  We present a novel numerical method to calculate periodic orbits for
dynamical systems by an iterative process which is based directly on the action
integral in classical mechanics. New solutions are obtained for the planar
motion of three equal mass particles on a common periodic orbit with finite
total angular momentum, under the action of attractive pairwise forces of the
form $1/r^{p+1}$. It is shown that for $-2 < p \le 0$, Lagrange's 1772 circular
solution is the limiting case of a complex symmetric orbit. The evolution of
this orbit and another recently discovered one in the shape of a figure eight
is investigated for a range of angular momenta. Extensions to n equal mass
particles and to three particles of different masses are also discussed
briefly.
