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### Session S03 - Geometric and Variational Methods in Celestial Mechanics

Monday, July 12, 12:30 ~ 13:10 UTC-3

## Celestial Mechanics tools for studying the hydrogen atom

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian depending on one parameter $K>0$, which is a perturbation of the standard Kepler problem. The associated Hamiltonian system has two equilibria: $L_1$ (center-saddle for all $K$) and $L_2$ (center-center for small $K$ and complex-saddle otherwise). Associated to $L_1$ there is a family of Lyapunov periodic orbits that form a normally hyperbolic invariant manifold (NHIM). In this talk, we compute the primary transversal homoclinic orbits to the NHIM (and therefore the associated scattering maps) combining Poincaré-Melnikov methods with numerical methods. It should be noted that the transversality of these homoclinic orbits is exponentially small in $K$ (in analogy with the libration point $L_3$ of the R3BP).