Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a gravitational field suspended at a certain height, that ball has a potential energy that associated with (Q), which is conserved over time.

And if there were certain translations to occur, such as the ball falling, that quantity of potential energy (Q) would be the same in the form of kinetic energy while reducing it's initial potential energy.

It seems to me, that the initial quantity (Q) has to defined based on certain physics then Noether's theorem is applied, but what defines the quantity (Q)? and how is it certain that it cannot be created nor destroyed, rather conserved?

For me, there is a gap of understanding that is not clear.

Isn't Noether's theorm reliant on a few axioms?

  1. Principle of least action.
  2. Lagrangian mechanics.
  3. Hamiltonian mechanics.

I'd like to use an analogy to illustrate the "gap" that's in my understanding:

Imagine driving down a road (R), the speed limit is always and forever at (v).

In order to satisfy the conditions above, you'd have to be on a certain vehicle capable on driving on the R and reaching or being within the range of v.

The latter seems to be the "filler" of the gap for me.

I would love to hear your thoughts on this.

  • $\begingroup$ I think this needs a good, slow, explicative answer. There may be a larger gap in someones understanding ( but also in mine's - concerning the question) $\endgroup$ – jaromrax Feb 12 at 15:22
  • $\begingroup$ It seems that you're essentially looking for a proof of Noether's Theorem, as that explains both how $Q$ is defined and why it is conserved. It is indeed dependent on the principle of least action, since it is defined within the framework of Lagrangian/Hamiltonian mechanics. $\endgroup$ – probably_someone Feb 12 at 15:37

In a nutshell:

  1. Without Noether's theorem:

    • For a Lagrangian $L(q,v,t)$, the momentum $p$ and the energy $h$ are defined as $$ p~:=~\frac{\partial L}{\partial v} \qquad\text{and}\qquad h~:=~pv-L,$$ respectively.

    • Momentum $p$ and energy $h$ are conserved if $L$ does not depend explicitly on $q$ and $t$, respectively. This follow directly from Lagrange equation.

  2. With Noether's theorem:

    • If the action $S=\int dt~L$ has a quasisymmetry under infinitesimal translations of $q$ and $t$, then one may show that the formula for the corresponding conserved Noether charge $Q$ is $p$ and $h$, respectively.

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