The Three-Body Problem and its Classical Integration
Many astronomy enthusiasts have already encountered the "two-body problem" (the "Kepler’s Three Laws" we learned in high school are part of this). Since the masses of planets in the solar system are relatively small and their distances are relatively large, applying the solutions of the "two-body problem" for celestial calculations and predictions can satisfy certain approximation requirements. However, if higher precision is required, the gravitational influence of other planets cannot be ignored. This leads to the so-called N-Body Problem, which concerns the laws of motion of N point masses acting solely under their mutual gravitational attraction. The simplest case, the two-body problem, has been completely solved. In contrast, the three-body problem (or problems involving more bodies) is vastly different. Poincaré proved that the three-body problem cannot be solved analytically in the general case; furthermore, it is a chaotic system, where any tiny perturbation can lead to unpredictable effects.
According to Newtonian mechanics, choosing an inertial reference frame and letting the three point masses be M_1, M_2, M_3 with position vectors \vec{r}_1, \vec{r}_2, \vec{r}_3 respectively, we can list the equations of motion (where derivatives are taken with respect to time t):
M_1\ddot{\vec{r}}_1 = \frac{GM_1 M_2(\vec{r}_2-\vec{r}_1)}{|\vec{r}_2-\vec{r}_1|^3} + \frac{GM_1 M_3(\vec{r}_3-\vec{r}_1)}{|\vec{r}_3-\vec{r}_1|^3} \tag{23} M_2\ddot{\vec{r}}_2 = \frac{GM_1 M_2(\vec{r}_1-\vec{r}_2)}{|\vec{r}_1-\vec{r}_2|^3} + \frac{GM_2 M_3(\vec{r}_3-\vec{r}_2)}{|\vec{r}_3-\vec{r}_2|^3} \tag{24} M_3\ddot{\vec{r}}_3 = \frac{GM_1 M_3(\vec{r}_1-\vec{r}_3)}{|\vec{r}_1-\vec{r}_3|^3} + \frac{GM_2 M_3(\vec{r}_2-\vec{r}_3)}{|\vec{r}_2-\vec{r}_3|^3} \tag{25}
We may have seen descriptions elsewhere stating: "The equations of motion for the general three-body problem are 18th-order equations, requiring 18 integrals to obtain a complete solution." I believe this statement is not very rigorous, or at least not easy to understand. To be precise, the above is a system of second-order ordinary differential equations. Each \vec{r}_i can be projected onto three dimensions, resulting in 9 second-order differential equations. Each second-order equation requires two independent constants of integration upon solving; thus, we need to find 18 integration constants for the problem to be completely resolved. In 1843, Jacobi proved that for the N-body problem, if all but two integrals have been found, those last two can be determined using a special method.
For the N-body problem, there are 10 classical integrals of the same form. Taking the three-body problem as an example, BoJone will attempt to derive these ten integrals with the readers. First is the Conservation of Momentum. Adding equations (23), (24), and (25), we get: M_1 \ddot{\vec{r}}_1 + M_2 \ddot{\vec{r}}_2 + M_3 \ddot{\vec{r}}_3 = 0
Integrating this yields: M_1 \vec{r}_1 + M_2 \vec{r}_2 + M_3 \vec{r}_3 = \vec{C}_1 t + \vec{C}_2 \tag{26} How many integration constants are here? Two? No, there are actually 6 integration constants here (when projected into a 3D coordinate system)! Furthermore, there is a special case: if we choose their common center of mass as the reference point, we have \vec{C}_1 = \vec{C}_2 = \vec{0}. In this case, the problem becomes relatively simpler.
