The Beauty of Laplace’s Equation, Mathematical Key to … Everything
Physics has its own Rosetta Stones. They’re ciphers, used to translate seemingly disparate regimes of the universe. They tie pure math to any branch of physics your heart might desire. And this is one of them:
It’s in electricity. It’s in magnetism. It’s in fluid mechanics. It’s in gravity. It’s in heat. It’s in soap films. It’s called Laplace’s equation. It’s everywhere.
Laplace’s equation is named for Pierre-Simon Laplace, a French mathematician prolific enough to get a Wikipedia page with several eponymous entries. In 1799, he proved that the the solar system was stable over astronomical timescales—contrary to what Newton had thought a century earlier. In the course of proving Newton wrong, Laplace investigated the equation that bears his name.
It has just five symbols. There’s an upside-down triangle called a nabla that’s being squared, the squiggly Greek letter phi (other people use psi or V or even an A with an arrow above it), an equals sign, and a zero. And with just those five symbols, Laplace read the universe.
Phi is the thing you’re interested in. It’s usually a potential (something physics majors confidently pretend to understand), but it can be plenty of other things. For now, though, let’s say that it represents the height above sea level of every point on a landscape. On a hilltop, phi is large. In a valley, it’s low. The nabla-squared is a set of operations collectively called the Laplacian, which measures the balance between increasing and decreasing values of phi (heights) as you move around the landscape.
From the top of a hill, you descend no matter which direction you walk. This is what makes it the top of the hill, but it also makes the Laplacian negative: the going-down options entirely outweigh the going-up. It’s positive in a valley for the same reason: you can’t go anywhere but up. Somewhere between these two, there’s going to be a place where a step can take you uphill as much as it can down. At that point, where up and down are exactly balanced, the Laplacian is zero.
In Laplace’s equation, the Laplacian is zero everywhere on the landscape. That has two related consequences. First, from anywhere on the land, you have to be able to go up as much as you can go down. Second, the highest and lowest values of phi are restricted to the edges of the landscape. This is simply a result of the first part: If there’s any variation in phi, it has to happen before the crest of the hill or the trough of the valley. So you have to stop looking where land starts to level out.
Real places are too bumpy to satisfy Laplace’s equation. But soap is more cooperative. Dunk a contorted wire hanger into soapy water and you’ll notice that the film doesn’t have any bumps. Play around a bit and you’ll see that you can never position the hanger so that the soap seems to go higher than the hanger’s highest point or lower than its lowest point. From any perspective, the highest and lowest parts are on the wire boundaries.
The shape of that film is caused by surface tension. But it’s perfectly described and predicted by Laplace’s equation—reminder, an equation that he studied because it described the solar system.
Or imagine a charged piece of metal out in empty space. Usually, space has no voltage, but in this case the space very close to the metal is going to have a voltage very similar to the metal itself. Far away, the voltage will be small—but only infinitely far away will it truly be zero. As you move away from the metal, there won’t be any sharp peaks or troughs because no other charges are around to cause voltage spikes, so the voltage will gradually drop off.
And that brings us back to Laplace. To find the voltage anywhere in space due to this piece of metal, you just need to solve Laplace’s equation.
Actually, no you don’t. That’s the beauty of the Rosetta Stones of physics: When you solve Laplace’s equation for soap films, you only specify anything about wire hangers at the last step. Everything before that is completely independent of the soap, so it’s perfectly applicable here to the voltage. You don’t need to change a thing.
That same solution can be applied all over the place, and all you ever need to do is change the last step. Gravity is large at a mass and asymptotically approaches zero—and you’re back to Laplace. Water’s velocity is zero where something’s in its way and unperturbed far away—and you’re back to Laplace. The head of a drum tightly fits its rim and the surface tension keeps it taut and flat—and you’re back to Laplace. So it goes throughout the universe, through classes and research alike. Laplace pops up wherever you look, and you only ever have to solve it the one time.
Until someone decides to hit the drum, as people are wont to do. But that’s a perturbation for another time.
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