How to Prove the Classical Angular Momentum Commutation Relation?

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To prove the classical angular momentum commutation relation, the expression {Li, Lj} = εijkLk can be derived using the Levi-Civita symbol and the definition of the Poisson bracket. It is important to express Li in terms of the generalized coordinates qi and momenta pi. The discussion highlights that using square brackets can lead to confusion in classical mechanics. A participant expresses a desire to extend the relation to a general vector Vi, questioning how to construct such a vector while maintaining the relation {Vi, Lj} = εijkVk. The conversation emphasizes the need for clarity in notation and the proper formulation of vectors in classical mechanics.
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[Li,Lj]=εijkLk

how can I prove this expression classically?
 
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Classically, L is not an operator, so you cannot define a commutator.

You can show that {Li, Lj}=εijkLk. I don't know if that's what you meant by saying "Classically". If so, just write out Li in terms of qi and pi. If you write the correct expression for it using Levi-Civita symbol and apply definition of Poisson bracket, it should be a trivial matter.
 
Yes, exactly. Thank you very much. Using square brackets may be confusing in classical mechanics. I figured out to make this with levi civita symbol. But there is another problem I have now. if I replace the Li with some general vector Vi, it should still be hold

{Vi,Lj}=εijkVk

how should I constract a general V vector?
 

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