The "Third" Integral in the Restricted Three-Body Problem Revisited
In 1964 M. Hénon and, independently, V. Szebehely with G. Bozis presented the first numerical results, indicating the existence of a “new” local integral of motion in the circular restricted three-body problem. The first terms of an asymptotic expansion of this integral were later calculated by Contopoulos . Several years later, the Celestial Mechanics astronomical community started to develop a very successful theory on local integrals of motion in the restricted three-body problem, which however in the jargon of this field are called proper elements and are related to known analytical approximate solutions. The calculation of proper elements is based on the implicit assumption that the orbit traced by a planet (major or minor) is nearly-regular. Here we show that this method is also applicable, albeit partly, in a special case of chaotic motion in the Solar System, known as “stable chaos”. Thus, the existence of an additional local integral of motion in the elliptic restricted three-body problem is responsible for the phenomenon of stable chaos.
KeywordsChaotic Motion Periodic Trajectory Local Integral Chaotic Trajectory Orbital Resonance
Unable to display preview. Download preview PDF.