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 Exercise [20.15] 
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Joined: 12 Jul 2010, 07:44
Posts: 154
Post Exercise [20.15]
Laura's solution is here (as a .GIF image):


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:

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

is incorrect. Furthermore the equation:

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:

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!

18 May 2012, 14:59
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