Foreword
After more than two years of development, Typecho (the engine used by this site) has finally released a new version. Although it is still in beta, I couldn’t wait to upgrade. Of course, there are no changes to the front end, but almost the entire program has been refactored, and the back end is much cleaner. This is the first article written using the new version, composed using Markdown syntax.
Newton vs. Hooke
Among all mechanical systems, the simplest is perhaps simple harmonic motion. It is described by the simplest system of linear differential equations with constant coefficients: \ddot{\boldsymbol{x}}+\omega^2 \boldsymbol{x}=0
This describes the motion of an object under the force described by Hooke’s Law for elastic deformation. We can quickly write the exact solution to this equation using trigonometric functions. In contrast, the solution to the two-body problem is much more complex. Although the two-body problem is also exactly solvable, it is clearly not as simple and clear as simple harmonic motion. However, besides both being central force problems, they share another commonality: their motion orbits are both ellipses! (Strictly speaking, they are conic sections, as there could also be parabolas and hyperbolas, but without loss of generality, this article only analyzes elliptical orbits). Is there some kind of connection between the two? If the two-body problem could be transformed into simple harmonic motion, would the analysis process be greatly simplified?
Celestial Mechanics
In this article, we only consider the planar case. Therefore, the above equation can be represented using complex numbers as: \ddot{z}+\omega^2 z=0
Using complex numbers is very convenient here. We can directly write its solution as: z=pe^{i\omega t}+qe^{-i\omega t}. By expanding both sides, it is easy to verify that this is indeed an elliptical equation with the origin as the center of symmetry. (We may consider only the case where p and q are real, as complex p and q merely rotate the ellipse by an angle around the origin).
Consider the transformation z \mapsto z^2. What does the orbit become? z^2=p^2 e^{2i\omega t}+q^2 e^{-2i\omega t}+2pq. This is not some strange curve, but another ellipse—an ellipse with the origin at one of its foci! (Please verify this). Before the transformation, it was the shape of a simple harmonic motion orbit; after the transformation, it is the orbit of a two-body problem. We have reason to believe that the transformation z \mapsto z^2 links the two-body problem and simple harmonic motion together.
In fact, the answer is that through the transformation w=z^2 and d\tau=|z^2|dt, the simple harmonic motion equation \ddot{z}+\omega^2 z=0 can be converted into the two-body problem equation: \frac{d^2 w}{d\tau^2}=-\frac{C w}{|w|^3}. This is known as Bohlin’s Theorem. To complete this proof, one only needs to perform simple calculus calculations:
\begin{aligned} \frac{dw}{d\tau} &= 2z\frac{dz}{d\tau} = 2z\frac{dz}{dt}\frac{1}{|z^2|} = \frac{2}{\bar{z}}\frac{dz}{dt} \\ \frac{d^2 w}{d\tau^2} &= \frac{d}{|z^2|dt}\left(\frac{2}{\bar{z}}\frac{dz}{dt}\right) = \frac{2}{z\bar{z}}\left(\frac{1}{\bar{z}}\frac{d^2 z}{dt^2} - \frac{1}{\bar{z}^2}\frac{dz}{dt}\frac{d\bar{z}}{dt}\right) \\ &= - \frac{2}{z\bar{z}}\left(\frac{\omega^2 z}{\bar{z}} + \frac{1}{\bar{z}^2}\frac{dz}{dt}\frac{d\bar{z}}{dt}\right) = - \frac{2}{z\bar{z}^3}\left(\omega^2 |z|^2 + \left|\frac{dz}{dt}\right|^2\right) \\ &= - \frac{4E}{z\bar{z}^3} = - \frac{4Ez^2}{z^3\bar{z}^3} \\ &= - \frac{4Ew}{|w|^3} \end{aligned}
In the last step, E=\frac{1}{2}\omega^2|z|^2+\frac{1}{2}|\frac{dz}{dt}|^2 is the law of conservation of energy for the simple harmonic motion \ddot{z}+\omega^2 z=0. This is essentially the complete result for the two-body problem; it reveals the mutual dual relationship between the law of universal gravitation and Hooke’s Law. Newton already preliminary revealed this result in his Principia (he used geometric analysis, as he did not yet understand the tools of complex analysis at that time). This indicates that modern complex functions can establish relationships between motions in different central force fields.
In fact, Euler discovered very early on that for the one-dimensional two-body problem equation \frac{d^2 x}{dt^2}=-\frac{c}{x^2}, one can use u=\sqrt{x} and dt=xd\tau to transform it into the simple harmonic motion \frac{d^2 u}{d\tau^2}+2Eu=0. The aforementioned Bohlin’s Theorem is the two-dimensional generalization. Bohlin, E. Kasner, and V.I. Arnol’d were among the first discoverers of this theorem. What is shown above is basically what Bohlin discovered, while Kasner and Arnol’d discovered the more general result below.
Duality of Mechanical Laws
A point z in the complex plane moves in a central force field where the force is proportional to the a-th power of the distance from the center. Under the transformation w=z^{\alpha}, the orbit of z will transform into the trajectory of motion in a central force field proportional to the A-th power of the distance to the center, where A is determined by the following formula: (a+3)(A+3)=4, \quad \alpha=\frac{a+3}{2}
In this way, every power-law central force field has another central force field dual to it. Self-duality corresponds to a=-1 and a=-5. This is also mentioned in Newton’s Principia. As for the proof of the theorem, one only needs to repeat the calculations above to obtain:
\begin{aligned} \frac{d^2 z}{dt^2} &= -z|z|^{a-1}, \quad d\tau=|z|^{a+1} \\ \frac{d^2 w}{d\tau^2} &= -Cw|w|^{A-1}, \quad C=2E\alpha(\alpha-1) \\ 2E &= \left|\frac{dz}{dt}\right|^2 + \frac{2|z|^{a+1}}{a+1} \end{aligned}
(The above results are taken from Huygens and Barrow, Newton and Hooke: Early Steps in Mathematical Analysis and Catastrophe Theory, from Evolutes to Quasicrystals)
Furthermore, using quaternions as a tool, these duality laws can be generalized to higher dimensions. If the reader feels that this seems to be just a mathematical transformation game, that would be incorrect. In fact, this is not only a transformation game but also has great practical value in computation. For example, in the transformation of the two-body problem, if one directly performs numerical calculations on the two-body problem, the term -\frac{4Ew}{|w|^3} has a singularity at the origin, which brings difficulties to the calculation. However, the transformed simple harmonic motion equation has no singularity and can be calculated conveniently.
Additionally, in quantum mechanics, the dual transformation of the two-body problem corresponds to the Kustaanheimo-Stiefel transformation, which is the basis for Feynman’s path integral calculation of the hydrogen atom problem! More content will be discussed in the next article.
References
Visual Complex Analysis (Section 5.10 Celestial Mechanics)
Mathematical Methods of Classical Mechanics (Section 8. Investigation of Motion in Central Force Fields)
Huygens and Barrow, Newton and Hooke: Early Steps in Mathematical Analysis and Catastrophe Theory, from Evolutes to Quasicrystals
The Kustaanheimo-Stiefel transformation in geometric algebra
When reprinting, please include the original address: https://kexue.fm/archives/2121
For more details regarding reprinting, please refer to: Scientific Space FAQ