Another is the Conservation of Angular Momentum. From (23), (24), and (25), we have: \begin{aligned} M_1\vec{r}_1 \times \ddot{\vec{r}}_1 &= \frac{GM_1 M_2(\vec{r}_1 \times \vec{r}_2)}{|\vec{r}_2-\vec{r}_1|^3} + \frac{GM_1 M_3(\vec{r}_1 \times \vec{r}_3)}{|\vec{r}_3-\vec{r}_1|^3} \\ M_2\vec{r}_2 \times \ddot{\vec{r}}_2 &= \frac{GM_1 M_2(\vec{r}_2 \times \vec{r}_1)}{|\vec{r}_1-\vec{r}_2|^3} + \frac{GM_2 M_3(\vec{r}_2 \times \vec{r}_3)}{|\vec{r}_3-\vec{r}_2|^3} \\ M_3\vec{r}_3 \times \ddot{\vec{r}}_3 &= \frac{GM_1 M_3(\vec{r}_3 \times \vec{r}_1)}{|\vec{r}_1-\vec{r}_3|^3} + \frac{GM_2 M_3(\vec{r}_3 \times \vec{r}_2)}{|\vec{r}_2-\vec{r}_3|^3} \end{aligned}
Adding these three equations and using the property \vec{a} \times \vec{b} = -\vec{b} \times \vec{a}, we get: M_1\vec{r}_1 \times \ddot{\vec{r}}_1 + M_2\vec{r}_2 \times \ddot{\vec{r}}_2 + M_3\vec{r}_3 \times \ddot{\vec{r}}_3 = 0
Since (\vec{r} \times \dot{\vec{r}})' = \vec{r} \times \ddot{\vec{r}}, we can obtain a new integral: M_1\vec{r}_1 \times \dot{\vec{r}}_1 + M_2\vec{r}_2 \times \dot{\vec{r}}_2 + M_3\vec{r}_3 \times \dot{\vec{r}}_3 = \vec{C}_3 \tag{27} This provides another three integration constants! However, readers might have noticed that during the integration process, we have not yet used the "Law of Universal Gravitation" specifically. In other words, in the 9 integrals derived so far, we haven’t truly utilized the denominator terms like |\vec{r}_2-\vec{r}_3|^3. That is to say, whether the form of gravity is F=\frac{GM_1M_2}{R^2} or F=\frac{GM_1M_2}{R^3}, these results hold (as long as the force is central). Therefore, the next effort in integration must focus on the denominator terms.
\begin{aligned} M_1\ddot{\vec{r}}_1 \cdot \dot{\vec{r}}_1 &= \frac{GM_1 M_2(\vec{r}_2-\vec{r}_1) \cdot \dot{\vec{r}}_1}{|\vec{r}_2-\vec{r}_1|^3} + \frac{GM_1 M_3(\vec{r}_3-\vec{r}_1) \cdot \dot{\vec{r}}_1}{|\vec{r}_3-\vec{r}_1|^3} \\ M_2\ddot{\vec{r}}_2 \cdot \dot{\vec{r}}_2 &= \frac{GM_1 M_2(\vec{r}_1-\vec{r}_2) \cdot \dot{\vec{r}}_2}{|\vec{r}_1-\vec{r}_2|^3} + \frac{GM_2 M_3(\vec{r}_3-\vec{r}_2) \cdot \dot{\vec{r}}_2}{|\vec{r}_3-\vec{r}_2|^3} \\ M_3\ddot{\vec{r}}_3 \cdot \dot{\vec{r}}_3 &= \frac{GM_1 M_3(\vec{r}_1-\vec{r}_3) \cdot \dot{\vec{r}}_3}{|\vec{r}_1-\vec{r}_3|^3} + \frac{GM_2 M_3(\vec{r}_2-\vec{r}_3) \cdot \dot{\vec{r}}_3}{|\vec{r}_2-\vec{r}_3|^3} \end{aligned}
Adding the three terms gives: \begin{aligned} M_1\ddot{\vec{r}}_1 \cdot \dot{\vec{r}}_1 + M_2\ddot{\vec{r}}_2 \cdot \dot{\vec{r}}_2 + M_3\ddot{\vec{r}}_3 \cdot \dot{\vec{r}}_3 = -GM_1 M_2 \frac{(\vec{r}_2-\vec{r}_1) \cdot (\vec{r}_2-\vec{r}_1)'}{|\vec{r}_2-\vec{r}_1|^3} \\ -GM_2 M_3 \frac{(\vec{r}_3-\vec{r}_2) \cdot (\vec{r}_3-\vec{r}_2)'}{|\vec{r}_3-\vec{r}_2|^3} - GM_1 M_3 \frac{(\vec{r}_3-\vec{r}_1) \cdot (\vec{r}_3-\vec{r}_1)'}{|\vec{r}_3-\vec{r}_1|^3} \end{aligned}
Using the identity \vec{r} \cdot \dot{\vec{r}} = r \dot{r}, we transform this into: \begin{aligned} M_1\ddot{\vec{r}}_1 \cdot \dot{\vec{r}}_1 + M_2\ddot{\vec{r}}_2 \cdot \dot{\vec{r}}_2 + M_3\ddot{\vec{r}}_3 \cdot \dot{\vec{r}}_3 \\ = -GM_1 M_2 \frac{d|\vec{r}_2-\vec{r}_1|}{dt|\vec{r}_2-\vec{r}_1|^2} - GM_2 M_3 \frac{d|\vec{r}_3-\vec{r}_2|}{dt|\vec{r}_3-\vec{r}_2|^2} - GM_1 M_3 \frac{d|\vec{r}_3-\vec{r}_1|}{dt|\vec{r}_3-\vec{r}_1|^2} \end{aligned}
Both sides are integrable, thus: \frac{1}{2}M_1\dot{\vec{r}}_1^2 + \frac{1}{2}M_2\dot{\vec{r}}_2^2 + \frac{1}{2}M_3\dot{\vec{r}}_3^2 = \frac{GM_1 M_2}{|\vec{r}_2-\vec{r}_1|} + \frac{GM_2 M_3}{|\vec{r}_3-\vec{r}_2|} + \frac{GM_1 M_3}{|\vec{r}_3-\vec{r}_1|} + C_4 \tag{28}
