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 $h_{max} = \frac{1}{2} v^2/ g \approx 5.1$ 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.

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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 $x$ axis.  If the cylinder is infinitely long, then no matter where you stand along the $x$ axis the system will look the same.  The second symmetry is with respect to rotations along the $x$ 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 $x$ direction (which is related to the translational symmetry) and the total angular momentum around the $x$ 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 $\xi$ without changing the environment, then there is a conserved quantity $P_{cons}$ equal to

$P_{cons} = \partial(K.E.)/\partial \dot{\xi}$,

where $K.E.$ is the kinetic energy and $\dot{\xi}$ is the time rate of change of $\xi$.  As an example, the kinetic energy can be written $\frac{1}{2} m ( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 )$, so that if the system is unchanged by translations along the $x$ direction, then the conserved quantity is $P_{cons} = m \dot{x} = m v_x$, which is the momentum in the $x$ direction.

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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 $x, y$ or $z$ directions, nor by small rotations around any of the axes.  It is, however, unchanged by a particular combination of translation and rotation.  Specifically, if $d$ is the distance between coils of the helix then the environment is unchanged when you simultaneously rotate 360 degrees around the $x$ axis and translate by $d$ in the $x$ 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 $L_x + \frac{d}{2\pi} P_x$ is conserved, where $L_x$ is the angular momentum around the $x$ axis and $P_x$ 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.

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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.

January 20, 2011 10:59 pm

Noether’s theorem is a great topic, so it’s always fun to see blog posts describing its virtues, just be a little careful with the “symmetries => conserved quantity” statement, as it obviously has caveats. Check out Bad Physics for comments on Noether’s 1st, and why it’s not *quite* that straightforward (but still simple).

Love the blog.

January 20, 2011 11:11 pm

Thanks for the warning! The huge caveat, as you pointed out, is that the forces involved have to be conservative . This means that there can’t be any energy-dissipating forces involved, like friction.

And of course you shouldn’t go crazy with liberal uses of the terms “symmetry” and “conservation”, as in the hilarious “Journal of Public Relations” example on your Bad Physics link.

January 20, 2011 10:59 pm

Very nice piece of intelligible exposition

January 21, 2011 8:05 am

Good news: Well, there actually IS a good biography. Bad news: You’ve gotta learn German first.

January 21, 2011 8:10 pm

seems to be the tranlation of the German book published in 1970. saw references to this in:
http://www.weylmann.com/weyl+noether.pdf

February 2, 2011 1:03 am

Wow, that pdf link is awesome. I particularly enjoyed reading Weyl’s eulogy at Emmy Noether’s funeral.

January 29, 2011 10:46 pm

As a current AP Physics student, I found your article very interesting and helpful! I followed your first example but am still working on the second. I, too, am grateful for Noether’s Theorem and look forward to implementing it during some challenging momentum problems. It’s amazing how simply a seemingly complicated problem can be…

January 30, 2011 1:29 pm

I definitely agree with the idea that energy conservation is just like accounting. As an AP Physics student, I have learned that bascially every problem can be solved in more than one way (using kinematics or conservation of energy or other options) and that it is important to be able to recognize different approaches to any given problem. I look forward to learning how to apply Noether’s Theorem to some of the problems involving energy and momentum that we do.

January 30, 2011 2:13 pm

I always love learning about new ways to do physics problems, and Noether’s Theorem has a lot of promise for simplifying otherwise complex ones. I’m curious, though–what if a nonconservative force were present? Would the the theorem become completely useless? Would there be some other way to account for the dissipated energy? Or would the presence of a nonconservative force make it necessary to return to traditional kinematics methods?

January 30, 2011 4:31 pm

I totally agree that Noether’s theorem simplifies a lot of physics conditions by looking at the symmetric aspect. It helps to identify conserved quantities and therefore solve problems faster. I’m currently learning Work and Energy in Physics and Understanding Conservation of Energy is ideed a significant learning goals. I look forward to put Noether’s theorem into more practice:)

January 30, 2011 5:33 pm

I like the symmetrical aspect of conservation of energy and it definitely helps simplify physics problems (such as in the first example given). I am curious to learn about more of the applications of Noether’s Theorem so that I can solve problems faster and with more than one method. One of the upcoming units in that AP Physics class that I am taking is rotational motion and I am looking forward to learning about possible applications of Noether’s Theorem in that unit.

January 30, 2011 6:37 pm

As an AP Physics student, I have become quite reliant on Newton’s Second Law when attempting to solve physics problems. This reliance, however, often puts me in a difficult situation whenever a problem does not provide sufficient information about the forces present in a situation (as pointed out in the post). Thus, I can greatly appreciate the usefulness of Noether’s Theorem and how simple it can make seemingly complex problems. As I recall problems I grappled with earlier in the year, it is amazing to think how much easier using Noether’s Theorem could have made them and I look forward to applying the theorem as I work on physics problems in the future. I am also interested in the varying applications of Noether’s Theorem in one scenario as you change your definition of the system you’re examining, for while some symmetries may exist for one definition of the system, they may not for other definitions.

January 30, 2011 7:52 pm

As an AP physics student, it is interesting to read about another method of solving physics problems because I have learned that problems are often solvable in many different ways. I definitely agree with the usefulness of thinking of energy conservation as accounting, leading you to figure things out by keeping track of a conserved quantity. As mentioned in the post, I think it is an important question to ask What quantities are conserved. (I’m not quite sure I understand how you would know that.) I also thought it was interesting to think about situations in which quantities are symmetric with respect to time, such as energy conservation. It was also neat to realize that this theorem was developed by an early 20th century woman!

January 30, 2011 8:45 pm

Noether’s Theorem is fascinating. Someone tried to describe it to me early in the week and I didn’t understand, but this article does a good job of explaining what the symmetrical forces are and how to find them. What I do not yet quite understand is how that actually makes problems simpler. I would guess that keeping the reference frame along the symmetrical force–such as along the x-axis or the helix–would make the problem reduce very quickly. With the example of the ball traveling down the hallway, I am not sure how the problem would simplify with Noether’s Theorem. All in all though, Noether’s Theorem is very interesting and an inspiring new way to look at physics

13. January 31, 2011 9:11 am

I like the ideas of that energy conservation is just like accounting. I found how convenient to use different aspect to think/rethink about physics problems. It is fascinating to think find the corresponding conserved quantity; however, I need more practice. It is much simpler and easier. The Noether’s Theorem is inspiring and I am looking forward to applying it in practice.

February 2, 2011 12:54 am

Wow, which physics teacher referred their entire class to my blog?