Gian Marco Canneori

Universit`a degli studi di Torino, Italy 

Abstract: 

Mechanical systems driven by anisotropic potentials have gradually become an object of interest in the theory of dynamical systems, initially for their applications in atom theory of crystal semicon- ductors, but also because they arise after particular reductions by symmetries of the classical N-body problem. The archetype model is the anisotropic Kepler problem, introduced by Gutzwiller in the 70s and further studied by Devaney in a series of remarkable papers (see [G, D1, D2, D3]). The peculiarity of such potentials resides in the presence of a non-constant angular component, which results in a complete loss of rotational invariance. As a consequence, the orbits structure is completely different from the case of the classical Kepler problem, the integrability is destroyed by the anisotropy and regularization methods for collision orbits inevitably fail. 

In this course we will deepen the study of collision orbits for the anisotropic Kepler problem based on the Gutzwiller definition (see [G]). In particular, we will introduce McGehee coordinates to extend the flow beyond the singularity and we will analyse the dynamical features of the new flow (see [D1]). An exhaustive description of collision orbits will then be given, in order to show that they cannot be regularized. In fact, we will see how a particular method of regularization introduced by Easton in [E] cannot be employed in this situation and actually prove that the anisotropic Kepler problem is not regularizable (see [D2]).

For the purposes of this course we will mainly follow the notes of Devaney [D3].

[D1] R.L. Devaney. Collision orbits in the anisotropic Kepler problem. Invent. Math., 45(3):221–251, 1978.

[D2] R.L. Devaney. Nonregularizability of the anisotropic Kepler problem. J. Differential Equations, 29(2):252–268, 1978.

[D3] R.L. Devaney. Singularities in classical mechanical systems. In Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), volume 10 of Progr. Math., pages 211–333. Birkhäuser, Boston, Mass., 1981.

[E] R. Easton. Regularization of vector fields by surgery. J. Differential Equations, 10:92–99, 1971.

[G] M.C. Gutzwiller. The anisotropic Kepler problem in two dimensions. J. Mathematical Phys., 14:139–152, 1973.