The parametric orbits and the form invariance of three-body in one-dimension

In this paper, the differential equations of motion of a three-body interacting pairwise by inverse cubic forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') are expressed in Ermakov formalism in two-dimension polar coordinates, and the Ermakov invariant is obtained. By rescaling of the time variable and the space coordinates, the parametric orbits of the three bodies are expressed in terms of relative energy H₁ and Ermakov invariant. The form invariance of the transformations of two conserved quantities are also studied.