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

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Noether's Theorem connects symmetries in a system's Lagrangian to conserved quantities, typically involving cyclic coordinates. However, the discussion seeks a "non-trivial" example where the Lagrangian lacks cyclic coordinates yet still allows for the application of Noether's Theorem. The Runge-Lenz vector in a 1/r potential is cited as a conserved quantity, illustrating a more complex scenario. This example demonstrates that even without cyclic coordinates, conservation laws can emerge from symmetries. The complexity of such derivations highlights the depth of Noether's Theorem 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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