Other
Scientific paper
Dec 1998
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998dps....30.5522c&link_type=abstract
American Astronomical Society, DPS meeting #30, #55.P22; Bulletin of the American Astronomical Society, Vol. 30, p.1451
Other
Scientific paper
The Jacoby or energy integral of the equations of motion for the restricted three body problem is the only isolating integral now available for them in terms of elementary functions. The complexity of this problem makes the discovery of additional isolating integrals using elementary functions unlikely. However, it is possible to find two new time independent integrals for the motion in the orbit plane P of the large bodies, expressible in terms of elementary functions, if the motion equations are linearized about L_4 and L_5. The first new integral i_Q, is a quadratic in the kinematic variables as is i_J, the Jacoby integral specialized for the above context, and together they are a complete set of isolating integrals for motion in plane P. The second new integral is not a quadratic nor is it isolating. When 3 dimensional motion is considered in this linearized context, there is another known quadratic isolating integral i_Z which expresses conservation of energy for motion perpendicular to plane P. A general statistical treatment of the linearized motion near L_4 and L_5 in the restricted three body problem either in plane P or in 3 dimensions is now possible since the methods of stellar dynamics guarantee that the general distribution function need only involve an independent set isolating integrals. It is also shown for motion in plane P that there are 2 independent linear combinations of the quadratic integrals which are positive definite as is i_Z. This property is necessary if a most probable distribution in plane P or in the 3 dimensional case is to have a cutoff. An initial analysis shows that this linearized theory correctly predicts the orientation of the major and minor axes of the Trojan groups but exaggerates their ratio.
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