Exercise [20.15]

deant · 18 May 2012, 14:59

Laura's solution is here (as a .GIF image):

http://camoo.freeshell.org/20.15.gif

However, it seems to me (on a casual reading) that it contains several major errors.
In particular, $S_{ab}$ is supposed to be antisymmetric, so the statement:

Quote:

$S_{cb}$ is nonzero only if c is the p_d

index

is incorrect. Furthermore the equation:

Quote:

$S_{cb}\nabla_a\left\{\mathcal{H},\right\}^c = -\frac{\partial^2\mathcal{H}}{\partial p_d \partial q^d}$

manifestly makes no sense, as there are two free indices on the left, and none (or one, if d is unsummed) on the right. Finally:

Quote:

$S_{ab}=\frac{1}{2}\mbox{d}p_d\wedge\mbox{d}q^d$ and $S_{ba}=-\frac{1}{2}\mbox{d}q^d\wedge\mbox{d}p_d$

is also wrong, since the two right hand sides are equal, but $S_{ab}\neq S_{ba}$ .

I have posted my solution as Exercise [20.15]b; hopefully that is error-free!