4

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?

Qmechanic
  • 220,844
eSurfsnake
  • 206
  • 1
  • 2

1 Answers1

1

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:

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

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

  3. 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.

Daddy Kropotkin
  • 2,790