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I know Noether's theorem can be applied to symmetries of the action, which is defined within the Lagrangian approach to mechanics.

However, does the same theorem still apply if I use a Newtonian approach with vectorial laws of motion? I know that these laws of motion have the same form for all inertial observers, and this sounds like a symmetry to me. However, this is NOT a symmetry of the action as well (different inertial observers will calculate different kinetic energy --> different actions for the same system).

So in conclusion we have a symmetry in Newtonian approach that is not a symmetry in Lagrangian approach. Should I deduce that Noether's theorem does not apply to Newtonian approach, but only to Lagrangian approach?

Federico Toso
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  • FWIW, the original Noether's theorem (and its Noether current formula) assume an action formulation. Related: https://physics.stackexchange.com/q/439235/2451 – Qmechanic Sep 25 '19 at 21:17
  • "we have a symmetry in Newtonian approach that is not a symmetry in Lagrangian approach" - This is incorrect. "different inertial observers will calculate different kinetic energy" in either formulation. There is no inconsistency between Newton and Lagrange. – safesphere Sep 25 '19 at 22:49
  • @safesphere yes but the difference in kinetic energy doesn't affect the simmetry of newtonian laws (in particular the second one), because kinetic energy doesn't appear in them. On the contrary, it appears in the action calculation, so the action symmetry in compromised. Do you agree? – Federico Toso Sep 25 '19 at 23:43
  • It affects the first law. A body at rest in one frame is not at rest in another. – safesphere Sep 26 '19 at 02:43
  • @safesphere I'm not sure about that... the first law says that a free object keeps its state of motion, both if it's moving and if it is at rest. But even if you were right, we could simply focus on Newton's second law and say that it is simmetric for inertial frames. Anyway, Noether's theorem does not seem to apply to this summetry – Federico Toso Sep 26 '19 at 06:00
  • Only in the same frame, but then the energy is also the same. – safesphere Sep 26 '19 at 06:03

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