Equations of Motion and Energy Integral of the Plane
Circular Restricted Three-Body Problem
plane circular restricted three-body
problem
Important correction on February
4th!!
A major goal for this winter break was to achieve some breakthroughs regarding the periodic orbits of the three-body problem. To this end, I deployed theoretical “nuclear weapons” such as “vectors,” “complex analysis,” and “differential equations.” Unfortunately, as the saying goes, “The flowers one purposefully plants do not bloom, but the willow seeds one accidentally drops grow into shade.” To this day, there hasn’t been much progress on that front. However, looking back today at the “Circular Restricted Three-Body Problem” in Introduction to Celestial Mechanics, and after some reflection, I have re-derived its equations of motion and energy integral using the methods I’ve been considering lately. This can be considered an “unexpected harvest,” which I share with everyone here as a Spring Festival gift.
The so-called “Circular Restricted Three-Body Problem” refers to a situation where two massive bodies (point masses) move in circular orbits under their mutual gravitational attraction, and a third body (with mass tending to 0) is assumed to be affected only by the gravity of these two bodies without influencing their motion. Since the general three-body problem cannot be integrated, this “restricted model” simplifies the problem significantly (though it still cannot be solved analytically), and thus it has found certain applications. Its application conditions are: the third body must have a small mass (such as a spacecraft relative to the Earth and the Sun) and the motion must be short-range. Note that being short-range is also a very important condition; note that being short-range is also a very important condition. The smaller the mass, the larger the range of application. If the mass is large, one cannot calculate for too long a distance.
The research method in this article synthesizes vectors, complex numbers, and simple calculus. For some extensions regarding vectors, you may refer to: https://kexue.fm/archives/1188. It is important not to confuse the three multiplication operations of complex numbers. We primarily study the planar case first.
First, let the position of the third body be r. Let the orbital radius of the second body be l_2, its angular velocity be \omega, and its mass be m. Let the first body have mass M and orbital radius l_1. The reference point is their common center of mass, which gives Ml_1 + ml_2 = 0 and |l_1| + |l_2| = l. The reason for choosing the common center of mass rather than either body as the reference point is that the common center of mass is an inertial frame, allowing for simple composition of forces. We can represent the trajectories of the first and second bodies as l_1 \exp(i\omega t) and l_2 \exp(i\omega t). (Regarding \exp(z), it is somewhat similar to e^z, but not entirely identical. We still have Euler’s formula \exp(i\theta) = \cos\theta + i\sin\theta, which is consistent with the form of circular motion equations.) Based on force analysis, we have:
\ddot{r} = -GM\frac{r - l_1 \exp(i\omega t)}{|r - l_1 \exp(i\omega t)|^3} - Gm\frac{r - l_2 \exp(i\omega t)}{|r - l_2 \exp(i\omega t)|^3} \tag{29}
Studying the above equation directly is quite troublesome because the right-hand side explicitly contains t. To eliminate t, we multiply each term by \exp(-i\omega t):
\ddot{r}\exp(-i\omega t) = -GM\frac{r \exp(-i\omega t) - l_1}{|r \exp(-i\omega t) - l_1|^3} - Gm\frac{r \exp(-i\omega t) - l_2}{|r \exp(-i\omega t) - l_2|^3} \tag{30}
Readers will notice that we also multiplied the modulus terms by \exp(-i\omega t). This is to unify the form, and since |\exp(-i\omega t)| = 1, according to the definition of complex multiplication, the modulus does not change after multiplication.
