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I want to calculate the poisson bracket of two angular momentum components: $$\{L^i,L^j\}=\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial q^k} \frac{\partial\epsilon_{\:\:s}^{\:j\:r}q^sp_r}{\partial p_k}-\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial p_k} \frac{\partial\epsilon_ {\:\:s}^{\:j\:r} q^sp_r}{\partial q^k}$$ Which gives: $$\{L^i,L^j\}= \epsilon^{\:i\:m}_{\:\:k} \epsilon_{\:\:s}^{\:j\:k}p_mq^s- \epsilon^{\:i\:k}_{\:\:l} \epsilon_{\:\:k}^{\:j\:r}p_rq^l $$ But then I get problems with the index positions if I want to convert the Levi-Civita-Tensors into Kronecker-Deltas. Is angular momentum defined different in general coordinate systems or where else is the problem?

Qmechanic
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Silas
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    In flat space upper and lower index positions are not distinguishable. Are you comfortable with contracting Levi-Civita symbols? – Cosmas Zachos Mar 26 '22 at 16:42
  • @CosmasZachos My problem is that if I contract the Levi-Civita tensors in the first term I would get $\epsilon^{::im}{k} \epsilon{::s}^{:k:j}=\delta^i_s\delta^{mj}-\delta^{ij}\delta^{m}_s $ but then the Kronecker deltas have not always one upper and one lower index. – Silas Mar 27 '22 at 08:59
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    Raising and lowering indices is meaningless here for a flat metric. Try ignoring the distinction first, to get to the important result, and finesse the upper-lower (non)issue later! – Cosmas Zachos Mar 27 '22 at 16:43
  • @CosmasZachos Thanks but deriving the result is no problem. My question is just how I have to write the angular momentum components that I get no problems with the index position. – Silas Mar 28 '22 at 12:17
  • Upper and lower indices are equivalent in this space. Use flat metrics $\delta_jk$ to take them all up and follow the contractions. Are you completely comfortable with the WP page? It has mixed $\delta$s. – Cosmas Zachos Mar 28 '22 at 15:21
  • @CosmasZachos so does that mean that the definition $L_i:=\epsilon_{ijk}q^jg^{lk}p_l$ is only true in flat spaces or is the result of the poisson bracket different in curved spaces ? – Silas Mar 29 '22 at 16:31
  • Ach, if the issue is non-flat phase-spaces, you might as well have gone there up front. A couple of linked questions here include this and that.... – Cosmas Zachos Mar 29 '22 at 16:58
  • Besides generalizing the Levi-Civita tensor/symbol to curved space, another issue is that in curved space the position coordinates $q^j$ are no longer components of a vector. – Qmechanic Jun 27 '23 at 06:44

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