The Kapitza pendulum

Rarely does a ‘problem’ come along that makes you think more than casually about the question of mathematics’s reality, and problems in mathematical physics are full of them. I came across one such problem for the first time yesterday, and given its simplicity, thought I should make note of it.

I spotted a paper yesterday with the title ‘The Inverted Pendulum as a Classical Analog of the EFT Paradigm’. I’ve never understood the contents of such papers without assistance from a physicist, but I like to go through them in case a familiar idea or name jumps up that warrants a more thorough follow-up or I do understand something and that helps me understand something else even better.

In this instance, the latter happened, and I discovered the Kapitza pendulum. In 1908, a British mathematician named Andrew Stephenson described the problem but wasn’t able to explain it. That happened at the hands of the Russian scientist Pyotr Kapitsa, for whom the pendulum is named, who worked it out in the 1950s.

You are familiar with the conventional pendulum:

Here, the swinging bob is completely stable when it is suspended directly below the pivot, and is unmoving. The Kapitza pendulum is a conventional pendulum whose pivot is rapidly moved up and down. This gives rise to an unusual stable state: when the bob is directly above the pivot! Here’s a demonstration:

As you can see, the stable state isn’t a perfect one: the bob still vibrates on either side of a point above the pivot, yet it doesn’t move beyond a particular distance, much less drop downward under the force of gravity. If you push the bob just a little, it swings across a greater distance for some time before returning to the narrow range. How does this behaviour arise?

I’m fascinated by the question of the character of mathematics because of its ability to make predictions about reality – to build a bridge between something that we know to be physically true (like how a conventional pendulum would swing when dropped from a certain height, etc.) and something that we don’t, at least not yet.

If this sounds wrong, please make sure you’re thinking of the very first instantiation of some system whose behaviour is defying your expectations, like the very first Kapitza pendulum. How do you know what you’re looking at isn’t due to a flaw in the system or some other confounding factor? A Kapitza pendulum is relatively simple to build, so one way out of this question is to build multiple units and check if the same behaviour exists in all of them. If you can be reasonably certain that the same flaw is unlikely to manifest in all of them, you’ll know that you’re observing an implicit, but non-intuitive, property of the system.

But in some cases, building multiple units isn’t an option – such as a particle-smasher like the Large Hadron Collider or the observation of a gravitational wave from outer space. Instead, researchers use mathematics to check the likelihood of alternate possibilities and to explain something new in terms of something we already know.

Many theoretical physicists have even articulated that while string theory lacks experimental proof, it has as many exponents as it does because of its mathematical robustness and the deep connections they have found between its precepts and other, distant branches of physics.

In the case of the Kapitza pendulum, based on Newton’s laws and the principles of simple harmonic motion, it is possible to elucidate the rules, or equations, that govern the motion of the bob under the influence of its mass, the length of the rod connecting the bob to the pivot, the angle between the line straight up from the pivot and the rod (θ), acceleration due to gravity, the length of the pivot’s up-down motion, and how fast this motion happens (i.e. its frequency).

From this, we can derive an equation that relates θ to the distance of the up-down motion, the frequency, and the length of the rod. Finally, plotting this equation on a graph, with θ on one axis and time on the other, and keeping the values of the other variables fixed, we have our answer:

When the value of θ is 0º, the bob is pointing straight up. When θ = 90º, the bob is pointing sideways and continues to fall down, to become a conventional pendulum, under the influence of gravity. But when the frequency is increased from 10 arbitrary units in this case to 200 units, the setup becomes a Kapitza pendulum as the value of θ keeps shifting but only between 6º on one side and some 3º on the other.

The thing I’m curious about here is whether mathematics is purely descriptive or if it’s real in the way a book, a chair or a planet is real. Either way, this ‘problem’ should remind us of the place and importance of mathematics in modern life – by virtue of the fact that it opens paths to understanding, and then building on, parts of reality that experiences based on our senses alone can’t access.

Featured image: A portrait of Pyotr Kapitsa (left) in conversation with the chemist Nikolai Semyonov, by Boris Kustodiev, 1921. Credit: Kapitsa Collection, public domain.