Problems you can solve just by looking at them: The meaning of Noether’s Theorem

I am grateful for Emmy Noether. 

Emmy Noether was an extremely prolific (and fairly heroic) German mathematician from the early 20th century; Einstein once called her the most important woman in the history of mathematics.  (She is also the subject of two hilariously bad “biographies”: one for children and one for young adults).

The great majority of Noether’s contributions to mathematics and to physics are well beyond my ability to understand, but the one I can appreciate is “Noether’s Theorem.”  Noether’s Theorem is an idea that has has made my life in physics significantly easier, because it allows one to turn difficult “dynamics” problems into simple “accounting” problems.

Let me explain what I mean.  Consider the following intro-level physics problem: You throw a rock vertically upward with initial speed 10 m/s.  What is the maximum height reached by the rock?

Isaac Newton taught us how to solve this problem:

While in flight, the force of gravity acts on the rock and dictates how the rock’s velocity is changing at any given moment.  Using the mathematical description of the gravitational force gives you a (very simple) differential equation for the rock’s position as a function of time.  You can solve this differential equation using the known initial conditions to find a full description of the rock’s trajectory over time.  Now you find the trajectory’s maximum height (perhaps by taking a derivative and setting it equal to zero) and you have your answer.

This is what I call a “dynamics” problem: you have a situation where you know the various forces at work and how they affect the trajectory of an object at any given point in space, so you use it to calculate the trajectory.  While the “throwing a rock into the air” problem is easy for anyone who’s taken a class or two in physics, such dynamics problems can get notoriously difficult very quickly (say, when you have more than one force at work or more than one interacting object in motion).

Of course, there is a much faster way of solving the “rock in the air” problem.  Namely, you make use of energy conservation, which says that the total energy of the rock is identical at every moment in time.  This allows you to figure out the maximum height by equating the energy at the bottom of the trajectory (purely kinetic) to the energy at the top of the trajectory (purely potential).  The answer is meters.

When I was a TA, I tried to teach my students that using energy conservation is like accounting.  You know how much energy you started with and you need to keep track of where all the energy is going.  Energy doesn’t disappear; it only moves around from one place to another, just like money that can get saved or spent in one way or another.  If you know where an object has “spent” its energy you can figure out nearly every important fact about it.

These “accounting” approaches are pretty powerful in physics: they allow you to quickly figure things out by keeping track of a conserved quantity.  And there are a number of different quantities that can be worth “accounting” for: energy, momentum, and angular momentum, to name a few.

The problem is that the momentum and angular momentum of a particular object are not necessarily conserved — that depends on what kind of environment the object is moving through and what kind of forces are present.  For example, if you throw a bouncy ball down a long, frictionless hallway then the forward momentum of the ball is conserved but the vertical momentum is not (because of gravity) and neither is the lateral momentum (because the ball can bounce off the walls).

So when you begin to solve a physics problem, one of the first and most important questions to answer is this: When I have an object moving through a given environment, what quantities are conserved?

Noether’s Theorem gives an answer to this question.  What’s more, it provides a way to identify other conserved quantities that you might not even have thought to look for.  And the theorem is so simple that you can usually figure out the conserved quantities just by drawing a picture.

Noether’s Theorem and what it looks like

Noether’s Theorem can be stated this way:

For every continuous symmetry that an environment has, there is a corresponding conserved quantity.

The theorem then gives a simple recipe for calculating what these conserved quantities are, which I’ll discuss in a bit.  But first I should give you a sense of what it means to have a “continuous symmetry”.

Imagine, as an example, that you are trying to describe the behavior of one (or multiple) objects in the vicinity of a very long force-emitting cylinder.  Something like this:

I’m not going to tell you what kind of force the cylinder is emitting because it doesn’t matter — it could be gravitational attraction, or electric repulsion, or nuclear radiation, or simply contact force associated with hitting the cylinder’s hard walls.

Without knowing what kind of force is being emitted, we apparently don’t know much about how the objects around the cylinder will behave.  Will they get stuck to the cylinder, or will they be flung away, or will they orbit it, or what?  What we can say immediately though, just after drawing the picture, is that the environment in which the objects are moving has two important symmetries.  The first is symmetry with respect to motion along the axis.  If the cylinder is infinitely long, then no matter where you stand along the axis the system will look the same.  The second symmetry is with respect to rotations along the axis: you can spin everything around the x axis and nothing will change.

Noether’s Theorem guarantees that for each of these symmetries there is a conserved quantity.  In this case, the two conserved quantities are the total momentum in the direction (which is related to the translational symmetry) and the total angular momentum around the axis (which comes from the rotational symmetry).  As a general rule, translational symmetries always produce conserved linear momentum and rotational symmetries produce conserved angular momentum.

More exactly, Noether’s theorem says that if you can continuously change some coordinate variable without changing the environment, then there is a conserved quantity equal to

,

where is the kinetic energy and is the time rate of change of .  As an example, the kinetic energy can be written , so that if the system is unchanged by translations along the direction, then the conserved quantity is , which is the momentum in the direction.

Noether’s Theorem also allows you to identify less obvious conserved quantities.  For example, imagine that the force-emitting object is a cylinder with a helical coil wrapped around it, like this:

This environment is no longer unchanged by small translations in the or directions, nor by small rotations around any of the axes.  It is, however, unchanged by a particular combination of translation and rotation.  Specifically, if is the distance between coils of the helix then the environment is unchanged when you simultaneously rotate 360 degrees around the axis and translate by in the direction.  Any small translation/rotation done in that same proportion also leaves the environment unchanged.  Noether’s Theorem therefore guarantees that a particular combination of linear momentum and angular momentum will be conserved forever.  Specifically, you can work out from the equation above that is conserved, where is the angular momentum around the axis and is the linear momentum.

Probably the most profound insight of Noether’s Theorem comes from its view of the principle of energy conservation itself.  Energy conservation appears naturally from Noether’s Theorem when you assume that the environment is symmetric with respect to translations in time.  That is, saying that energy is conserved is equivalent to saying that the laws of physics are unchanging in time.

I never cease to be amazed how mathematics can guide our way of thinking philosophically about the universe.

In the end, I really haven’t done justice to Noether’s Theorem, which has tremendous consequences in field theory and pure mathematics as well in as the normal mechanics of particles.  But this simplified version of the theorem is enough to make me grateful, because it allows me to solve hard problems just by drawing pictures.

Now if only I could get someone to write a decent biography of Emmy Noether.