Reconstructing Euler's work on collinear solutions of the 3-body problem and identifying their corresponding Lagrange points (Q6584210)

In this paper, a slightly different perspective on Euler's solution are presented, which are well-known central configurations characterizing all collinear solutions of the 3-body problem. The author, here, introduces an additional body with zero mass and classifying the resulting central configurations of the circular restricted 4-body problem. The 4th body with zero mass moves on a circle with the same center and the same angular velocity as that of the primaries, and they called as Euler-Lagrange solutions or Euler-Lagrange points. There are 6 possible positions for the body with zero mass, four of them collinear to the three massive bodies. In this paper, not only explicitly described, the possible values for the masses of the primaries, but also, shows their relation to the six possible positions of the body with zero mass. For each set of possible positions for the three primaries, the author shows a starting and ending point for each one of these six positions of the body with zero mass. The possible points between the starting and ending points depend on the masses of the primaries. The work, here, generalizes the work for the particular Euler solution with equal masses to all possible collinear solutions of the 3-body problem. Since, by previous authors, it has been showed that all six solutions are unstable, the author, here, expects that none of the solutions described in this paper are stable.