Least Action

A Score for Every Path

Throw a ball to a friend. The usual explanation of its arc is a force: gravity pulls down, so the path curves. Here is the other one. Picture any path the ball might take from your hand to your friend’s, leaving and arriving at the same moments as the real throw: a lower curve, a higher one, a lopsided one. At each moment along it the ball has energy of motion and energy of height. Take the first minus the second, and add that up over the whole flight. The total is called the action, and every possible path has its own.

The path the ball really takes has the smallest action of all. A higher path keeps the ball up where its energy of height is large, which lowers the total. But to get up there and back in the same time it must move faster, which raises it. Start from a flat path and lift it: the gain grows in step with the lift, and the cost of the extra speed grows with its square. A little lift pays and too much does not, and the real arc sits where the two balance. Change it a little, in any way at all, and the total does not change at first; it only starts to grow. The real path sits where the ground is flat.

That is the whole rule, and it is called the principle of least action. Work it through, and out comes the exact arc of the throw, the same one Newton’s laws give. The two say the same thing in different words.

Why motion minus height, and not the two added, as in the total energy? For gravity, clocks give the deepest answer. Of all the paths between your hand and your friend’s, the ball takes the one on which its own clock ticks the most: higher up a clock runs faster, and speed slows it. For anything much slower than light, the ticks lost to speed follow the energy of motion, and the ticks gained by height follow the energy of height. The action is that count turned around, so least action is most ticking.

The rule can seem to need the ball to know where it will land. It does not. Newton’s way starts from where the ball is and how fast it moves, and runs forward. The action’s way starts from both ends of the trip and asks which path between them is flat. Two ends, each at its moment, fix a throw as well as a start, a speed and a direction do, just as two points fix a line as well as one point and a direction. The rule did not foresee where the ball lands. You told it.

Stationary, Not Least

For a short trip, “least” is always right. For a long one it can fail, when a force keeps pulling the moving thing back, as a pendulum’s does. Take a pendulum hanging still, and ask for its path from the bottom back to the bottom, some time later. Its real path is to stay where it is, and its score is zero: no motion, no height. A rival swings out and comes back in the same time. On a short trip it must swing fast, and its speed costs more than its height takes off. Given longer, it can swing out slowly, paying little for speed, while its time up the side, where height lowers the score, keeps adding up. Once the trip lasts longer than a free swing out and back to the bottom, about a second for a pendulum a meter long, the rival scores below zero. Staying still is no longer the cheapest path. It is still the real one.

The real path was never picked for being lowest. It is picked for being flat. For a long trip, the flat place is a mountain pass: downhill one way, uphill another, level right at the top. Each time the trip grows by another such swing, one more way down opens. “Least action” is the name that stuck; “stationary action” is the exact one.

Light keeps the same rule. Inside a curved mirror, the route light takes can even be the slowest of its near neighbors. The sky shows it on a grand scale. Near a cluster of galaxies time runs slower and space is stretched, and together they hold passing light back a little, much as water does. A route round the cluster can then be longer and still quicker, so one object far behind can appear several times, each image carried along a different route. Each route takes a stationary time: some are the quickest routes near them, and others sit at a pass. Their times differ, so the images arrive at different times.

Astronomers saw four images of an exploding star round one galaxy of a cluster. They arrived within a few weeks of each other, and the star stayed bright for months, so all four were seen together. Models of the cluster said a fifth image would appear about a year later, somewhere else. It did, on schedule, 376 days after the first of the four, and that delay has since been used to measure how fast universe is expanding.

i Five images of one exploding star, each by its own route ×

One Rule for All of Physics

The rule does not stop at balls and beams, though each theory writes its own score. Write the action of a planet round its star, make it stationary, and the orbits come out. For electric and magnetic fields, a path is the field’s whole story, its value at every point from one moment to the next, and the action adds up a share from every bit of space and every moment. Make it stationary, and out come the laws of electricity, magnetism and light. Write the action of the shape of space and time itself, and out comes Einstein’s theory of gravity. The Standard Model, the theory of every known particle and every force but gravity, is written down as one action.

i Orbits, light, bent spacetime: each from a stationary action ×

For a ball or a planet the action predicts nothing that Newton’s laws do not: they are the same physics. It still saves work. Describe a double pendulum or a robot arm by whatever angles suit it, write its energies in those, and the forces in the rods and joints that only hold the parts together never have to be found. What falls outside the simple rule is motion whose energy drains into the jiggle of atoms, as it does through friction and drag. A forced motion still has an action, even when the force changes with time, like a parent pushing a swing. A block sliding to a stop, its motion turning into warmth in the block and the floor, has no action of the simple kind; the usual way to write one brings in the atoms that take up that energy.

One action for every theory changes what a law of physics is. A new theory is usually proposed today as an action, not as a list of forces. One reason is what an action shows at a glance: its symmetries. Emmy Noether proved that every smooth symmetry of an action comes with a quantity that never changes. If the action is the same today as tomorrow, energy is kept. If it is the same here as there, momentum is kept. If it is the same facing any way, angular momentum, the amount of turning motion, is kept.

How Does Light Know?

Back to the question least time raised: does light compare routes before it sets off? It does not. Light travels as a wave, and its waves spread along every route to the target at once. What reaches the target is all of them added together. Every route’s wave rises and falls at the same rate; what differs is when its crests arrive, and that is set by its travel time.

Near the quickest route, the travel time hardly changes from one route to the next. A whole band of neighboring routes arrives in step, crest with crest, and they add up. Far from it, the time changes fast from route to route, so neighbors arrive out of step, crest against trough, and cancel. The light that arrives has come along the quickest route and its close neighbors. That is also why flat is what counts, not lowest: at a pass, too, the time is level, and neighbors keep step.

Take away the routes that cancel, and light goes where least time says it should not. A mirror with fine strips of its silver removed sends light off at angles a whole mirror never would, split into colors: the rainbow on the underside of a video disc comes the same way, from its fine, evenly spaced tracks. Physicists have even measured what looped routes add, in through one slit, round to another and out again. Behind three slits cut in a thin metal film, counting those routes changes the stripes slightly. Light turned into ripples of the metal’s electrons, running along its surface between the slits, made the change large enough to see, and it was there.

Matter does the same. In quantum theory a ball, too, is worked out by adding up every path it could take, each path’s wave turned by its action. Where the action is flat, neighboring paths agree, and elsewhere they cancel. For a ball the band of paths that agree is far too narrow to see, so it seems to follow one exact arc. Least action is what that sum over every path looks like for anything as big and heavy as a ball.