Renato Calleja (IIMAS-UNAM)

Title: Symmetries and choreographies in the N-body problem

Abstract:

N-body choreographies are periodic solutions to the N-body equations in which N equal masses chase each other around a fixed closed curve. In my talk I will describe numerical and rigorous continuation and bifurcation techniques in a boundary value setting used to follow Lyapunov families of periodic orbits. These arise from the polygonal system of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies. I will present a sample of the many choreographies that we have determined numerically along the Lyapunov families and bifurcating families. I will also talk about the computer assisted proofs that validate some of theses choreographies. This is joint work with Eusebius Doedel, Carlos García Azpeitia, Jason Mireles-James and Jean-Philippe Lessard.