Torus knot choreographies in the n-body problem. (14th January 2021)

• Record Type: Journal Article
• Authors: Calleja, Renato; García-Azpeitia, Carlos; Lessard, Jean-Philippe; James, J D Mireles
• Is Part Of: Nonlinearity. Volume 34:Number 1(2021)
• Page Start: 313
• Page End: 348
• Publication Date: 2021-01-14
• Subjects: celestial mechanics -- choreographies -- delay differential equations -- computer-assisted proofs -- contraction mapping; 70F10 -- 70F15 -- 34K13 -- 37C27 -- 37J45 -- 65G20 -- 47H10; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abcb08
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We develop a systematic approach for proving the existence of choreographic solutions in the gravitational n body problem. Our main focus is on spatial torus knots: that is, periodic motions where the positions of all n bodies follow a single closed which winds around a two-torus in R 3 . 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 constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial choreographies for n = 4, 5, 7, and 9 bodies.