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  1. Assume a gravitational field, with area A having some gravitational forces while area B having no gravitational force. The Lagrangian of particle moving along this field is obviously time invariant, but the mechanical energy (Hamiltonian) is by no means conserved if you go from region A to region B horizontally, moving vertically in region B, then go horizontally to region A. You gain energy after moving to the point with higher potential energy. This breaks conservation of energy but Noether's theorem fails to tell there either. I know as someone's answer under this question might say, such field would never be real because it's highly discontinuous (and thus not differentiable, so Lagrangian makes no sense), however you can always make the gravity continuously go to 0 and moves higher through space with 0 gravity then move back to points with non-zero gravity.

  2. Same issue occurs with presence of friction, because the Lagrangian doesn't depend on time explicitly (there is no point to assume friction depends on time), yet with presence of friction, no conservation of mechanical energy would occur, even though Lagrangian stays the same over time.

I think both examples have issues with having non-conservative forces, because least action principle only applies with conservative forces, but if Noether's theorem only applies to systems with conservative forces, what's the point then as energy in that system is already conserved by the definition of potential and kinetic energy?

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There are two ways to look at the Noether theorem in gravity. If gravity is a curvature of spacetime rather than a force field, then, without the force, the potential energy is not a meaningful concept. Instead, we must consider the gravitational time dilation. In gravity, time is not uniform, but moves slower in stronger gravity. Thus, according to the Noether theorem (for non-uniform time and no potential energy), energy does not conserve. Specifically, the kinetic energy is higher in stronger gravity.

For example, when an apple falls down from a tree, its kinetic energy increases, because its time becomes more dilated. The Noether theorem holds, energy in a non-uniform time does not conserve. When I pick up the apple and lift it up, I spend energy to move the apple to the area of faster moving time. This energy is gone and not conserved (remember, there is no poteential energy in this view). The Noether theorem holds again.

The Noether theorem as applied to energy conservation is a reflection of a deeper symmetry of nature between time and energy as Fourier conjugates. The often overlooked non-conservation side of this theorem is even more important than just a specific case of conservation. The key conclusion here is that the described process is reversible. When the apple fell, the same amount of energy was released without conservation as the amount of energy lost without conservation when the apple was lifted back. Thus overall the total energy over the full cycle was in fact conserved. This fact, as a result of the symmetry, allows us to simplify things by removing the time dilation from consideration and replacing its effect by the concept of "potential energy" (again according to the non-conservation part of the theorem).

This approach works with real static gravitational field where the overall energy is conserved due to the reversibility of the processes. However, in a dynamic gravitational field (e.g. the expanding universe), the processes are no longer reversible and consequently we no longer can consistently define "potential energy", so energy does not conserve... unless the universe starts shrinking again. Then all redshifted photons become blueshifted and gain their energy back. So energy would be conserved again over a full cycle of a reversible process.

Your first example represents a gravitational field that is impossible in nature, as it would violate the general relativity equations. However, your point is valid that this fact must have no bearing on the validity of the Noether theorem. The catch here is that in this case you cannot use the concept of "potential energy", because it becomes inapplicable just as in the case of a non-static spacetime (your processes are not reversible). You must go back to gravitational time dilation. Then you would clearly see that the Noether theorem perfectly holds, because energy is not conserved in non-uniform time of your example.

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  • $\begingroup$ It's nice that you talked about general relativity and reversibility, and indeed it holds under the condition of time dilation. Yet when the theorem was proved, there was no theory of relativity, only classical mechanics. If just thinking under scheme of classical mechanics, what is the problem here? Non conservative forces? $\endgroup$ – Ca Parvulus Lee Jul 19 '18 at 17:34
  • $\begingroup$ You cannot write a classical Lagrangian for this case, because potential energy cannot be consistently defined in your example. If you make a full circle, the potential energy would not return its initial value. Thus the potential energy at any point does not have a definite single value. It is a result of the fact that your field cannot be created by real sources. This is easier to analyze for the electric field with no time dilation. Your field would violate the U(1) symmetry of electromagnetism and thus be impossible to be described by the Maxwell equations or defined with potential energy. $\endgroup$ – safesphere Jul 19 '18 at 18:16
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  1. In example 1 without friction, the potential is not necessarily discontinuous and the mechanical energy conservation is not necessarily violated. Take e.g. a potential $V(x)=x_+=\max(x,0)$ where regions A and B are $x>0$ and $x<0$, respectively.

    Unless of course OP has a non-conservative scenario in mind. If so, see pt. 2.

  2. In example 2 with friction, it is a non-conservative force. Non-conservative forces generically have no action principle, and hence no Noether's theorem (NT).

  3. The point of NT is that it guarantees a conservation law for every symmetry of the action. Of course, if the conservation laws have already been found by other means, e.g. energy conservation, then NT yields nothing new.

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  • $\begingroup$ basically NT only works for systems where least action principle holds? I saw it was proved by Euler-Lagrangian equation so I think so, then how to deal with those with no action principles? $\endgroup$ – Ca Parvulus Lee Jul 19 '18 at 17:29
  • $\begingroup$ Yes, the main assumptions in NT is a symmetry of an action. A symmetry of eq. of motions is not enough. $\endgroup$ – Qmechanic Jul 19 '18 at 17:38
  • $\begingroup$ $\uparrow$ Saw where?? $\endgroup$ – Qmechanic Jul 19 '18 at 17:44

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