What is the shortest/simplest/most elegant proof that the orbital period of the 2-body problem depends only on the length of the major axis, and NOT on the eccentricity of the ellipse? I'm looking for a Feynmann style proof; perhaps Feynmann himself came up with an elegant proof? I've read "Feynmann's lost lecture", in which Feynmann reproduces Maxwell's elegant proof that Kepler's Laws are equivalent to the inverse square law and elliptical orbits, but I don't recall that Feynmann addressed this particular question by itself. In any case, this fact means that one can have simultaneous bodies orbiting with the same period in the same plane, which should be able to produce some pretty animations. For example, one might color a circular orbit with the colors from a color wheel, and then deform that circle into an ellipse of the same major axis.