| Given a system of differential equations which admits a continuous group of symmetries and possesses a periodic solution, we show that under certain nondegeneracy assumptions there always exists a continuous family containing infinitely many periodic and quasi-periodic trajectories. This generalizes the continuation method of Poincare to orbits which are not necessarily periodic. We apply these results in the setting of the Lagrangian N-body problem of homogeneous potential to characterize an infinite family of rotating nonplanar "hip-hop" orbits in the four-body problem of equal masses, and show how some other trajectories in the N-body theory may be extended to infinite families of periodic and quasi-periodic trajectories. |
