Is there something more to Noether's theorem?

Question

From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics completely equivalent to the conservation laws? Or does Noether's theorem add something?

If yes, then I deduce that this theorem does not provide any more insight in why are the invariants mass, momentum and energy. Indeed, in Lagrangian mechanics, you still have to define your conjugate variables (hence the momentum) and the Lagrangian itself (hence the energy). This means you cannot deduce these invariants from more fundamental postulates. Am I correct?

Now, bonus question. Is there any profound reason why these invariants are all proportional to mass? And proportional to powers of velocity?

Answer 1

Within the Newtonian framework of mechanics conservation laws are tricky to develop and are not obvious at first glance. Lagrangian mechanics generalises the concept of conservation laws by exploiting "symmetries". The connection between symmetries and conservation laws is made by Noether's theorem.

An object has a symmetry if it is invariant under a transformation. The transformation could be discrete or continuous, local or global and the object could be the action, Lagrangian, equations of motion or even the coordinates themselves.

The relationship between symmetries and conservation laws in Noether's theorem holds only for continuous symmetries, however it encompasses both global and local transformations through the first and second Noether theorems.

The benefit of this result is that we can quickly spot symmetries and therefore are guaranteed a conservation law. Conservation laws are useful for reducing the complexity of a problem through reduction procedures.

Edit

I think the main part of your question is as follows:

Is there any additional information learned about the system from employing Noether's theorem as opposed to using the Newtonian framework?

The conservation law itself will contain no extra information. To say that an object is conserved in time is simply to observe the vanishing of it's time derivative. However Noether's theorem does in fact allow us to gain extra information about our system.

As an example

Consider a Hamiltonian system $(M,\omega ,H)$ where $M$ is a symplectic manifold, $\omega$ is a symplectic 2-form and $H$ is a Hamiltonian function. A continuous symmetry of the Hamiltonian system is a vector field $X$ on $M$ such that the Lie derivative (denoted $\mathcal L_X$) of $\omega$ and $H$ vanishes, \begin{equation} \mathcal L _X\omega=\mathcal L_XH=0 \end{equation} By the Poincar\'e lemma if $\iota _X\omega$ is closed then locally a scalar function $F:M\mapsto \mathbb R$ can be found, meaning $\iota _X\omega =dF$, (where $\iota _X$ is the interior product) i.e if $X$ is symplectic then in the neighbourhood it is Hamiltonian and hence $X_F(H)=\{H,F\}=0$ meaning $F$ is in involution indicating that it is constant along integral curves of $X_H$, a conserved quantity.

With analytical mechanics comes an abstraction. Newtonian physics could never really tell me about the properties of $\omega$ or Liouville's theorem etc. So from that angle, yes, Noether's theorem gives us an awful lot more insight into the physics of the system than simply stating a conservation law.

However at the same time, physics is physics, no matter how you choose to describe a system, all results should tie together and of course we shouldn't expect new physics by re-phrasing the problem. We are instead appreciative that we can learn more.

I hope that helps and that I have understood your question properly?

Answer 2

From the definition of lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space.

Yes, that's what we can read on websites like this. But note that we define our time using the motion of light, and our space too.

Is the reverse true? Are Lagrangian mechanics completely equivalent to the conservation laws?

They aren't completely equivalent, they're based on the difference in kinetic and potential energy. And the potential energy of a pendulum is just hidden kinetic energy. Gravity converts internal kinetic energy into external kinetic energy.

Or does Noether's theorem add something?

I'd say it adds something. But I've never felt it's particularly profound. Think of Compton scattering, where you can convert some of the photon energy into the motion of an electron. In theory you could do another Compton scatter on the residual photon, and another and another. In the limit there's no E=hf photon wave energy left, it has been entirely converted into the motion of electrons. And yet, and yet: you can make an electron (and a positron) out of a photon in pair production. So in a way an electron is made of motion. I think that's profound. Or energy-momentum if you prefer.

If yes, then I deduce that this theorem does not provide any more insight in why are the invariants mass, momentum and energy.

I tend to agree. But note that invariant mass isn't invariant. When you drop a brick some of its mass-energy is converted into kinetic energy. When you dissipate this, you're left with a mass deficit. Energy is conserved. IMHO it's conserved because everything is made of energy, it's fundamental. It's the one thing we can neither create not destroy. As to why, I don't know.

Indeed, in Lagrangian mechanics, you still have to define your conjugate variables (hence the momentum) and the Lagragian itself (hence the energy). This means you cannot deduce these invariants from more fundamental postulates. Am I correct?

I'd say no, because energy and momentum aren't really two different things. They're two aspects of energy-momentum. Imagine a cannonball travelling in space at 1000 m/s relative to you. You might say it has kinetic energy, and you might say it has momentum. But the former is in essence a measure of stopping distance, while the latter is a measure of stopping time. You can't reduce one without reducing the other. They are two sides of the same coin. And again, we define our time and our space using the motion of light. Motion is king.

Now, bonus question. Is there any profound reason why these invariants are all proportional to mass? And proportional to powers of velocity?

They're proportion to mass because the mass of a body is a measure of its energy-content. Remember the wave nature of matter and pair production, and atomic orbitals where electrons "exist as standing waves". Have a look at the photon in the mirror-box in this paper: http://arxiv.org/abs/1508.06478 (the 't Hooft is not the Nobel 't Hooft). Then think of photon momentum as resistance to change-in-motion for a wave propagating linearly at c, and think of electron mass as resistance to change-in-motion for a wave going round and round at c. And yes, the power of velocity is there in E=mc² where you divide by c to go from energy to momentum, and by c again for mass. It's also in ½mv² for the cannonball, to do with stopping distance. That's a bit like an inverse Compton, where you take photon energy out of a moving electron.