Some new concepts in then-body and 3-body problems

Abstract

Then-body problem in 3-space for point masses ofarbitrary magnitude is approached by introduction of a weightedharmonic mean separation and anrms velocity of the particles. It is shown how these averages may be expressed in terms of asingle parameter for each of the cases of positive and negative total energy. The general problems of escape and collision are classified by the introduction ofescape, rest andcollision polynomials. For systems with a non-null total angular momentum it is shown how anrms angular momentum may be constructed and used with aharmonic mean centroidal moment of inertia.

In the 3-body problem agraphical construction of the equipotentials equivalent to a numerical algorithm is given. Finally the possibility of referencing 3-body motions to theapex (point of least average separation) is discussed. In the apex frame the particles move radially along 3 equiangular rigid spokes which are rotating about the apex. Although this simple description is lost if any 2 bodies subtend a view angle 120° or more from the third body it is to be expected that this will never occur in some motions, e.g. the Lagrangian triangle case.