Why energy is conserved

Symmetry

Energy is conserved. But why? Recall the joke from Course 3: the world was surely made from one sheet of paper with a few simple rules on it, not from a stack of textbooks. Is conservation of energy one of the rules on that sheet? Physicists studied many systems, and found that conservation is tied to something deeper, called symmetry.

Start with a circle. Draw any line through its center, and the circle is symmetric about that line, whatever direction you choose. Now spin the circle around its center. If this picture were an animation, could you tell whether the circle was turning? You couldn’t. It looks the same at every angle. That, too, is a symmetry.

Now cut a small notch in the rim. Most of those symmetries vanish: turn the circle, and the notch gives the rotation away. Or draw two diameters at right angles inside it. Now the circle looks the same only when turned by 90 degrees, or 180, or 270. It still has some symmetry, but less than before. Adding detail lowers symmetry.

In physics, symmetry means the same thing: some operation leaves things unchanged. Slide a cart from one place to another. If nothing about its behavior changes, the laws are symmetric under that shift. Run Galileo’s ramp experiment today and again tomorrow. If you get the same result, distance growing with the square of the time, the law is symmetric in time. Run it at home, in a lab, or on a mountaintop, and get the same result again: the law is symmetric in space.

Noether’s theorem

Studying system after system, physicists noticed a pattern. Wherever a law was symmetric in time, energy was conserved. Symmetry and conservation seemed to be two faces of something deeper. The person who proved it was a mathematician.

In 1918 Emmy Noether proved what physicists now call Noether’s theorem: every continuous symmetry of a physical system corresponds to a quantity that is conserved. The pairings are remarkable:

• If the laws are the same today as tomorrow, energy is conserved.
• If they are the same here as there, momentum is conserved.
• If they are the same in every direction, angular momentum, the quantity that keeps a spinning skater turning, is conserved.

So energy is conserved because the laws of physics don’t change with time. When Noether died, Einstein wrote a tribute to her in the New York Times:

Pure mathematics is, in its way, the poetry of logical ideas.

Albert Einstein, letter to the New York Times, 1935

Symmetry first

This changed how physics works at its frontier. Remember C. N. Yang’s sketch from Course 4: experiments lead to a rough theory, which mathematics turns into a framework. Conservation laws and symmetries were found at the end of that road, hidden inside frameworks built from experiments. But at the edge of physics, many experiments are impossible. Nobody can put a black hole in a laboratory.

So physicists turned the process around. Faced with a new system they don’t understand, they first ask what symmetries it might have, since every symmetry brings a conservation law. Electric charge is conserved, but which symmetry does that correspond to? Working backward from that question led to what physicists call gauge symmetry, and in 1954 Yang and Robert Mills extended it into a theory that now underlies much of particle physics. Yang called the new approach “symmetry dictates interaction”: physicists no longer start from experiments to reach symmetry; they start from symmetry and build the equations it allows. In this reading, the first principles of physics have moved deeper, from Newton’s and Maxwell’s equations to symmetry and conservation themselves. The difficulty is that you often have to guess which symmetry a system has. Sometimes you guess right, and sometimes you don’t.

The path of least action

There is another way to reach the same deep level, starting with a lifeguard. Someone is drowning offshore, and the lifeguard is on the sand some way down the beach. Should she run straight at the swimmer? She runs much faster than she swims, so the straight line isn’t fastest. She should run farther along the sand and cut her time in the water. What matters is the total time, not the distance.

In 1662 the French mathematician Pierre de Fermat proposed that light always takes the path of least time. That single principle explains why light bends when it enters water. There is an everyday version. Look up a route across a big city, and a maps app asks whether you want the fastest route, the shortest route, or the one with fewest transfers. Each is “least” of something, and you have to know which quantity matters for your problem.

The fastest slide

Now a puzzle about Galileo’s ramps. A ball rolls from a high point to a lower point off to one side. Which track gets it there fastest: a straight ramp, a gentle curve, or a deep curve? Galileo considered the question in 1638. In 1696 the Swiss mathematician Johann Bernoulli challenged Europe’s best minds to solve it. Newton, Leibniz, and Jakob Bernoulli all sent solutions. Newton’s arrived anonymously, and Bernoulli is said to have recognized its author “as the lion by its claw.” The answer is a curve called the cycloid, steeper than a straight ramp at first, so the ball picks up speed early and spends less time going slowly.

Every ball ends at the same speed, since each falls the same height and turns the same potential energy into kinetic energy. What differs is how they spend their time. The winning curve does its slow part quickly and its fast part at length. In the eighteenth and nineteenth centuries mathematicians generalized this into the principle of least action. For every possible path, add up a quantity called the action, roughly kinetic energy minus potential energy, accumulated over time. The path nature actually takes is the one that makes the action least. Nature, it seems, always acts in the most economical way it can.

A law of laws

Noether’s theorem ties the threads together. Every system follows the principle of least action. The form of its action is shaped by its symmetries. And every symmetry comes with a conservation law. Least action, symmetry, and conservation form a triangle, which this course calls a “law of laws.” Ordinary laws are enough for ordinary problems like a planet’s orbit, a cart, or a pot of water. At the frontier, physicists need these deeper tools.

If physics seems like a mountain of books, that just means you haven’t reached the top yet. Up there you find the single sheet of paper. How well you can use it depends on your own paradigm. The lesson closes with a line Einstein is reported to have said to a student:

I want to know how God created this world. I am not interested in this or that phenomenon. I want to know His thoughts; the rest are details.

Albert Einstein, as recalled by Esther Salaman, 1955