On the Lagrange's equilateral homographic flat motion of three bodies interacting with the Newton gravitational force

Abstract

We present the equilateral three-body motion with three different masses according to Newton gravitational force, which was discovered by Lagrange, tracing conic trajectories. We will extend to several bodies the generalization of the equilateral triangle solution discovered by Lagrange. The flat n-body problem of several different masses can be solved in closed and elementary form if we assume that the polygon formed by several celestial bodies always remains similar to itself. The Lagrange proof was simplified by C. Caratheodory and we extend ´ without problem this proof to several bodies. The bodies move on a fixed plane with two independent coordinates: one rotation around the center of mass, and one radial expansion. At any time the position vector of each body is the same multiple of the acceleration vector of the body. Bodies move tracing similar conics with the pole of each conic at the center of mass. For the three-body Lagrange’s case, a rigid triangle function of the masses discovered by Simo, is described with very simple geometry. Which we should place in a particular position ´ imposed by the Lagrange’s solution. We present a set of mathematical properties which are not well known.