I really like Wikipedia's illustrated explanation of Noether's theorem [1]. Euler-Lagrange equations can also be proved using a similar trick - make an infinitesimal smooth 'bump' in a realizable path and consider the change in velocity on both sides of the bump and how it is cancelled out by the change in position so that the net change of action is 0.
I think that integration by parts can also be derived in a similar way but I would have to make a proper blog post with illustrations to make it clear.
Sometimes I also think of Noether's theorem as an application of the Euler-Lagrange equations in a coordinate system where the continuous symmetry changes only one coordinate; the conserved quantity is the associated generalized momentum (because the lagrangian is independent of that coordinate).
> I think that integration by parts can also be derived in a similar way but I would have to make a proper blog post with illustrations to make it clear.
I think that integration by parts can also be derived in a similar way but I would have to make a proper blog post with illustrations to make it clear.
Sometimes I also think of Noether's theorem as an application of the Euler-Lagrange equations in a coordinate system where the continuous symmetry changes only one coordinate; the conserved quantity is the associated generalized momentum (because the lagrangian is independent of that coordinate).
[1] https://en.wikipedia.org/wiki/Noether%27s_theorem#Brief_illu...