stark–Zeeman problem. We describe the results of numerical experiments: the Arnold web is clearly highlighted and, looking at the behaviour of the KAM frequencies on orbits of 10⁸ revolutions, evidence for the diffusion existence is reached.

The aim of this paper is twofold. First, we want to find angle-action variables suitable for the study of a generic perturbed Kepler problem: indeed, the unperturbed problem is degenerate, since its Hamiltonian depends on only one action variable (instead of three), and only a circle (instead of a three-dimensional torus) is intrinsically defined. Fortunately, the manifold of the orbits is compact, so the perturbed averaged system has always elliptic equilibrium points: nearby these points the reduced system behaves like a two-dimensional harmonic oscillator, which bears naturally the variables we seek. Second, we will apply the method of Numerical Frequencies Analysis in order to detect the transition from order to chaos. Four numerical examples are examined, by means of the free programs KEPLER and NAFF.

Following the same central idea of Féjoz [9] [10] [8], we study the planar averaged $3$-body problem without making use of series developments, as is usual, but instead we perform a global geometric analysis: the space of the orbits for a fixed energy is reduced under the rotational symmetry to a $2$-dimensional symplectic manifold, where the motion is described by the level curves of the reduced Hamiltonian. The number and location of the critical points are investigated both analytically and numerically, confirming a conjecture of Féjoz.