# Mathematical Sciences Research Institute

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# Seminar

Celestial Mechanics: Global instability in the elliptic restricted three body problem November 01, 2018 (02:00 PM PDT - 03:00 PM PDT)
Parent Program: Hamiltonian systems, from topology to applications through analysis MSRI: Simons Auditorium
Speaker(s) Amadeu Delshams (Polytechnical University of Cataluña (Barcelona))
Description No Description
Video
The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold. In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity, which turns out to be topologically equivalent to a normally hyperbolic invariant manifold (TNHIM). On this TNHIM, it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e., orbits parabolic both in the future and the past. A non-canonical symplectic structure still persists close this TNHIM and extends naturally to a $b^3$-symplectic structure. Such singular structures appear also in other problems of Celestial Mechanics. Since the inner dynamics inside the TNHIM is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the ERTBP whose angular momentum varies in any predetermined fashion.