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\begin{document}

\centerline{\bf Solution for assignment 14 set~4} \medskip

\centerline{\bf Poisson Brackets}
\bigskip

\noindent Let us consider functions $g=g(q_k,p_k,t)$ and 
$h=h(q_k,p_k,t)$. The Poisson bracket is defined by
$$ [g,h]\ \matrix{\rm def \cr =}\ \sum_k\left(
\frac{\partial g}{\partial q_k}\,\frac{\partial h}{\partial p_k} -
\frac{\partial h}{\partial q_k}\,\frac{\partial g}{\partial p_k}\right)
\,.$$
The properties of the assignment are shown in the following: \smallskip

\noindent 1. Resulting in the definition of the total time derivative:
\begin{eqnarray} \nonumber
  \frac{d g}{dt}\ =\ [g,H] + \frac{\partial g}{\partial t}&
  =& \sum_k\left( \frac{\partial g}{\partial q_k}\,\frac{\partial H}
  {\partial p_k} - \frac{\partial H}{\partial q_k}\,
  \frac{\partial g}{\partial p_k}\right)+\frac{\partial g}{\partial t}
  \\ \nonumber &=& \sum_k\left( \frac{\partial g}{\partial q_k}\,
  \dot{q}_k + \frac{\partial g}{\partial p_k}\,\dot{p}_k\right) +
  \frac{\partial g}{\partial t}\,.
\end{eqnarray}
2. Using $\partial q_j/\partial p_k =0$:
$$ \dot{q}_j\ =\ [q_j,H]\ =\ \sum_k\left(\frac{\partial q_j}{\partial q_k}\,
   \frac{\partial H}{\partial p_k} - \frac{\partial H}{\partial q_k}\,
   \frac{\partial q_j}{\partial p_k}\right)\ =\ \sum_k \delta_{jk}\,
   \dot{q}_k\ =\ \dot{q}_j\,, $$
3. Using $\partial p_j/\partial q_k =0$:
$$ \dot{p}_j\ =\ [p_j,H]\ =\ \sum_k\left(\frac{\partial p_j}{\partial q_k}\,
   \frac{\partial H}{\partial p_k} - \frac{\partial H}{\partial q_k}\,
   \frac{\partial p_j}{\partial p_k}\right)\ =\ \sum_k \delta_{jk}\,
   \dot{p}_k\ =\ \dot{p}_j\,, $$
4. Commutators of generalized coordinates and momenta:
\begin{eqnarray} \nonumber
  [q_i,q_j]\ =\ \sum_k\left(\frac{\partial q_i}{\partial q_k}\,
   \frac{\partial q_j}{\partial p_k} - \frac{\partial q_j}{\partial q_k}\,
   \frac{\partial q_i}{\partial p_k}\right)\ =\ 0
\end{eqnarray}
as $\partial q_j/\partial p_k=0$ and $\partial q_i/\partial p_k=0$.
Similarly
\begin{eqnarray} \nonumber
  [p_i,p_j]\ =\ \sum_k\left(\frac{\partial p_i}{\partial q_k}\,
   \frac{\partial p_j}{\partial p_k} - \frac{\partial p_j}{\partial q_k}\,
   \frac{\partial p_i}{\partial p_k}\right)\ =\ 0\,,
\end{eqnarray}
while
\begin{eqnarray} \nonumber
  [q_i,p_j]\ =\ \sum_k\left(\frac{\partial q_i}{\partial q_k}\,
   \frac{\partial p_j}{\partial p_k} - \frac{\partial p_j}{\partial q_k}\,
   \frac{\partial q_i}{\partial p_k}\right)\ =\ \sum_k\left( \delta_{ik}\,
