# GDMSeminar: Abstracts of the Talks (2021/2)

**Renato Calleja **(UNAM, Mexico City)

*Choreographies in the n-body problem and a conjecture of Marchal *

Tuesday, March 16, 2021, 16:00 GMT

**Abstract**: N-body choreographies are periodic solutions to the N-body equations in which equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for continuing and proving the existence of choreographies in the gravitational body problem with the help of the digital computer. These arise from the polygonal system of bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. For odd numbers of bodies between n = 3 and n = 15 we find numerically that the figure eight choreography can be reached starting from the regular n-gon. Based on these calculations we extend a conjecture claiming the n-gon and the eight are in the same continuation class for all odd numbers of bodies. This is joint work with Carlos García-Azpeitia, Jason Mireles James, Jean-Philippe Lessard and is a continuation of work with Eusebius Doedel and C. García-Azpeitia.