** Authors: **

Renato Calleja, Carlos Garcia-Azpeitia, J.P. Lessard and J.D. Mireles James

** Abstract:** We develop a systematic approach for proving the existence of spatial choreogra- phies in the gravitational n body problem. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is construc- tive and makes extensive use of the digital computer. We provide all the necessary analytic estimates, and give an implementation which works for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial torus knot choreographies for n = 4, 5, 7, and 9 bodies.

(submitted).