Periodic Orbits of the N-Body Problem

We give a fast, accurate, and highly convergent algorithm for computing periodic orbits of the N-body problem. The Lindstedt-Poincaré technique from perturbation theory, Fourier interpolation, the dogleg strategy devised by Powell for trust region methods in unconstrained optimization, proper handling of the symmetries of the Hamiltonian, and a simple mechanism to eliminate high frequency errors are the main elements of the algorithm. Some of the examples show periodic orbits with a heavy primary and lighter masses circling around it in imitation of the motion of the planets around the sun. Most of our computations are as accurate as exact formulas would be if such formulas were to be available. Some of the periodic orbits are linearly stable. We give a brief discussion of the possibility of finding stellar bodies trapped in such periodic orbits.