In this talk, as a continuation of my talk in the Members' Colloquium but with a specialized audience in mind, I will discuss in more detail some of the general geometric and dynamical structures underlying the theoretical aspects of the restricted 3-body problem, and outline new research directions.
Despite the fact that the 3-body problem is an ancient conundrum that goes back to Newton, it is remarkably poorly understood, and is still a benchmark for modern developments. In this talk, I will give a (very) biased account of this classical problem, both from a modern theoretical perspective, i.e. outlining possible lines of attack coming from symplectic geometry, holomorphic curves and Floer theory; as well as comment on practical and numerical aspects, within the context of finding orbits for space mission design and ocean worlds exploration.
In his search for closed orbits in the planar restricted three-body problem, Poincaré’s approach roughly reduces to:
1. Finding a global surface of section;
2. Proving a fixed-point theorem for the resulting return map.
This is the setting for the celebrated Poincaré-Birkhoff theorem. In this talk, I will discuss a generalization of this programme to the spatial problem.
For the first step, we obtain the existence of global hypersurfaces of section for which the return maps are Hamiltonian, valid for energies below the first critical value and all mass ratios. For the second, we prove a higher-dimensional version of the Poincaré-Birkhoff theorem, which gives infinitely many orbits of arbitrary large period, provided a suitable twist condition is satisfied. Time permitting, we also discuss a construction that associates a Reeb dynamics on a moduli space of holomorphic curves (a copy of the three-sphere), to the given dynamics, and its properties.
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