Can anyone provide a simple, inuitive explanation for Noether's Theorem? I recently came across this theorem for the first time.  As I understand it, what she showed was that conservation 'laws' are often simply an artifact of symmetry or invariance.  
For example, the source I read said that conservation of momentum flows naturally if all the physics gives the same answer at all places - e.g., Newton's laws are the same here as they are there.  It also said that conservation of energy derives from the laws of physics being invariant over time.
I sort of see the math in the proof, but don't quite get the intuition.  Can anyone provide a brief explanation?
 A: The essence of Noether's Theorem(s) is that if your system (with well defined generalized coordinates and energy) has a continuous symmetry then you are guaranteed to have a corresponding conserved quantity. By "continuous symmetry" I vaguely mean a real-valued, differentiable, linear transformation between the generalized coordinates. 
Some examples:


*

*the energy functional of the system is conserved when time translation symmetry is respected. 

*the linear momentum of the system is conserved when spatial translation symmetry is respected. 

*the angular momentum of the system is conserved when angular translation symmetry is respected. 
By "is respected" I mean that the dynamics of the system are unaffected by the respective transformation. 
Edit: I did not understand the questioner was asking for intuition behind the $\bf{proof}$ of Noether's theorem. Of course, this depends on how the theorem is formally stated and on how one goes about proving that statement (there are numerous ways). But a simple proof is provided here beginning at the bottom of page 1. The intuition for this proof is similar to what is stated above: begin with the general statement of variation, $\delta L = 0$, and find a symmetry/transformation that depends on some parameter. Now apply the transformation to your variation of $L$, and try to $\it{find}$ a conserved quantity. The more abstract the proof becomes, the more abstract intuition is required, which I can provide a more detailed statement/proof of the theorem if the questioner desires.