This is the final integral, representing the Conservation of Energy. Note that C_4 here is a scalar, so unlike integral (27), it cannot be projected into three dimensions; it counts as only one integral. At this point, all ten first integrals of the three-body problem have been derived. For problems with more than three bodies, these same ten integrals exist; one simply adds more terms to (26), (27), and (28). Tragically, as of now, these are the only 10 integrals known to exist.
In 1887, Bruns proved that if Cartesian coordinates and velocity components are used as basic variables, no new algebraic integrals (integrals that are algebraic functions of the variables) exist for the three-body problem. In 1889, Poincaré further proved that if combinations of orbital elements are used as variables, new single-valued analytic integrals also do not exist. In 1898, Painlevé demonstrated that new integrals in the form of algebraic functions of velocity components do not exist either. In 1941, Siegel also proved that for the planar circular restricted three-body problem, no new algebraic integrals exist besides the Jacobi integral. Although various pessimistic conclusions emerged in the search for new integrals of the three-body problem, these conclusions are conditional and not absolute. The path is not completely blocked, but there is not yet a single glimmer of light. Since the 1950s, two new avenues for studying the integrals of the three-body problem have been proposed.
One avenue is seeking new integrals in the form of series. For example, in 1965, the Greek astronomer Contopoulos found an integral expressed as a series. This series is expanded using the mass of the smaller finite body in the planar circular restricted three-body problem as a small parameter. The coefficients of the series can, in principle, be determined step-by-step, though they are in integral form. However, the convergence of this series has not yet been proven, so it cannot be formally established. The other avenue is using numerical methods to prove the existence of new integrals, with preliminary results already obtained for the planar circular restricted three-body problem. For instance, Woltjer and others used numerical methods to find the intersection lines between the hypothetical integral surfaces and coordinate planes, known as invariant curves. Based on these invariant curves, the existence of hypothetical integrals is inferred, though they have not been explicitly found.
References
Yi Zhaohua, Introduction to Celestial
Mechanics
Y. Hagihara, Celestial Mechanics, Vol. I, MIT Press,
Cambridge, 1970.
The content above was summarized by BoJone while reading "Introduction to Celestial Mechanics." Of course, the original author did not use vectors to describe the proof process but instead used partial differential functions. BoJone felt that readers might find vector derivations easier to understand and thus employed the vector approach. In fact, these ten first integrals are among the most fundamental contents of celestial mechanics and can be found in almost any introductory work on the subject. However, BoJone searched the internet for a long time and could not find this specific material. It is evident that currently, the work of digitizing and networking mathematical theories is still lacking, and the sharing of theoretical resources is insufficient. Therefore, I have derived them here for the reference of those in need. If there are any errors, please do not hesitate to point them out.
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