“A Variational Approach to Frozen Planet Orbit”
Martedì 12 Marzo 2024, ore 17:00 - Aula 1BC45 - Stefano Baranzini (Postdoc Università degli studi di Torino)
Abstract
In this talk I will report on an ongoing project, jointly with G. Canneori and S. Terracini. We study a one dimensional model for the Helium atom in which the two electrons and the nucleus are collinear and subject to electric attraction/repulsion. The nucleus is fixed at the origin and the system is governed by a system of two non-linear singular differential equations.
Using a mountain pass type argument and a suitable smoothing of the system, we show the existence of a particular family of periodic solutions called frozen planet orbits for all negative values of the energy. For energy close to zero, one electron keeps collapsing into the nucleus whereas the outer one oscillates slowly and far from the other particles.
In the second part of the talk we investigate the case in which the repulsive force disappears and explore the connection with billiards dynamics. The ODE system decouples in two Kepler-like 1-dimensional equations and solutions converge to a period 2 billiard trajectory.
