I have a problem which I am finding difficult to derive. I think I am missing something.
Assuming that the Poisson Bracket for two functions $(u, v)$ is defined on the canonical coordinates and momenta $(q_{1},\cdots, q_{n}, p_{1},\cdots,p_{n})$ as follows
$$ (u,v)=\sum\limits_{r=1}^{r=n}\left( \frac{\partial u}{\partial q_{r}}\frac{\partial v}{\partial p_{r}}-\frac{\partial u}{\partial p_{r}}\frac{\partial v}{\partial q_{r}} \right), $$
we need to prove the following statement:
The problem: if $F$ and $\phi$ are functions of $(f_{1},f_{2},\cdots,f_{k})$, which in turn are functions of the canonical set $(q_{1},\cdots, q_{n}, p_{1},\cdots,p_{n})$, show that
$$ (F,\phi)=\sum\limits_{r,s} \left( \frac{\partial F}{\partial f_{r}}\frac{\partial \phi}{\partial f_{s}}-\frac{\partial F}{\partial f_{s}}\frac{\partial \phi}{\partial f_{r}} \right) (f_{r},f_{s}) \ \ \ \ ............. (*).$$
My attempt: Whenever I try to work this out, starting from the definition of the bracket for $(F,\phi)$ and then writing the terms like $\frac{\partial F}{\partial q_{k}}$ as $\sum_{r}\frac{\partial F}{\partial f_{r}}\frac{\partial f_{r}}{\partial q_{k}}$, and $\frac{\partial \phi}{\partial p_{k}}$ as $\sum_{s}\frac{\partial \phi}{\partial f_{s}}\frac{\partial f_{s}}{\partial p_{k}}$, and so on, then collecting terms, I end up with the result
$$ (F,\phi)=\sum\limits_{k}\left( \frac{\partial F}{\partial q_{k}}\frac{\partial \phi}{\partial p_{k}}-\frac{\partial F}{\partial p_{k}}\frac{\partial \phi}{\partial q_{k}} \right)=\sum\limits_{k}\left(\sum_{r} \frac{\partial F}{\partial f_{r}}\frac{\partial f_{r}}{\partial q_{k}}\sum_{s}\frac{\partial \phi}{\partial f_{s}}\frac{\partial f_{s}}{\partial p_{k}}-\sum_{s}\frac{\partial F}{\partial f_{s}}\frac{\partial f_{s}}{\partial p_{k}}\sum_{r}\frac{\partial \phi}{\partial f_{r}}\frac{\partial f_{r}}{\partial q_{k}} \right)$$
$$= \sum\limits_{r,s} \left( \frac{\partial F}{\partial f_{r}}\frac{\partial \phi}{\partial f_{s}}-\frac{\partial F}{\partial f_{s}}\frac{\partial \phi}{\partial f_{r}} \right)\sum\limits_{k} \frac{\partial f_{r}}{\partial q_{k}}\frac{\partial f_{s}}{\partial p_{k}} .$$
But, as you can see when comparing with equation (*) above, only the term in the large bracket is correctly reached, and there is a missing term to complete the bracket for $(f_{r},f_{s})$. I don't seem able to trace its origin here. Any help would be appreciated.
This exercise is found in Whittaker's book on Analytical Dynamics (p. 300).
This is an error in the exercise; it’s missing a factor $\frac12$.
This sort of thing is often most easily cleared up by substituting a simple example instead of trying to chase the error through the complicated general expressions. If you substitute $F=f_1$ and $\phi=f_2$, the result should be $(f_1,f_2)$, but the formula you quote from the exercise yields $2(f_1,f_2)$ (one contribution each for $(r,s)\in\{(1,2),(2,1)\}$). (Alternatively, the sum should be over $r\lt s$ instead of $r,s$.)
You can see that your result is half their result by noting that the first factor is antisymmetric in $r$ and $s$, so you can double the result by subtracting the same expression with $r$ and $s$ exchanged, which completes the second factor.