Let R = r \exp(-i\omega t). We can write: \begin{aligned} \dot{R} &= \dot{r}\exp(-i\omega t) - i\omega r \exp(-i\omega t) = \dot{r}\exp(-i\omega t) - i\omega R \\ \ddot{R} &= \ddot{r}\exp(-i\omega t) - i\omega \dot{r} \exp(-i\omega t) - i\omega \dot{R} \\ &= \ddot{r}\exp(-i\omega t) - i\omega(\dot{R} + i\omega R) - i\omega \dot{R} \\ &= \ddot{r}\exp(-i\omega t) - 2i\omega \dot{R} + \omega^2 R \end{aligned} \tag{31}
Substituting (31) into (30), we get:
\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}
Equation (32) is the differential equation commonly used to study the circular restricted three-body problem; it does not explicitly contain t. Through coordinate projection, it can be converted into Cartesian coordinate form (3D works as well). One can also set appropriate units for mass, length, time, and angle such that G, l, \omega are all 1. What might be confusing is l_1, l_2; previously they were defined as scalars (orbital radii), so how can they be subtracted from a vector? Actually, l_1, l_2 here are real numbers, which can be viewed as vectors where the y and z components are both 0.
The above derivation uses complex numbers, saving us from the dazzling process of coordinate transformations. BoJone believes that this derivation method makes it easier for everyone to understand. This method stems from recent reflections on periodic orbits of the three-body problem and the further integration of complex numbers and vectors, which has deepened my understanding of complex numbers. Now, let’s continue to derive the energy integral.
Taking the “dot product” of each term in (32) with \dot{R}, we get: \dot{R} \cdot \ddot{R} = \omega^2 R \cdot \dot{R} - GM\frac{(R - l_1) \cdot (R - l_1)'}{|R - l_1|^3} - Gm\frac{(R - l_2) \cdot (R - l_2)'}{|R - l_2|^3}
Forms change, but identities remain invariant. The identity \vec{r} \cdot \dot{\vec{r}} = |\vec{r}| |\vec{r}|' still holds...
|\dot{R}| |\dot{R}|' = \omega^2 |R| |R|' - GM\frac{|R - l_1|'}{|R - l_1|^2} - Gm\frac{|R - l_2|'}{|R - l_2|^2}
Integrating gives: \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} Equation (33) is the energy integral, also known as the “Jacobi integral.”
Note that \dot{R} is not the true velocity; \dot{r} is. We have: \begin{aligned} |\dot{R}|^2 &= [\dot{r}\exp(-i\omega t) - i\omega r \exp(-i\omega t)] \cdot [\dot{r}\exp(-i\omega t) - i\omega r \exp(-i\omega t)] \\ &= [\dot{r} - i\omega r] \cdot [\dot{r} - i\omega r] = |\dot{r}|^2 + \omega^2 |r|^2 - 2\omega(ir) \cdot \dot{r} \end{aligned} (Please refer to the complex dot product operation). In the second equality, \exp(-i\omega t) was canceled from both vectors, meaning both were rotated by an angle of \omega t; thus, the angle between them naturally remains unchanged, and the modulus does not change either, so the dot product remains invariant. Note that in the third equality, since r is a complex number, the result is not yet fully determined, so we retain the dot product symbol “\cdot”. Since |R|^2 = |r|^2, substituting into (33) yields:
\frac{1}{2}|\dot{r}|^2 = \frac{GM}{|R - l_1|} + \frac{Gm}{|R - l_2|} + \omega(ir) \cdot \dot{r} - C
It can be seen that the energy in the plane circular restricted three-body problem is not simply the conservation of kinetic energy and gravitational potential energy; it has an “extra” \omega(ir) \cdot \dot{r} term. The plane circular restricted three-body problem is just a 2nd-order differential equation in 2D, so it should require 4 constants to be integrated. One has been integrated above, leaving three, but those three seem to be very, very difficult...
The above process is merely a retelling of existing results. Of course, there are many ways to “tell” a story. Professionals at the forefront of science are the creators of most advanced achievements. However, original content (especially theory) is inevitably accompanied by “dryness.” Therefore, popular science workers are more like “translators”; they do not just translate foreign languages into Chinese, but also describe seemingly profound scientific content in a relaxed and lively (at least relatively so) yet rigorous language. In BoJone’s view, this is also the reason for the rise of the Science Squirrel Improvement Association (Songshuhui). BoJone is also willing to be such a “translator” and continue to share the beauty of science with everyone.