Can somebody show me a non-trivial exmple of Noether Theorem?

[kof9595995]

Can somebody show me a "non-trivial" exmple of Noether Theorem?

Noether Theorem states that if the Lagrangian of a system is invariant under some continuous coordinate transformation, then there's a conserved quantity.But does it simply mean the Lagrangian has to have a cyclic coordinate? Or a cyclic coordinate is just a special "trivial" case of Noether theorem? If so could somebody show me a "non-trivial" example? I mean a Lagrangian with no cyclic coordinates but we can apply Noether Theorem on.

 

[Vanadium 50]

You can show the Runge-Lenz vector is a conserved quantity in a 1/r potential. However, I warn you - after wrestling with this derivation, you might wish for the trivial examples again.