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Given the Runge-Lenz vector

$$\vec{A}=\vec{p}\times\vec{L}-mk\frac{\vec{r}}{r}$$

and the angular momentum

$$\vec{L}=\vec{r}\times\vec{p}$$

We can get

$$\{L^i,A^j\}=\epsilon^{ijk}A^k.$$

I think it is useful to write both as

$A^j=\epsilon^{abc}p_bL_c-mk\frac{r_n}{\sqrt{r^{\alpha}r_{\alpha}}}$ (maybe I'm wrong with the notation)

and

$L^{i}=\epsilon^{ijk}r_{j}p_{k}$

So the Poisson Bracket is expressed by {$L^{i},A^{j}$}$=${$\epsilon^{ijk}r_{j}p_{k},\epsilon^{abc}p_bL_c-mk\frac{r_n}{\sqrt{r^{\alpha}r_{\alpha}}}$}

and using the properties, I can distribute the components such that

{$L^i,A^j$}={$\epsilon^{ijk}r_{j}p_{k},\epsilon^{abc}p_bL_c$}$-${$\epsilon^{ijk}r_{j}p_{k},mk\frac{r_n}{\sqrt{r^{\alpha}r_{\alpha}}}$}

Doing so, I'm getting for the first bracket

$\epsilon^{ijk}\epsilon^{abc}${$r_jp_k,p_bL_c$}=$\epsilon^{ijk}\epsilon^{abc}\epsilon^{c\mu\nu}(-r_jp_bp_{\nu}\delta^{k\mu}+p_kr_{\mu}p_{\nu}\delta^{jb}+p_kp_br_{\mu}\delta^{j\nu})$ (by defining $L_c=\epsilon^{c\mu\nu}r_{\mu}p_{\nu}$)

but for the second one, I have

$\epsilon^{ijk}mk${$r_jp_k,\frac{r_n}{\sqrt{r^{\alpha}r_{\alpha}}}$}=$-r_nr_k(r^{\alpha}r_{\alpha})^{-3/2}-(r^{\alpha}r_{\alpha})^{-1/2}\delta^{kn}$

I'd appreciate if you can tell me what I'm doing wrong and help me to do the math!

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  • $\begingroup$ What happens to the second term of $A^j$ when $\alpha\neq\beta$? $\endgroup$ Apr 27 '18 at 19:29
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    $\begingroup$ I'm using Einstein notation, which implies a summation over the indices. So, if $\alpha\neq\beta$ the term wouldn't be added. So we have, for example in 3 dimensions, $mk\frac{r_n}{\sqrt{{r_{1}}^2+{r_2}^2+{r_3}^2}}$ $\endgroup$
    – Kali
    Apr 27 '18 at 23:32
  • $\begingroup$ When $\alpha\neq\beta$, we have that $\frac{r_n}{\sqrt{\delta^{\alpha\beta}r_\alpha r_\beta}}=\frac{r_n}{0}$. You're adding a bunch of infinities to each other. $\endgroup$ Apr 27 '18 at 23:35
  • $\begingroup$ And if that's not, in fact, what you're doing, then you need to fix your notation, as the summation over indices occurs at the term level, not within the square root. $\endgroup$ Apr 27 '18 at 23:37
  • $\begingroup$ I need to express $\frac{1}{\sqrt{{r_1}^2+{r_2}^2+{r_3}^2}}$ in terms of indices, how should I write it then? $\endgroup$
    – Kali
    Apr 27 '18 at 23:37