How to Prove the Classical Angular Momentum Commutation Relation?

[ssamsymn]

[L_i,L_j]=ε_ijkL_k

how can I prove this expression classically?

 

[K^2]

Classically, L is not an operator, so you cannot define a commutator.

You can show that {L_i, L_j}=ε_ijkL_k. I don't know if that's what you meant by saying "Classically". If so, just write out L_i in terms of q_i and p_i. If you write the correct expression for it using Levi-Civita symbol and apply definition of Poisson bracket, it should be a trivial matter.

 

[ssamsymn]

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 L_i with some general vector V_i, it should still be hold

{V_i,L_j}=ε_ijkV_k

how should I constract a general V vector?