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

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    Noether Theorem
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Noether's Theorem establishes that the invariance of a system's Lagrangian under continuous coordinate transformations leads to conserved quantities. A non-trivial example of this theorem is demonstrated through the Runge-Lenz vector in a 1/r potential, which illustrates conservation without relying on cyclic coordinates. This example highlights the complexity of deriving conserved quantities in systems where cyclic coordinates are not present, emphasizing the theorem's broader applicability beyond trivial cases.

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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.
 
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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.
 

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