A canonical transformation is a transformation from one set of coordinates $q,p$ to a new one $Q(q,p), P(q,p)$. For a function $f(Q,P)$ using the chain rule and using summation notation
$$\frac{\partial f}{\partial q_{i}}=\frac{\partial f}{\partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{i}}+\frac{\partial f}{\partial P_{j}}\frac{\partial P_{j}}{\partial q_{i}}$$
$$\frac{\partial f}{\partial p_{i}}=\frac{\partial f}{\partial Q_{j}}\frac{\partial Q_{j}}{\partial p_{i}}+\frac{\partial f}{\partial P_{j}}\frac{\partial P_{j}}{\partial p_{i}}$$
The Poisson bracket:
\begin{eqnarray} \left\{f,g\right\}&=&\frac{\partial f}{\partial q_{i}}\frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q_{i}}\\
&=& \left(\frac{\partial f}{\partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{i}}+\frac{\partial f}{\partial P_{j}}\frac{\partial P_{j}}{\partial q_{i}}\right)\left(\frac{\partial g}{\partial Q_{k}}\frac{\partial Q_{k}}{\partial p_{i}}+\frac{\partial g}{\partial P_{k}}\frac{\partial P_{k}}{\partial p_{i}}\right) – \left(\frac{\partial f}{\partial Q_{j}}\frac{\partial Q_{j}}{\partial p_{i}}+\frac{\partial f}{\partial P_{j}}\frac{\partial P_{j}}{\partial p_{i}}\right)\left(\frac{\partial g}{\partial Q_{k}}\frac{\partial Q_{k}}{\partial q_{i}}+\frac{\partial g}{\partial P_{k}}\frac{\partial P_{k}}{\partial q_{i}}\right)\\
&=&
\end{eqnarray}
For me, it is not very clear how to obtain the relations which are written in the attached image (the first two lines). What do these relations mean? What is it about? Is it about the chain rule, it is just simple multiply? How can I obtain the first two lines from image?
Thanks!
UPDATE
$\displaystyle \frac{\partial f}{\partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{i}}$ – it is about the derivative
In the calculation of Poisson bracket we have:
$\displaystyle \frac{\partial f}{\partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{i}} \cdot \frac{\partial g}{\partial P_{k}}\frac{\partial P_{k}}{\partial p_{i}} $ – it is about the composition and about product which is marked with $\cdot$
when we try to make a arrangement it about what….
$\displaystyle \frac{\partial f}{\partial Q_{j}} \frac{\partial g}{\partial P_{k}} \left(\frac{\partial Q_{j}}{\partial q_{i}} \frac{\partial P_{k}}{\partial p_{i}} \right)$ here it about what?
Best Answer
It's just a rearrangement. Multiply out the products before and after the step and check that you get the same terms in each case.