Families of Symmetric Exchange Orbits in the Planar $$(1+2n)$$ ( 1 + 2 n ) -Body Problem

We study a particular $$(1+2n)$$\n -body problem, conformed by a massive body and 2n equal small masses, since this problem is related with Maxwell’s ring solutions, we call planet to the massive body, and satellites to the other 2n masses. Our goal is to obtain doubly-symmetric orbits in this problem. By means of studying the reversing symmetries of the equations of motion, we reduce the set of possible initial conditions that leads to such orbits, and compute the 1-parameter families of time-reversible invariant tori. The initial conditions of the orbits were determined as solutions of a boundary value problem with one free parameter, in this way we find analytically and explicitly a new involution, until we know this is a new and innovative result. The numerical solutions of the boundary value problem were obtained using pseudo arclength continuation. For the numerical analysis we have used the value of $$3.5 \\times 10 ^{-4}$$\n as mass ratio of some satellite and the planet, and it was done for $$n=2,3,4,5,6$$\n . We show numerically that the succession of families that we have obtained approach the Maxwell solutions as n increases, and we establish a simple proof why this should happen in the configuration.