Stability and eccentricity of periodic orbit for two planets in a 1:1   resonance
  The nonlinear stability domain of Lagrange's celebrated 1772 solution of a three-body problem is obtained numerically as a function of the masses of the bodies and the common eccentricity of their Keplerian orbits. This domain shows that this solution may be realized in extra-solar planetary systems similar to those that have been discovered recently with two Jupiter-size planets orbiting a solar-size star. For an exact 1:1 resonance, the Doppler shift variation in the emitted light would be the same as for stars which have only a single planetary companion. But it is more likely that in actual extra-solar planetary systems there are deviations from such a resonance, raising the interesting prospect that Lagrange's solution can be identified by an analysis of the observations. The existence of another stable 1:1 resonance solution which would have a more unambiguous Doppler shift signature is also discussed. 
Michael Nauenberg