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Could God make atoms so flawed that they could break?

Could God make atoms so perfect that they would defy His power to break them? It was only one of the difficulties thrown up by God’s omnipotence, even before relativity placed a precise upper limit on velocity and before quantum mechanics placed a precise upper limit on certainty. The natural philosophers wished to affirm the presence and power of God in every corner of the universe.

Yet even more fervently they wished to expose the mechanisms by which planets swerved, bodies fel , and projectiles recoiled in the absence of any divine intervention. No wonder Descartes appended a blanket disclaimer: “At the same time, recal ing my insignificance, I affirm nothing, but submit al these opinions to the authority of the Catholic Church, and to the judgment of the more sage; and I wish no one to believe anything I have written, unless he is personal y persuaded by the evidence of reason.”

The more competently science performed, the less it needed God. There was no special providence in the fal of a sparrow; just Newton’s second law, f = ma . Forces, masses, and acceleration were the same everywhere. The Newtonian apple fel from its tree as mechanistical y and predictably as the moon fel around the Newtonian earth.

Why does the moon fol ow its curved path? Because its path is the sum of al the tiny paths it takes in successive instants of time; and because at each instant its forward motion is deflected, like the apple, toward the earth. God need not choose the path. Or, having chosen once, in creating a universe with such laws, He need not choose

again. A God that does not intervene is a God receding into a distant, harmless background.

Yet even as the eighteenth-century philosopher scientists learned to compute the paths of planets and projectiles by Newton’s methods, a French geometer and philosophe , Pierre-Louis Moreau de Maupertuis, discovered a strangely magical new way of seeing such paths. In Maupertuis’s scheme a planet’s path has a logic that cannot be seen from the vantage point of someone merely adding and subtracting the forces at work instant by instant.

He and his successors, and especial y Joseph Louis Lagrange, showed that the paths of moving objects are always, in a special sense, the most economical. They are the paths that minimize a quantity cal ed action —a quantity based on the object’s velocity, its mass, and the space it traverses. No matter what forces are at work, a planet somehow chooses the cheapest, the simplest, the best of al possible paths. It is as if God—a parsimonious God—

were after al leaving his stamp.

None of which mattered to Feynman when he encountered Lagrange’s method in the form of a computational shortcut in Introduction to Theoretical Physics. Al he knew was that he did not like it. To his friend Welton and to the rest of the class the Lagrange formulation seemed elegant and useful. It let them disregard many of the forces acting in a problem and cut straight through to an answer. It served especial y wel in freeing them from the right-angle coordinate geometry of the classical reference frame required by Newton’s equations. Any reference

frame required by Newton’s equations. Any reference frame would do for the Lagrangian 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 al the forces. The problems got harder and harder as the class advanced through classical mechanics. Bal s rol ed 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 bal 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 bal can have taken only one path.

Bader told Feynman to make two familiar calculations of the bal ’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 al high-school physics students Feynman was used to adding those energies together. An airplane, accelerating as it dives, or a rol er coaster, sliding down the gravity wel , trades its potential energy for kinetic energy: as it loses height it gains speed. On the way back up, friction aside, the airplane or rol er coaster makes the same conversion in

the airplane or rol er 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 bal ’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 fal , but when the bal arrived at its destination, the path it had fol owed 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 bal seems to choose its path. It seems to know al 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 bil iard bal s 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 bal or a planet or a ray of light makes its way instant by instant, not that it fol ows a preordained path. From the Lagrangian point of view the forces that pul and shape a bal ’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 supreme Being, but in phenomena whose universality suffers no exception and whose simplicity lays them quite open to our sight.” The universe wil s simplicity.

Newton’s laws provide the mechanics; the principle of least action ensures grace.

The hard question remained. (In fact, it would remain, disquieting the few physicists who continued to ponder it, until Feynman, having long since overcome his aversion to the principle of least action, found the answer in quantum mechanics.) Park phrased the question simply: How does the bal know which path to choose?