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

January 20, 2011

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.

25 Comments leave one →
  1. Nice but... permalink
    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.

    • gravityandlevity permalink*
      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.

    • Brian permalink*
      July 31, 2015 10:42 am

      Unfortunately, Bad Physics seems to be mostly gone. But the crazy paper that was being linked to was this one: http://dx.doi.org/10.1207/s1532754xjprr1604_4

  2. Avery Andrews permalink
    January 20, 2011 10:59 pm

    Very nice piece of intelligible exposition

  3. Bense permalink
    January 21, 2011 8:05 am

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

  4. gaddeswarup permalink
    January 21, 2011 8:10 pm

    http://www.amazon.com/Emmy-Noether-1882-1935-Auguste-Dick/dp/3764330198
    seems to be the tranlation of the German book published in 1970. saw references to this in:
    http://www.weylmann.com/weyl+noether.pdf

    • gravityandlevity permalink*
      February 2, 2011 1:03 am

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

  5. Emily Liptow permalink
    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…

  6. Ellen R permalink
    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.

  7. nRT permalink
    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?

  8. Angel C permalink
    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:)

  9. Melissa C permalink
    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.

  10. Anne Jordan permalink
    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.

  11. Martha Gilchrist permalink
    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!

  12. Sonja P permalink
    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.

  14. gravityandlevity permalink*
    February 2, 2011 12:54 am

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

    Let me try to address a couple comments briefly:
    1) You are actually very unlikely to encounter a formal teaching of Noether’s Theorem before graduate-level classes in physics. Most people aren’t really taught it until they get to Quantum Field Theory, which is kind of like seventh-year physics. The basic idea behind it is pretty cool, though, so I decided to write about it here.
    2) In problems with dissipation, there isn’t much hope to using “accounting” approaches. The problem is that nothing at all is conserved: not energy and not momentum. So if you throw that bouncy ball down the hallway but the entire hallway is submerged in water, you can’t use any of the conservation principles to help you. This is why physicists deal with dissipation problems only when they absolutely have to.
    3) The main take away message for someone in first or second year physics is that if you can draw the picture, you can immediately identify which of the various momenta are conserved (linear and angular momentum). This might be able to help you quite a bit. I wouldn’t worry too much about funny examples like the helix one, where only some particular combination is conserved.
    4) Well, maybe the real take away message for someone in first or second year physics is that physics is beautiful, and that women are perfectly capable of making revolutionary contributions to it.

  15. February 28, 2012 1:43 pm

    Very well drafted piece of writing. I appreciate the simplicity and clarity of thought with which it is explained. Thank you.

  16. jessica permalink
    March 17, 2013 7:10 pm

    Thank you for posting this.I am a 7th grader and I am doing a report on Emmy Noether and for a full 100 (required for me) I’ll have to explain the concept with full understanding.This article makes it simplier for me to understand this concept,thank you.🙂

  17. March 31, 2015 4:37 pm

    I highly recommend this wonderful book on the topic:
    http://www.amazon.com/Noethers-Wonderful-Theorem-Dwight-Neuenschwander/dp/0801896940

  18. May 27, 2015 2:56 am

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

    Me too. Math is oracular. Fits physics (and so many other things) like a glove.

  19. June 23, 2015 2:53 pm

    Hmm. The one time I used Noether’s theorem very long ago, it was in a blind, plug-and-play way. I might have to go back and try and understand what the hell I actually did in light of this explanation. If I still have my notes that is🙂

Trackbacks

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  2. Happy Pi Day, Emmy! | A Woman in Time
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