[scottmsul]

A simpler way to describe Noether's Theorem is that for everything you can change about the coordinate system which leaves the physics the same, it's possible to derive a real physical quantity that is conserved. For instance, if we redefined the origin from (0,0,0) to (dx,0,0), all the x's would get dx added to them, but the actual motions of everything would still be the same. This means there's a symmetry with respect to X. Using Noether's Theorem, it's possible to derive a physical quantity that must be conserved, which in the case of positional symmetry, happens to be momentum. Similarly, redefining time to t'=t+dt doesn't change the motions, which can be used to derive something that is conserved, which happens to be energy. Some other ones include multiplying everything be e^(i*theta) in quantum mechanics, which derives conservation of probability, and changing the angle of the (x,y,z) axes at the origin, which derives conservation of angular momentum.