n-gon Equilibria of the Discrete -body Problem

Abstract We prove that the discrete-time general <?CDATA $(1+n)$?> -body problem (d-G <?CDATA $(1+n)$?> BP) proposed by Minesaki can exactly trace the orbits of elliptic relative equilibrium solutions in the original general <?CDATA $(1+n)$?> -body problem (G <?CDATA $(1+n)$?> BP). These orbits include the orbits of relative equilibrium solutions that have already been discovered. Before this proof, no discrete-time system had been shown to retain the orbits of elliptic relative equilibrium solutions in <?CDATA ${\rm{G}}(1+n)$?> BP. d-G <?CDATA $(1+n)$?> BP can also precisely reproduce doubly symmetric orbits of the general <?CDATA $(1+4)$?> -body problem, each of which passes near a square equilibrium solution over a long time interval.