Noether and the derivative of the Action

[nemuritai]

I know that the Action has units Energy·time or Momentum·position. A second fact is that the derivative of the action with respect to time is Energy and similar with momentum-position, consistent with a units ie. dimensions check.Is it a coincidence that both are Noether conserved quantities? For example will the derivative of the Action with respect to phase be electric charge?

 

[anuttarasammyak]

Angular momentum is a conserved quantity, but action is not though the both has same dimension.

 

[PeroK]

I know that the Action has units Energy·time or Momentum·position. A second fact is that the derivative of the action with respect to time is Energy and similar with momentum-position, consistent with a units ie. dimensions check. Is it a coincidence that both are Noether conserved quantities?

If the Lagrangian is independent of time then energy is conserved. And, if it is indepenent of a spatial coordinate, then momentum in that direction is conserved. I don't see that as a coincidence.

For example will the derivative of the Action with respect to phase be electric charge?

I don't understand this question.

 

[vanhees71]

It's of course not a coincidence, because if the Lagrangian doesn't depend on some coordinate $q$ (but only its generalized velocity $\dot{q}$), then the canonical momentum is conserved for the solutions of the equation of motion, i.e., it is a conserved quantity:
$$p_q=\frac{\partial L}{\partial \dot{q}} \; \Rightarrow \; \dot{p}_q = \frac{\partial L}{\partial q}=0.$$
This canonical momentum is the generator of a symmetry (Noether's theorem).