How does Noether's theorem prove that laws of physics are invariant with regards to time?

Question

Disclaimer: my interest in this topic is recreational, and I know as little about the mathematics involved here as I do in the relevant physics, outside of high school material.

To me, for instance, conservation of energy is a principle which I should just apply to formulaes involved in, say, physics of newtonian objects. Sure enough, they work, and I have practised them in classrooms.

But it only recently occurred to me that there was a lingering question in this regard: how come a calculus I made some decades ago (for some definition of "some") is still valid today?

And in the meantime (there is that "time" again) I learnt about special and general relativity which both force one to rethink what time is. And Noether's theorem in this regard still seems to hold true...

So, what is it that Noether demonstrated in the end? I have read many times that this theorem proves that the laws of physics are independent of time... But this seems to mean, to me, that this stands true whatever your referential is.

Answer 1

What Nother's theorem states is that if there exists a differential family of transformation of generalized coordiantes $\{q_{i}\}_{i\in\{1, ..., n\}}$, $\Omega:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}^{n}$ parameterized by a real parameter $\lambda$ $$q_{i}=\sum_{j=1}^{n}\Omega_{ij}(\lambda)\tilde{q}_{j}$$ With the identity element defined by $$q_{i}=\sum_{j=1}^{n}\Omega_{ij}(0)q_{j}$$ (or in vector notation $\vec{q}=\Omega(\tilde{\vec{q}}, \lambda)$), and this family of continous transformations leaves the Lagrangian of the system invatiant $$\mathcal{L}\Big(\vec{q}, \frac{d}{dt}\vec{q}, t\Big)=\mathcal{L}\Big(\Omega(\vec{q}, \lambda), \frac{d}{dt}\Omega(\vec{q}, \lambda), t\Big)$$ Then the following function of coordinates, velocities and time is a constant of motion $$\mathcal{I}\Big(\vec{q}, \frac{d}{dt}\vec{q}, t\Big)=\lim_{\lambda\rightarrow{0}}\sum_{k=1}^{n}\frac{\partial\mathcal{L}}{\partial{\dot{q}}_{k}}\frac{d}{d\lambda}\sum_{j=1}^{n}\Omega_{ij}(\lambda)\tilde{q}_{j}$$ It says nothing about time invariance and the energy conservation. Energy conservation is the manifistation of the Lagrangian being explictly time independant. I.e. consider the E-L equations for a time independant Lagrangian $$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial{\dot{q}}}=\frac{\partial\mathcal{L}}{\partial{q}}$$ You multiply by $\dot{q}$ using the chain rule and integration you have $$\dot{q}\frac{\partial\mathcal{L}}{\partial{\dot{q}}}-\mathcal{L}=E=const$$ For example $$\mathcal{L}\Big(\vec{q}, \frac{d}{dt}\vec{q}_{1}, t\Big)=\frac{m}{2}\Big(\frac{d}{dt}\vec{q}\Big)^{2}-U(|\vec{q}|, t)$$ Where $\vec{q}\in\mathbb{R}^{2}$. This Lagrangian is invariant under rotations i.e., in this case $\Omega\in{SO(2)}$, using the Nother's theorem you have $$I=m(q_{1}\dot{q}_{2}-q_{2}\dot{q}_{1})=const$$ Which is angular momentum conservation. However, as the potential energy is assumed to be explicily time dependent the energy is not conserved.

Answer 2

Concerning your more philosophical doubts:

The nice thing about mathematics is that once you prove that some statement is true, you know that it will always be true and there is no way of changing it. However, there is a cost for this and that is, that any non-trivial mathematical statement is essentially of the form "Under the condition that A holds, we know that B is true". It will not tell you anything useful about this condition A, instead you will need to check this one for yourself. But if you know A holds, then it will tell you that B also holds.

In case of Noether's theorem the statement, when applied to time invariance, roughly is "Under the condition that a system is time invariant, we know that Energy is conserved". It does not tell you anything about time invariance, only about what will happen if you have it.

In the specific case of physics, the only thing that you can do to overcome this is to start making assumptions and test them in experiments. We assume that physics in general is time invariant. We have done a lot of experiments and they all seem to verify this. (In fact every time you repeat an experiment that was already done at some other time and get the same result, you verify time invariance.) So it seems a prudent assumption. As a result of this assumption we expect conservation of energy for all future experiments.

Essentially this is as good as it gets, there is no way to prove any law of physics in a mathematical sense. We can only do experiments or derive it from other laws that we assume to hold because of experiments. Even when a physicist tries to show something "from first principles", which in philosophical terms would mean "without assuming anything", what he actually means is "assuming only what already has been verified by a lot of experiments".

There are of course other points. In this case, Noether's theorem allows us to reduce the number of assumptions from "time invariance" and "energy conservation" to only "time invariance", which reduces the number of things we need to check. Sometimes things also work the other way around. Let's say we had some theory that assumes "time invariance" and something like "unconserved energy". Then Noether's theorem will tell us that our theory is contradictory and we need to replace at least one of those assumptions. (A good example of such a situation would be special relativity, where Galilean invariance proved to be incompatible with Maxwell's equations and the former had to be replaced).

Finally of course note that Noether's theorem is much more powerful than just predicting energy conservation. However the full statement is a bit hard to appreciate without spending quite a bit of work on the mathematics and the general "symmetries imply conserved quantities" is a bit trite. Yet there are many easier special cases to consider. As the previous answer mentioned, invariance under rotation implies conservation of angular momentum. In the same way invariance under translation implies conservation of momentum, and so on. If you get to something like relativity, your symmetries get more complicated, but whenever you find one, it will always result in a corresponding conserved quantity.