Background required for Noether's Theorem?

[ibkev]

Can someone please explain the progression of topics I would need to study in order to tackle Noether's Theorem? I keep hearing how important it is and am setting a self-study goal for myself to eventually understand it with enough rigour that I can appreciate it's beauty.

I have a feeling I have a long way to go... :) I've done Calculus 1,2, and 3. The equivalent of 1st year university physics (Halliday/Resnick). Currently, I'm working through linear algebra.

 

[Sturk200]

You cover it on a basic level in intermediate mechanics. It's actually not very difficult and you probably have enough math to understand it tonight if you're so inclined. Basically you write down the energy of your system (really the Lagrangian), and then think of things that you might do to the expression. For instance, maybe you want to translate the system in space (so add something to the position coordinates). If you find that the expression remains unchanged after such an operation, then this indicates there is a conserved quantity present. We might find for instance, that the energy is invariant under a spatial translation: this indicates that momentum is conserved. If it's invariant under a rotation, then angular momentum is conserved. If it's invariant under a translation in time, then the energy itself is conserved. These are the basics. You may also discover that the energy is invariant under some more subtle operation, such as varying the z-coordinate by an integer multiple of 2pi. That would also indicate the presence of some kind of conserved quantity, though perhaps not as obvious or intuitive. In general we say that any such invariance of the energy (really the Lagrangian) implies a conservation law.

As an example, the Lagrangian for a free particle is just its kinetic energy, which depends only on its velocity (mv^2/2), not on its position. We can translate the particle in space, or rotate it about some axis without in any way changing the energy. By Noether's theorem this is equivalent to saying that linear and angular momentum are conserved.

Learn Lagrangian mechanics and it will make sense. That said, I'm sure it gets more complicated on a higher level, which I haven't yet been able to understand myself.

 

[anorlunda]

Leonard Susskind derives using only the calculus of variations it in his video course on Classical Physics. I don't remember which lecture, but all 10 lectures are enjoyable and rewarding.

 

[ibkev]

Fantastic - thanks for the thorough answer and the link!