technique. Feynman refused to employ it. He said he would not feel he understood the real physics of a system until he had painstakingly isolated and calculated all the forces. The problems got harder and harder as the class advanced through classical mechanics. Balls rolled down inclines, spun in paraboloids—Feynman would resort to ingenious computational tricks like the ones he learned in his mathematics-team days, instead of the seemingly blind, surefire Lagrangian method.
Feynman had first come on the principle of least action in Far Rockaway, after a bored hour of high-school physics, when his teacher, Abram Bader, took him aside. Bader drew a curve on the blackboard, the roughly parabolic shape a ball would take if someone threw it up to a friend at a second-floor window. If the time for the journey can vary, there are infinitely many such paths, from a high, slow lob to a nearly straight, fast trajectory. But if you know how long the journey took, the ball can have taken only one path. Bader told Feynman to make two familiar calculations of the ball’s energy: its kinetic energy, the energy of its motion, and its potential energy, the energy it possesses by virtue of its presence high in a gravitational field. Like all high-school physics students Feynman was used to adding those energies together. An airplane, accelerating as it dives, or a roller coaster, sliding down the gravity well, trades its potential energy for kinetic energy: as it loses height it gains speed. On the way back up, friction aside, the airplane or roller coaster makes the same conversion in reverse: kinetic energy becomes potential energy again. Either way, the total of kinetic and potential energy never changes. The total energy is conserved.
Bader asked Feynman to consider a less intuitive quantity than the sum of these energies: their difference. Subtracting the potential energy from the kinetic energy was as easy as adding them. It was just a matter of changing signs. But understanding the physical meaning was harder. Far from being conserved, this quantity—the action , Bader said—changed constantly. Bader had Feynman calculate it for the ball’s entire flight to the window. And he pointed out what seemed to Feynman a miracle. At any particular moment the action might rise or fall, but when the ball arrived at its destination, the path it had followed would always be the path for which the total action was least. For any other path Feynman might try drawing on the blackboard—a straight line from the ground to the window, a higher-arcing trajectory, or a trajectory that deviated however slightly from the fated path—he would find a greater average difference between kinetic and potential energy.
It is almost impossible for a physicist to talk about the principle of least action without inadvertently imputing some kind of volition to the projectile. The ball seems to choose its path. It seems to know all the possibilities in advance. The natural philosophers started encountering similar minimum principles throughout science. Lagrange himself offered a program for computing planetary orbits. The behavior of billiard balls crashing against each other seemed to minimize action. So did weights swung on a lever. So, in a different way, did light rays bent by water or glass. Fermat, in plucking his principle of least time from a pristine mathematical landscape, had found the same law of nature.
Where Newton’s methods left scientists with a feeling of comprehension, minimum principles left a sense of mystery. “This is not quite the way one thinks in dynamics,” the physicist David Park has noted. One likes to think that a ball or a planet or a ray of light makes its way instant by instant, not that it follows a preordained path. From the Lagrangian point of view the forces that pull and shape a ball’s arc into a gentle parabola serve a higher law. Maupertuis wrote, “It is not in the little details … that we must look for the
Feynman had first come on the principle of least action in Far Rockaway, after a bored hour of high-school physics, when his teacher, Abram Bader, took him aside. Bader drew a curve on the blackboard, the roughly parabolic shape a ball would take if someone threw it up to a friend at a second-floor window. If the time for the journey can vary, there are infinitely many such paths, from a high, slow lob to a nearly straight, fast trajectory. But if you know how long the journey took, the ball can have taken only one path. Bader told Feynman to make two familiar calculations of the ball’s energy: its kinetic energy, the energy of its motion, and its potential energy, the energy it possesses by virtue of its presence high in a gravitational field. Like all high-school physics students Feynman was used to adding those energies together. An airplane, accelerating as it dives, or a roller coaster, sliding down the gravity well, trades its potential energy for kinetic energy: as it loses height it gains speed. On the way back up, friction aside, the airplane or roller coaster makes the same conversion in reverse: kinetic energy becomes potential energy again. Either way, the total of kinetic and potential energy never changes. The total energy is conserved.
Bader asked Feynman to consider a less intuitive quantity than the sum of these energies: their difference. Subtracting the potential energy from the kinetic energy was as easy as adding them. It was just a matter of changing signs. But understanding the physical meaning was harder. Far from being conserved, this quantity—the action , Bader said—changed constantly. Bader had Feynman calculate it for the ball’s entire flight to the window. And he pointed out what seemed to Feynman a miracle. At any particular moment the action might rise or fall, but when the ball arrived at its destination, the path it had followed would always be the path for which the total action was least. For any other path Feynman might try drawing on the blackboard—a straight line from the ground to the window, a higher-arcing trajectory, or a trajectory that deviated however slightly from the fated path—he would find a greater average difference between kinetic and potential energy.
It is almost impossible for a physicist to talk about the principle of least action without inadvertently imputing some kind of volition to the projectile. The ball seems to choose its path. It seems to know all the possibilities in advance. The natural philosophers started encountering similar minimum principles throughout science. Lagrange himself offered a program for computing planetary orbits. The behavior of billiard balls crashing against each other seemed to minimize action. So did weights swung on a lever. So, in a different way, did light rays bent by water or glass. Fermat, in plucking his principle of least time from a pristine mathematical landscape, had found the same law of nature.
Where Newton’s methods left scientists with a feeling of comprehension, minimum principles left a sense of mystery. “This is not quite the way one thinks in dynamics,” the physicist David Park has noted. One likes to think that a ball or a planet or a ray of light makes its way instant by instant, not that it follows a preordained path. From the Lagrangian point of view the forces that pull and shape a ball’s arc into a gentle parabola serve a higher law. Maupertuis wrote, “It is not in the little details … that we must look for the
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