Episode 08 - Least action: Does nature optimise?

Tomlin Hawthorne

You're a lifeguard, standing on the sand, a little way back from the water. Out in the surf, well off to your right, a swimmer is in trouble. You can run much faster than you can swim. Every second counts. What route do you take? Decide. Somebody's drowning.

Tomlin Hawthorne

The straight line is wrong. It's the shortest path, but it has you swimming, slowly, for far too long. Running along the beach until you're directly opposite is wrong too. You've cut the swim to almost nothing and added a great deal of sand. The quickest route sits in between. You run at an angle, then turn at the water's edge and swim a shorter stretch. Your path has a bend in it, right at the shoreline. Now here's the peculiar part. A ray of light going from air into water takes exactly that kind of bend. And today's question is how on earth it knows to.

Tomlin Hawthorne

Last time, a figure skater pulled in her arms and spun faster, and we saw Emmy Noether tie that to the universe having no preferred direction. The same logic, no preferred moment in time, guarantees that energy is conserved. Today we keep those same energy accounts, kinetic and potential, and use them to tell the story of motion in a completely different way.

Tomlin Hawthorne

In 1662, Pierre de Fermat, a lawyer in Toulouse who did mathematics for the love of it, explained the bend. Light travels more slowly in water than in air. Nature, he said, doesn't minimise distance. It minimises time. So light covers extra ground in the air, where it's quick, and turns at the surface. That one rule gives you the bent straw in a glass of water. It explains how a lens focuses. It even explains the shimmering patch of water on a hot road. The air just above hot asphalt is thin, and light moves faster in thin air. So light from the sky finds it quicker to dip down through that fast layer and curve up into your eye. You're looking at sky that took the scenic route, because the scenic route was faster.

Tomlin Hawthorne

But feel the unease here. The lifeguard can see the swimmer. She can weigh up the routes. Light leaves the lamp with no idea where it will end up. How does it choose the fastest way to a destination it hasn't reached? It looks like foresight. And that is not how Newton's physics works. In Newton's picture, the forces acting right now decide the next instant. Nothing looks ahead.

Tomlin Hawthorne

The man who turned this hint into a principle was Pierre Louis Moreau de Maupertuis. Born 1698. Clever, vain, and physically brave. In 1736 and 1737 he led an expedition into Lapland, inside the Arctic Circle, to measure the curve of the Earth along a frozen river. Newton said a spinning planet should bulge at the equator and flatten at the poles. The leading French view said it should stretch toward the poles, like a lemon. Maupertuis's measurements backed Newton, and he came home a celebrity. Frederick the Great made him president of the Academy in Berlin, and in 1746 he published his principle of least action. Action, for him, was mass times speed times distance, and nature, he said, always makes it as small as possible. Then he went further. This economy, he claimed, was proof of a wise God.

Tomlin Hawthorne

In 1751 the mathematician Samuel König objected. The principle, he said, doesn't always give a minimum, and Leibniz had thought of it earlier, in a letter to Jakob Hermann. König only had a copy. The Academy, with Euler presiding, declared the letter a forgery. Voltaire, then at Frederick's court, couldn't let it pass. In 1752 he wrote the Diatribe of Doctor Akakia, and he mostly just quoted Maupertuis's own proposals back at him. Dig a hole to the centre of the Earth. Dissect Patagonian giants to study the mind. Found a city where only Latin is spoken. Frederick had the pamphlet burned by the public executioner. Voltaire left Prussia. Maupertuis died in Basel in 1759, the same year Voltaire published Candide, in which Doctor Pangloss insists that all is for the best in the best of all possible worlds.

Tomlin Hawthorne

Here's the awkward part. Voltaire won the argument and lost the science. König was right that it isn't always a minimum, and the modern version, Hamilton's, is stated differently. Take any imaginable path for a thrown ball, from your hand to the far wall. At every moment, take the kinetic energy and subtract the potential energy, and add that difference up over the whole journey. That total is the action. The path the ball really takes is the one where the action is stationary. Nudge the route slightly and the total doesn't change at first order, like the bottom of a valley or the top of a pass. Keep only that statement, and you can recover what Newton derived.

Tomlin Hawthorne

So now there are two complete descriptions of one world. Newton's is a chain: what's happening now causes what happens next. Hamilton's is a shape seen whole: of all the journeys there could have been, this one has a certain property. They always agree. Think of describing a novel sentence by sentence, or describing its shape, a tragedy bending toward its end. Both are accurate. But the second feels as though the ending is present from the first page. Ted Chiang built a story on exactly that feeling. In Story of Your Life, the basis for the film Arrival, a linguist meets aliens whose language and perception take in a whole path at once, the way Fermat's principle does. For them, time isn't a sequence. I won't say what it does to her. Read it. It's short, and it will teach you real physics and then quietly break your heart.

Tomlin Hawthorne

So how does the light know? I'll give you a promissory note, payable in episode thirty six, from Richard Feynman. In a sense we'll have to earn, light takes every path. The straight ones, the bent ones, the absurd ones via the Moon. Nearly all of them cancel each other out, like ripples meeting out of step. What survives is the bunch of paths around the stationary one. So there's no foresight and no scouting. The purpose was in the description, not in the light. Pangloss saw design. On the physics, at least, Voltaire was the better reader.

Tomlin Hawthorne

Your challenge is small. Put a coin in the bottom of an opaque mug. Step back and lower your head until the rim just hides the coin, and stay exactly there. Have someone pour water in gently. The coin rises into view. Neither you nor the coin has moved. The light is bending at the surface, taking the quickest route over the rim and into your eye. That's Fermat, in your kitchen, with small change. And the question to carry away is this. Two descriptions, one a chain and one a shape, agree on every fact. Is one of them what's really going on? Or is that a question with no meaning? Try asking it of a life, too.