On the existence of periodic solutions of Poincar�'s second sort in the general problem of three bodies moving in plane

Abstract

In his ‘Methodes Nouvelles de la Mecanique Celeste’ (Sections 42–47), Poincare envisaged the existence of a class (the ‘second sort’) of periodic solutions of the general problem of three bodies, arising by analytical continuation from unperturbed elliptic motion of two bodies about a primary, in which the two orbits are of commensurable period, and of non-zero eccentricity. Existence proofs have been given for the restricted problem case (in which the mass of the third body remains zero) by Arenstorf and Barrar, and for the lunar theory case by Arenstorf. A proof is offered here for the general problem, making use of the symmetry of the equations of motion (with use of the ‘Mirror Theorem’ of Roy and Ovenden) in an extension of the approach used by Barrar for the restricted problem, and using a device proposed by Poincare himself which enables the extension to the general problem to be made.