Invariant 4-tori in the co-orbital motion of three bodies

Speaker: Jesús Palacián (Universidad Pública de Navarra)

Abstract: We provide a qualitative explanation of the co-orbital motion of two small moons orbiting a planet. The two small bodies revolve about the planet in nearly circular orbits with almost equal radii. The system is modelled as a planar three-body problem whose Hamiltonian is expanded as a perturbation of two uncoupled Kepler problems. A combination of averaging, normal form, symplectic scaling and Hamiltonian reduction theories and the application of a KAM theorem for high-order degenerate systems allows us to establish the existence of quasi-periodic motions and KAM 4-tori related to the co-orbital motion of the moons. By conveniently selecting a suitable region of the reduced phase space (which is the Cartesian product of a two-dimensional sphere and one sheet of a two-sheet hyperboloid of revolution), we are able to establish the existence of these quasi-periodic motions that are valid for any value of an action variable, related to the angular momenta of the two moons. This is a joint work with Josep M. Cors and Patricia Yanguas.