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A variational approach to frozen planet orbits

Stefano Baranzini
Università degli Studi di Torino
Thursday, March 21, 2024 - 14:00 to 16:00

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 of the type:

\[{}\ddot {q}_1 = f'(q_1) + g′(q_2 − q_1),\\
{}\ddot{q}_2 = f ′(q_2) − g′(q_2 − q_1).\]

Here $f ′$ and $g′ $represent the electric attractive/repulsive forces and have a Newtonian like singularities at the origin.

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.

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