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blaisepascal
(I tried to write this last night, got in over my head, and realized I was doing it backwards. Either way, I'll start with a cut to shield people's friends-pages from my physics blather.



March 24th was Ada Lovelace Day, a day picked to celebrate women in technology, science, and engineering. One of the women discussed was Emmy Noether, who is best known for her eponymous theorem.

Noether's Theorem is one of those odd things. On the one hand, it's simple but very powerful, providing insight into the character of physical law. It draws a connection between the symmetries of nature and the conservation laws physicist hold dear. It's operation is straightforward and clear. But on the other hand, it isn't mentioned in any of the physics texts I used in school, nor is it mentioned in Feynman's Lectures. Ask a college undergrad who is not a physics major if he or she has ever heard of it, and you'll probably get blank stares. It's based on Lagrangian Mechanics, which is also likely to get you blank stares.

I never even heard of Lagrangian Mechanics before talking to anarchist_nomad at a wedding long ago (before he was Dr Nomad, possibly before he was Master Nomad). It was much later before I'd ever heard of Noether's Theorem.

To give you a flavor of what Noether's Theorem means, and how to apply it, let's use an example: two freely-floating identical masses connected by a spring of strength k and natural unstretched length l.

The procedure is:

First express the kinetic energy T(q1,q2,q1',q2',t) of the system in terms of the positions q1,q2 and velocities q1',q2' of the particles as well as time. In this case, we have T = m(q1'^2 + q2'^2)/2 (it's 1/2 mv^2 for each mass, summed).

Next express the potential energy V(q1,q2,q1',q2',t) of the system in the same terms. We get V = -k(|q1-q2|-l).

Then the Lagrangian L(q1,q2,q1',q2',t) = T-V = m(q1'^2 + q2'^2)/2+k(|q1-q2|-l).

In this case the q's are not necessarily 1-dimensional. The Lagrangian is the same for 2 or three dimension. In fact, it's not uncommon to simply see the Lagrangian written as L(q,q',t) where the generalized parameters q are assumed to have as many dimensions as needed to capture the dynamics of the situation.

Noether's Theorem works with differentiable symmetries of L, meaning that (in the simplest case) you replace q with q(s), where s is a parameter describing the symmetry, then L doesn't change as you change s. We can verify that q(s) = q+s is a symmetry of our Lagrangian easily enough: q'(s) = q', so T(q1'(s),q2'(s)) doesn't change, and since q1(s)-q2(s) = (q1+s)-(q2+s)= q1-q2 V doesn't change.

Noether's Theorem states that if q(s) is a symmetry of L, then (dL/dq')(dq(s)/ds) is a conserved quantity.

Let's take this in parts for our example: dL/dq' = dT/dq'-dV/dq' = dT/dq' (since V doesn't depend on q') dT/dq' = m(q1'+q2'). (I'm playing a little fast and loose with the rigor here, since I'm using a form of Noether's theorem that applies to one q on a problem that has 2. But the theorem can be generalized appropriately so it works). dq(s)/ds is also easy, since d/ds q(s) = d/ds q+s = 1. So the quantity mq1' + mq2' is conserved, which is linear momentum.

Basically, Noether's Theorem shows that in any system where a Lagrangian can be found which is symmetric in relation to translation of the coordinates, it means linear momentum is conserved. A more profound way of saying it is: If there is no origin, momentum must be a conserved quantity. Similarly, Noether's Theorem shows that if there's no preferred direction (i.e., the Lagrangian is symmetric under rotations), angular momentum must be a conserved quantity, and also that if there is no preferred now (i.e., the Lagrangian is symmetric under time translation), then energy is conserved. [1]

Three things taught as fundamental assumptions of physical law are that the laws of nature are constant across all space, directions, and time. Noether's theorem show that these imply the three great conservation laws of momentum, angular momentum, and energy.

This is deep, insightful, and not all that difficult to grasp. Yet this connection is not taught at the non-specialist undergrad level.

I think the problem is, in part, a historical perspective on teaching mechanics. Students are first taught Isaac Newton's Laws (circa 1680-1700), but not taught Joseph Lagrange's reformulation (circa 1788) or William Hamilton's reformulation of Lagrange's (1833) until more "advanced" classes. Newton was first, so he gets taught first, even if it means teaching it in terms not known in Newton's time -- or Lagrange's or Hamilton's, for that matter.

But it may also be the math. One of the key expressions in Lagrangian Mechanics is the Euler-Lagrange equation: dL/dq = d/dt (dL/dq'), which in general yields a 2nd order differential equation, which is not something to throw at freshmen and sophomore college students who haven't taken a course in differential equations. The theoretical underpinning of Lagrangian Mechanics, Hamilton's Principle[2], is based on the Calculus of Variation, which again is a bit tough to throw at freshmen and sophomores.

I've rambled enough. I hope it was accurate and entertaining.

[1] Incidentally, both Galileo, Newton, and Einstein agreed that the laws of physics should be symmetrical to "boosts", where the frames of reference are related by a constant linear velocity. I'm not sure how to write that symmetry as q(s), and I'm not familiar with what conserved quantity it represents. Can any of the physicists in the audience clue me in?

[2] Since Hamilton was born 17 years after Lagrange published, I'd love to know how Lagrange's mechanics got to be based on Hamilton's principle. Of course, Hamilton's principle is a generalization of Fermat's Principle, so it may be one of these "let's rewrite and generalize everything so that the guy it's named after won't recognize it anymore" things which happen in physics.

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