On the problem op n-stationary centers

Abstract

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes. If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the ( n+ l)-th body). In addition to the problem of central motion ( n = 1), soluble dynamics problems of this category include Euler's case [1] of two ( n= 2) stationary Newtonian attracting centers. This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4]. The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5]. We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed. Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].