N body spatial Parabolic orbits asymptotic to collinear central configurations

Abstract

An orbit for the Newtonian n body problem is called Parabolic when all the mutual distances between particles approach infinity while the velocities tend to zero as time t goes to infinity. The asymptotic growth rate for these mutual distances turns out to be constant multiples of t raised to the 2/3 power (or to the power 2/(p + 1) for the inverse p central force law, 1 < p < 3) [ 111. Indeed, when the position vectors are scaled by dividing by t213, or by the square root of the moment of inertia, then the scaled position vectors tend to the set of central configurations. (A central configuration is formed whenever for all the particles a fixed constant multiple of the mass times the position vector is equal to the corresponding force vector.) It turns out that central configurations can be identified with the critical point set of a certain function. Then, the configurations are classified as being either degenerate or nondegenerate to correspond with the classification of the critical point of this function. Should the central configuration be nondegenerate, as is known to be the case for collinear central configurations, the equilateral triangle configuration for the three body problem, and the equilateral tetrahedron configuration for the four body problem, then the limiting configuration of a given motion must be unique. (For a discussion of central configurations, see [ 15, Chap. 5; 8; 121). This uniqueness follows by combining two facts. First, the nondegeneracy implies that the orbit of the central configuration by the W(3) action must be isolated in the set of all central configurations. (The SO(3) action on configuration space is rotation about the center of mass of the system.) Secondly, there can be no rotation of the limiting configuration in the sense of motion along the SO(3) action on configuration space. (See [ 71 for the planar three body problem and [ 13 ] for the general case. The lack of