More Ado About Noether

I didn't say exactly how Noether connects e.g. conservation of energy to time-isotropy. Here's John Baez's explanation of Noether's theorem for the Lagrangian (a nice quantity tracking every position and velocity in a physical system): Noether's Theorem in a Nutshell

Baez gives the formula for the conserved quantity in a one-particle system:

C = p dq(s)/ds

where p(t) is the momentum at time t, q(t) the position, and s is the parameter of the symmetry - s measures how much the system was rotated or slid. If the Lagrangian has an s-isotropy, dC/ds = 0.

Let's throw caution to the wind (since we're laymen here) and talk about a Lagrangian that is invariant under changes of scale. Take s > 0 to be any real positive number, the scaling factor - this acts as our "symmetry parameter". For instance, if s = 2, we're doubling the size of the universe. The new position is s times the old position:

q(s,t) = sq(t)
What's C?
C = p dq(s)/ds

= p d(sq)/ds

= p q ds/ds

= p q

Well, that was easier than I hoped - and interesting. The conserved quantity in our scale-invariant system is the product of position and momentum. I wonder what that means....
- posted by Scott @ 5:28 PM