A few days ago, while browsing "The Heartstrings of the Universe," I discovered an updated and very interesting article titled "Stability of the Inverted Pendulum and the Ponderomotive Force" (as expected, those in the physics department encounter all sorts of fascinating phenomena). The article discussed how an inverted pendulum can be made stable by applying a specific motion to it. This piqued my interest, and I subsequently performed my own calculations.
A simple pendulum has two equilibrium points: one is the stable equilibrium point pointing vertically downward, and the other is the unstable equilibrium point pointing vertically upward. To put it figuratively, if you have a downward-pointing pendulum and you shake it slightly, it just swings back and forth; however, if the pendulum is pointing vertically upward and you shake it, it will quickly fall to the downward position. Can applying a motion to the pendulum make the inverted state stable? At first glance, it is difficult to judge whether such a motion exists. However, one only needs to imagine pulling the pendulum downward with an acceleration of 10\,\text{m/s}^2; in this case, the inverted pendulum naturally becomes stable because the effective force has reversed direction. This suggests that non-inertial motion can introduce an additional force field that stabilizes the inverted pendulum. Of course, the example of constant acceleration is not very useful because the range of motion is too large, eventually moving the pendulum beyond our control. But there is another type of motion that might stabilize the inverted pendulum: high-frequency oscillation. We will analyze this in detail.
Physixfan’s article also provided a mathematical analysis of this problem. However, I felt that the analytical process was not entirely satisfactory. He added a "high-frequency oscillation term" f to the original equation of motion m\ddot{x} = -dU/dx, changing it to: m\ddot{x} = -dU/dx + f But the meaning of f here is ambiguous. If x represents displacement, then f can be understood as a force; however, x could also be an angle or another variable, making it difficult to grasp. Furthermore, even if f is interpreted as a force, "adding a high-frequency oscillating force to the pendulum" is not clearly defined experimentally. A clearer approach is to apply a high-frequency oscillation directly to the original system—in other words, placing the pendulum in a high-frequency oscillating reference frame. (Imagine an extreme case: an elevator moving up and down at an extremely high speed, with the pendulum inside.)
Using the Principle of Least Action is the most convenient method for this analysis. Let the vertically upward direction be the positive x-axis and the horizontal rightward direction be the positive y-axis. The action of the pendulum is: S = \int \left[ \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - mgx \right] dt subject to the constraint x^2 + y^2 = l^2. From this, the equations of motion can be derived. Choosing this coordinate system is convenient for analyzing the inverted pendulum.
Now, we add an oscillation term x = h(t) to the original pendulum. That is, we let the fixed pivot of the pendulum move according to x = h(t). Consequently, the action remains: S = \int \left[ \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - mgx \right] dt but the constraint condition becomes: [x - h(t)]^2 + y^2 = l^2
Let x = l\cos\theta + h(t) and y = l\sin\theta. Substituting these into the action, we get: S = \int \left[ \frac{1}{2}m(l^2\dot{\theta}^2 + \dot{h}^2 - 2l\dot{h}\dot{\theta}\sin\theta) - mg(l\cos\theta + h) \right] dt
From the Euler-Lagrange equation, we obtain the equation of motion: \frac{d}{dt}(ml^2\dot{\theta} - ml\dot{h}\sin\theta) = mgl\sin\theta - ml\dot{h}\dot{\theta}\cos\theta which simplifies to: l\ddot{\theta} - (\ddot{h} + g)\sin\theta = 0
As we can see, the equilibrium point \theta = 0 still exists. When the h term is absent, the equation l\ddot{\theta} - g\sin\theta = 0 is unstable at the origin. To see this, use the approximation \sin\theta \approx \theta, yielding l\ddot{\theta} - g\theta = 0, which has the solution: \theta = Ae^{t\sqrt{g/l}} This is an exponential solution that diverges to infinity.
As long as (\ddot{h} + g) < 0, the equation can become a constrained (restoring) equation, making \theta = 0 stable. However, we want the "amplitude" of the change to the original pendulum to be small, so h can only be a small-amplitude oscillation. Let us assume: h(t) = h_0 \cos(\omega t) Then: \ddot{h} = -h_0 \omega^2 \cos(\omega t) Substituting this into the equation gives: l\ddot{\theta} + [h_0 \omega^2 \cos(\omega t) - g]\sin\theta = 0 Expanding near t=0 and \theta=0, we have the approximation: l\ddot{\theta} + (h_0 \omega^2 - g)\theta = 0 Evidently, for \theta = 0 to become a stable equilibrium point, we must have h_0 \omega^2 - g \gg 0, or: \omega \gg \sqrt{\frac{g}{h_0}} If h_0 = 0.01\,\text{m}, then we estimate \omega \gg 30\,\text{s}^{-1}. This is only about 5\,\text{Hz}. Since the frequency of our alternating current is 50\,\text{Hz}, the experiment in the video is entirely feasible. In summary, over long-term observation, high-frequency oscillation can generate a force that maintains the stability of the system. This force is called the Ponderomotive Force. This phenomenon has important applications in particle physics.
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