In our previous discussions regarding the restricted three-body problem, we derived the equations of motion in the plane: \ddot{R}+2i\omega \dot{R}=\omega^2 R-GM\frac{R-l_1}{|R-l_1|^3}-Gm\frac{R-l_2}{|R-l_2|^3} \tag{32} The energy integral is: \frac{1}{2}|\dot{R}|^2=\frac{1}{2} \omega^2 |R|^2+\frac{GM}{|R-l_1|}+\frac{Gm}{|R-l_2|}-C \tag{33} Based on these two equations, let us further explore some aspects of the restricted three-body problem...
(I) The Lagrange Point Problem
Lagrange originally derived the five Lagrange points while studying the circular restricted three-body problem. For the vast majority of differential equations, there exist several special solutions where the “function value = constant.” One simply needs to set all derivatives to zero and solve the resulting algebraic equation (provided a solution exists). In equation (32), setting \ddot{R} = 0 and \dot{R} = 0, we obtain: \omega^2 R-GM\frac{R-l_1}{|R-l_1|^3}-Gm\frac{R-l_2}{|R-l_2|^3}=0 Recalling that in the two-body problem, \omega^2=\frac{G(M+m)}{l^3}, we have: \frac{G(M+m)}{l^3} R-GM\frac{R-l_1}{|R-l_1|^3}-Gm\frac{R-l_2}{|R-l_2|^3}=0 \tag{34} The solutions to equation (34) are the five Lagrange points. I will not go into detail about the three collinear points, as they can be solved by considering different cases. As for L_4 and L_5, they are the solutions when |R-l_1|=|R-l_2|=l. This is a general solution, meaning it is independent of the specific values of M and m. I will not write out the specific derivation here; you can refer to https://kexue.fm/archives/874.
(II) The Third Cosmic Velocity Problem
The previous article mentioned that one of the application conditions for the restricted three-body model is short-range. Now, let’s look at the effect of long-range treatment—a “maximum range” problem, namely deriving the third cosmic velocity (escaping to infinity). In equation (33), R on the left side represents the trajectory relative to the rotating coordinate system (such as the Earth orbiting the Sun). \dot{R} represents the vector difference between the true velocity \dot{r} and the transverse coordinate velocity. Therefore, when the third body is near the Earth, \dot{R} can be regarded as the velocity relative to the Earth.
When flying to infinity, \dot{r}=0, but since the coordinate system itself is rotating, |\dot{R}|=\omega |R|, which leads to C=0. Thus: \frac{1}{2}|\dot{R}|^2=\frac{1}{2} \omega^2 |R|^2+\frac{GM}{ |R-l_1| }+\frac{Gm}{|R-l_2|} Applying this to the Sun-Earth system, where |l_1|\approx 0, |l_2|\approx l, |R|\approx l, and |R-l_2|=R_e (the radius of the Earth), substituting the data yields \dot{R}\approx 50 \text{ km/s}!! This is quite absurd—how did 16.7 get calculated as 50?
This clearly demonstrates the limitations of this model.
(III) The Copenhagen Problem (problem of Copenhagen)
Don’t immediately think of climate issues. Haha, there is an interesting matter regarding the restricted three-body problem: the study of a series of topics related to periodic solutions. The theory of periodic solutions established by Poincaré played a major role in solving difficult problems in the theory of asteroid motion and attracted significant attention. Strömgren and his colleagues at the Copenhagen Observatory did extensive work on the planar circular restricted three-body problem. They classified the periodic solutions that might appear near the five equilibrium points and the two finite masses P_1 and P_2, and studied prograde and retrograde periodic orbits as well as asymptotic orbits.
Based on their work, an international conference held at the Copenhagen Observatory in 1936 proposed a plan to study periodic solutions of the restricted three-body problem. The topic of study assumed that the two finite masses are equal and move in circular orbits around each other, while the third body has infinitesimal mass and moves in the same plane. The goal was to find three types of periodic solutions: (1) periodic orbits around one of the two finite masses; (2) periodic orbits around both finite masses simultaneously; (3) periodic orbits and asymptotic orbits around the Lagrange equilibrium points. Including these research topics on orbit evolution, they are collectively known as the Copenhagen Problem.
Since the 20th century, many mathematicians have used qualitative methods to study the N-body problem and have achieved many important results. For example, Wintner studied special solutions of the N-body problem and proved that under certain conditions, N particles can form a certain fixed shape (such as a line or a polyhedron), which only undergoes rotation and scaling during motion, while the shape remains unchanged. This type of special solution is called a central configuration, which is actually a generalization of the Lagrange special solutions in the three-body problem. Furthermore, the number of special solutions for N particles in relative equilibrium on the same straight line is \frac{1}{2}N! (Readers might want to try proving this?) This is consistent with the results of the three-body problem. Additionally, many people have generalized other results of the three-body problem, such as collision problems, regularization, and capture theory, to the N-body problem, obtaining similar conclusions.
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