   \delta_{j_k} - 0 \right)\ =\ \delta_{ij}\,.
\end{eqnarray}
5. Angular momentum commutator:
\begin{eqnarray} \nonumber
  [x_i,L_j] &=& \sum_k\sum_l\sum_m \epsilon_{jlm}\,\left(
  \frac{\partial x_i}{\partial x_k}\,\frac{\partial(x_lp_m)}{\partial 
  p_k} - \frac{\partial(x_lp_m)}{\partial x_k}\,
  \frac{\partial x_i}{\partial p_k}\,\right)\\ \nonumber &=&
  \sum_k\sum_l\sum_m \epsilon_{jlm}\,\delta_{ik}\,x_l\,\delta_{mk}\
  =\ \sum_l\sum_m \epsilon_{jlm}\,\delta_{im}\,x_l\ =\
  \sum_l \epsilon_{jli}\,x_l\ =\ \sum_l \epsilon_{ijl}\,x_l\,.
\end{eqnarray}
\begin{eqnarray} \nonumber
  [p_i,L_j] &=& \sum_k\sum_l\sum_m \epsilon_{jlm}\,\left(
  \frac{\partial p_i}{\partial x_k}\,\frac{\partial(x_lp_m)}{\partial 
  p_k} - \frac{\partial(x_lp_m)}{\partial x_k}\,
  \frac{\partial p_i}{\partial p_k}\,\right)\\ \nonumber &=& -
  \sum_k\sum_l\sum_m \epsilon_{jlm}\,\delta_{lk}\,p_m\,\delta_{ik}\
  =\ - \sum_l\sum_m \epsilon_{jlm}\,\delta_{li}\,p_m\ =\ -
  \sum_m \epsilon_{jim}\,p_m\ =\ \sum_m \epsilon_{ijm}\,x_m\,.
\end{eqnarray}
\begin{eqnarray} \nonumber
  [L_i,L_j]&=& \sum_k\sum_l\sum_m\sum_n\sum_r \epsilon_{ikl}
  \epsilon_{jmn}\left(\frac{\partial(x_kp_l)}{\partial x_r}
  \frac{\partial(x_mp_n)}{\partial p_r} - 
  \frac{\partial(x_mp_n)}{\partial x_r}
  \frac{\partial(x_kp_l)}{\partial p_r}\right) \\ \nonumber
  &=& \sum_k\sum_l\sum_m\sum_n\sum_r \epsilon_{ikl}
      \epsilon_{jmn}\left(\delta_{kr}\delta_{nr}x_mp_l -
      \delta_{mr}\delta_{lr}x_kp_n \right)\\ \nonumber 
  &=& \sum_k\sum_l\sum_m\sum_n \epsilon_{ikl}\epsilon_{jmn}\left(
      \delta_{kn}x_mp_l- \delta_{ml}x_kp_n\right)\\ \nonumber
  &=& \sum_k\sum_l\left(\sum_m \epsilon_{ikl}\epsilon_{jmk} x_mp_l 
      - \sum_n \epsilon_{ikl}\epsilon_{jln} x_kp_n \right)\\ \nonumber
  &=& \sum_k\sum_l\left(\sum_m \epsilon_{kli}\epsilon_{kjm} x_mp_l -
      \sum_n \epsilon_{lik}\epsilon_{lnj} x_kp_n\right)\\ \nonumber
  &=& \sum_l\sum_m(\delta_{lj}\delta_{im}-\delta_{lm}\delta_{ij})x_m p_l 
   -  \sum_k\sum_n(\delta_{in}\delta_{kj}-\delta_{ij}\delta_{kn})x_k p_n
  \\ \nonumber
  &=& x_i p_j-\delta_{ij}\,\vec{x}\cdot\vec{p} -
      x_j p_i+\delta_{ij}\,\vec{x}\cdot\vec{p}\ =\ x_i p_j - x_j p_i\,.
\end{eqnarray}
This agrees with
\begin{eqnarray} \nonumber
  \sum_k\epsilon_{ijk}L_k&=&\sum_k\sum_l\sum_m\epsilon_{ijk}
  \epsilon_{klm}x_l p_m\ =\ \sum_k\sum_l\sum_m \epsilon_{kij} 
  \epsilon_{klm} x_l p_m\\ \nonumber &=& \sum_l\sum_m
  (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\ x_lp_m\ 
  =\ x_i p_j - x_j p_i\,.
\end{eqnarray}

\end{document}

