Characteristics of Invariant Curves of Plane Orbits
Abstract
The general characteristics of the totality of orbits in a twodimensional potential, for a fixed value of energy, can be found by studying the invariant curves on a surface of section. The rotation number on each invariant curve is found as a function of its distance from the central invariant point, representing a stable periodic orbit. The tube orbits are represented by invariant curves of special forms which are called islands. There are no islands or tube orbits in separable potentials, while in the general case of non integrable systems it seems that there are infinite sets of islands and tube orbits. The rotation curve (rotation number versus distance from the center) can be found approximately by means of the third integral. The rotation number near a stable symmetric periodic orbit is equal to +aT/2ir, where +ja are the two nonzero characteristic exponents and T the period of the periodic orbit. The stability of the central periodic orbit (near the y axis) and the orbit y =0 was studied. It was found that in some cases the orbits remain stable even when the zero velocity curves are open. Some applications to galactic problems are mentioned.
 Publication:

The Astronomical Journal
 Pub Date:
 March 1968
 DOI:
 10.1086/110603
 Bibcode:
 1968AJ.....73...86C