If we only derive transformations from the perspective of Newton’s second law, the results regarding the duality of mechanical laws are undoubtedly elementary. For theoretical analysis, it is more convenient to start from the form of the Principle of Least Action. In fact, the computational effort required for this form is quite small, and it is even more convenient than directly substituting into the equations of motion for transformation.
In the previous article, we mentioned that the transformation z \mapsto z^2 maps an ellipse with its geometric center at the origin to an ellipse with a focus at the origin. We believe that this transformation can link Hooke’s Law with Newton’s Law of Universal Gravitation. We then immediately provided the transformation w=z^2, d\tau=|z^2|dt. However, this transformation itself is not obvious. If we only discovered the geometric meaning of z \mapsto z^2, how would we correspondingly derive the transformation d\tau=|z^2|dt? This article provides a preliminary solution to this problem.
Geometric Action
Let us review the Principle of Least Action in mechanics: S = \int_{{t_1}}^{{t_2}} L dt = \int_{{t_1}}^{{t_2}} {(T - U)} dt
Here we only consider conservative systems, where T=\frac{1}{2} m |\frac{d\mathbf{x}}{dt}|^2 is the kinetic energy of the system and U=U(\mathbf{x}) is the potential energy. It is not difficult to find that adding a constant to the Lagrangian L does not change any motion information or orbital information. Therefore, for the sake of generality, we write: S=\int_{t_1}^{t_2} Ldt=\int_{t_1}^{t_2} (T-U+c)dt
After reading this article, the reader will understand why we consider it this way. Since we are only concerned with the shape of the orbit, the above action is not suitable as it involves time. Using the law of conservation of energy: E=T+U=\frac{1}{2} m \left|\frac{d\mathbf{x}}{dt}\right|^2+U
We can rewrite the action into a form that contains only orbital information. First, note that L=T-U+c=(E+c)-2U, and dt=\sqrt{\frac{m}{2(E-U)}}|d\mathbf{x}|, so the action can be written as: S=\int \frac{(E+c)-2U}{\sqrt{2(E-U)/m}}|d\mathbf{x}|
This is essentially the result obtained by Jacobi in 1837. This is the Principle of Least Action with geometric properties, which transforms dynamical problems into geometric problems, consistent with the metric ideas in relativity. However, we will not extend this further here and instead return to our problem of mechanical duality.
Dual Action
We still use complex numbers as a tool. The action for simple harmonic motion can be written as: S=\int_{t_1}^{t_2} Ldt=\int_{t_1}^{t_2} \frac{1}{2}m \left(\left|\frac{dz}{dt}\right|^2-\omega^2 |z|^2\right)dt
Where energy conservation is |\frac{dz}{dt}|^2+\omega^2 |z|^2=2E. Using the geometric form of the action, we have: S=\int \frac{E-\omega^2 |z|^2}{\sqrt{2(E-\frac{1}{2}\omega^2 |z|^2)/m}}|dz|
Making the substitution w=z^2, then |dz|=\frac{|dw|}{2\sqrt{|w|}}, so: S=\int \frac{E-\omega^2 |w|}{\sqrt{2(E-\frac{1}{2}\omega^2 |w|)/m}}\left(\frac{|dw|}{2\sqrt{|w|}}\right)=\int \frac{E|w|^{-1}-\omega^2 }{\sqrt{2(E|w|^{-1}-\frac{1}{2}\omega^2 )/m}}|dw|
We know that constant factors do not affect the result of the variation, i.e.: S=\left(\frac{-1}{2i}\right)\int \frac{(\frac{1}{2} \omega^2+\frac{3}{2} \omega^2) -2E|w|^{-1} }{\sqrt{2(\frac{1}{2}\omega^2 -E|w|^{-1})/m}}|dw|
Ignoring the constant factor \left(\frac{-1}{2i}\right) and comparing this with S=\int \frac{(E+c)-2U}{\sqrt{2(E-U)/m}}|d\mathbf{x}|, it is not difficult to find that this is the action where c\mapsto \frac{3}{2} \omega^2, E\mapsto \frac{1}{2} \omega^2, U\mapsto E|w|^{-1}. Therefore, we say that the transformation w=z^2 transforms the mechanics of Hooke’s Law into the mechanics of the Law of Universal Gravitation.
Noting that dt=\frac{|d\mathbf{x}|}{\sqrt{2(E-U)/m}}, if we set t\mapsto \tau, then we should have d\tau= \frac{|dw|}{\sqrt{2(\frac{1}{2}\omega^2 -E|w|^{-1})/m}}, from which we can calculate d\tau \propto |w|dt.
As for the duality laws of central force fields with other powers mentioned in the previous article, they can be verified similarly without substantial difficulty. Moreover, after performing the calculations personally, the reader will find that using this method, one can quickly calculate the power of the duality law for a certain central force field and correspondingly find the value of \alpha for its orbital mapping z\mapsto z^{\alpha}. Compared to verifying by substituting into Newton’s second law, the action method appears even simpler. Furthermore, it can be found that these results can be generalized because they essentially rely on only two points:
1. The correspondence between complex numbers and plane vectors;
2. |z_1 z_2|=|z_1| |z_2|.
These are basic properties of quaternions and octonions; therefore, these conclusions can be easily generalized to higher dimensions.
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