The New Calculation Of Lagrangian Point 4, 5
In the previous installment, we solved for the Lagrangian points L_1, L_2, L_3, and expressed the hope of finding the points L_4 and L_5. Since L_4 and L_5 are no longer collinear with the "Earth-Sun" line, the previous method seemingly no longer applies. To obtain a general definition, we can adopt the following method to describe Lagrangian points: For a celestial body located at a Lagrangian point, the angle formed by its line to the Sun and the Earth’s line to the Sun is constant. (For convenience, the Earth-Sun system is used here for description, but it is the same for the general three-body problem.)
For L_4 and L_5, we might attempt to set a new vector to describe the position vector of these two points (such as \vec{R}). However, once we do this, we quickly find that it leads our calculation into a dead end. This is because we discover that given the angle between two vectors and one of the vectors, it is very difficult to express the other vector using an expression of the known vector. Without being able to do this, we cannot find the relationship between \vec{R} and \vec{r}, and thus cannot set up simultaneous equations to solve. Has our path come to an end? Initially, BoJone also pondered deeply without any clues, but...
A Fancy Divider
Wait, BoJone also mentioned at the end of the previous article: geometry can be studied very conveniently using complex numbers! Indeed, this problem becomes very simple when answered using complex numbers. First, let’s introduce complex numbers. To solve the problem of taking the square root of negative numbers, mathematicians introduced a class of numbers in the form z = a + bi, where i^2 = -1 and a, b \in \mathbb{R}. The set of such numbers is called complex numbers. At the time, they were introduced purely for computational convenience and were considered "imaginary numbers" with no practical meaning when b \neq 0. But later, scientists fell in love with this class of numbers because they can conveniently study practical problems in geometry, physics, and more...
Complex numbers have wonderful properties. Given some complex numbers, no matter what operations you perform, the result is always a complex number (that is, the result of complex number operations can always be written in the form z = a + bi, i^2 = -1, a, b \in \mathbb{R}). Thus, if two complex numbers a + bi and c + di are equal, it must be that a = c and b = d. (Do you notice that this is somewhat similar to vectors?). Complex numbers cannot be described on a single number line. Since there are two real numbers a and b, mathematicians use the "complex plane" method, establishing a coordinate system similar to XOY, where any point on the complex plane represents a definite complex number. This links complex numbers with geometry. (Remember? Any point in a Cartesian coordinate system represents a vector; the clever reader will connect the two.)
The "modulus" of a complex number (equivalent to the absolute value of a real number) is defined as r = |z| = \sqrt{a^2 + b^2}, which is very similar to the method for finding the length of a vector. The addition, subtraction, multiplication, and division of complex numbers also have their charms. (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i (doesn’t this look like vectors?); when two complex numbers are multiplied, the result is also a complex number, whose modulus is equal to the product of the moduli of the two complex numbers, and whose "argument" (the angle with the real axis) is equal to the sum of the arguments of the two complex numbers. (This seems to define a new type of vector operation?). This is a brief description of complex numbers; this article is not specifically about complex numbers, so I will not elaborate further. Readers who wish to study deeply can refer to the university textbook "Functions of a Complex Variable".
A Fancy Divider
Returning to our Lagrangian point problem, we know that the key to the problem is to find the relationship between \vec{R} and \vec{r}. Here, using the "complex plane" method, we represent vectors as complex numbers (i.e., each vector represents a complex number). Let k be a complex number, and \vec{R} = k\vec{r} (note that here \vec{r} represents a complex number rather than a vector, and k\vec{r} expresses a vector with a fixed angle relative to \vec{r}). We can still list an equation in the form of (22) based on (19), similar to (20) and (21). The difference is that here k is a complex number: (M+m)+m'\left(\frac{1-k}{|1-k|^3}+\frac{k}{|k|^3}\right)=\frac{M+m'}{|k|^3}-m\left(\frac{1-k}{k|1-k|^3}-\frac{1}{k}\right) \tag{22}
We have solved many real equations, but complex equations are rarely solved. BoJone was also confused for a while. However, we notice that if |1-k|^3 and |k|^3 are both equal to 1, then the terms in the above equation can cancel each other out (that is, a general solution is obtained). From this line of thought, let k = a + bi, then we have |k|^2 = a^2 + b^2 = 1 and |1-k|^2 = (1-a)^2 + b^2 = 1. Since a and b are both real numbers, we easily obtain a = 1/2 and b = \pm \frac{\sqrt{3}}{2}. That is to say, k = 1/2 \pm \frac{\sqrt{3}}{2}i is the general solution to equation (22) (independent of mass). The geometric meaning of this general solution is: Taking the Earth-Sun distance as the unit of length and the Sun as the origin, these two points are located at \frac{\sqrt{3}}{2} units vertically above or below the midpoint of the Earth-Sun line. These are the final two solutions for the Lagrangian points (the equilateral triangle solutions)!
At this point, readers who are encountering complex numbers for the first time might still feel that using complex numbers merely utilizes their operational properties, but that they still lack any physical meaning. Readers with such thoughts are mistaken. Deriving equation (22) from the perspective of complex numbers, the physical meaning of each step is very clear (keep in mind that, except for k itself, the result of each step of the operation is a real number, and each real number represents a physical quantity; the so-called complex number is just a "transitional quantity" between mathematics and physics, acting like a vector). It’s just that in our usual calculations, we generally use a Cartesian coordinate system, whereas here we use the complex plane. The advantage of this is that we only need to study one variable, rather than decomposing the vector into orthogonal components. This is why complex numbers are so convenient for studying geometry. Of course, it is not just geometry; "Complex Variables," which developed from complex numbers, has become an indispensable and powerful tool in fields such as astronomy, mechanics, and optics!
"When observing a small subject, one must examine its textures minutely; thus, one often finds interest beyond the physical world!" This is BoJone’s sentiment.
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