ABSTRACT
Certain “shades” always remain in the form of invariant solenoids carrying motions which can be described as quasi-periodic although not in the completely classical sense. Both periodic and quasi-periodic orbits can be derived from a certain variation principle which generalizes the one used by Birkhoff for the billiard ball problem. Both periodic and quasi-periodic orbits can be derived from a certain variation principle which generalizes the one used by Birkhoff for the billiard ball problem. The idea for the construction of a Birkhoff periodic orbit of type and then of the second compatible orbit is to consider the space of possible positions for angular coordinates for the desired orbit and then to introduce a certain functional whose critical points correspond to real orbits.
