So what I understand is that the law of conservation of energy, likes Newton's law of motion, can't be proved. However, by Noether's Theorem, if there is a time symmetry, the energy is conserved. It seems to me that we have actually "proven" the law of conservation of energy, is it true?

  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/94381/2451 and links therein. $\endgroup$ – Qmechanic Jun 2 '17 at 12:12
  • $\begingroup$ Noether's theorem and its implications are more profound and fundamental than just the statement that 'energy is conserved'. Energy conservation seems to be obvious from a logical standpoint. But still, there must be a rigorous proof. That is provided by Noether's theorem. In cases where time translational symmetry doesn't exist, the 'law' of energy conservation runs into difficulties. This is the case in, for example, an expanding universe in cosmology. $\endgroup$ – Avantgarde Jun 2 '17 at 12:36
  • $\begingroup$ Conservation of energy can be "proven" based on Newton's laws. I agree that the derivation from Noether's theorem has a more general character, it may be more satisfying. But to "prove" some statement you need to rely to some unproven assumptions. In physics we don;t usually talk about proof as in math. The validity of any statement relies on experimental evidence. $\endgroup$ – nasu Jun 2 '17 at 13:46
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    $\begingroup$ Possible duplicate of Noether Theorem and Energy conservation in classical mechanics $\endgroup$ – John Rennie Jun 2 '17 at 17:42

Sure, you've "proven" the conservation of energy - under the even stronger assumptions that the universe follows suitable Lagrangian dynamics and that those dynamics have a continuous time-translation symmetry. If you really feel those assumptions are easier to swallow, then yes, you've gone forward.

However, there are situations where there is no guarantee that there is time-translation invariance, such as general settings in GR, and there are situations where it is explicitly broken, like several standard cosmological models, so all you've really done is shift the weight of truth from one solid-but-not-quite-solid-enough pillar to another solid-but-not-quite-solid-enough pillar.


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