2
Using the deWitt condensed notation, we formally invert components of the 2-form
$$\Omega~=~\frac{1}{2} \mathrm{d}x^I \Omega_{IJ} \wedge \mathrm{d}x^J \tag{2.6.1}$$
into the Poisson bracket
$$ \frac{[\hat{A},\hat{B}]}{i\hbar}~\longleftrightarrow~\{A,B\}_{PB}~=~(\partial_IA) (\Omega^{-1})^{IJ} (\partial_JB).\tag{2.6.2}$$
[Ref. 1 puts Planck's constant $\hbar=...
2
Assuming from the notation
$$ \dot{x}^i~=~f^i(x,p,t), \qquad \dot{p}_i~=~g_j(x,p,t), \tag{1}$$
that the symplectic structure is the standard canonical symplectic structure $$\omega = \sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}x^i,\tag{2}$$
we get that
$$\begin{align}\mathrm{d}H(x,p,t)- \frac{\partial H(x,p,t)}{\partial t}\mathrm{d}t
~=~&\sum_{i=1}^n\...
2
Hints: The point is to construct the corresponding Hamiltonian formulation of the Lagrangian (2.86). The Dirac-Bergmann analysis reveals a primary constraint $$C\approx 0.\tag{2.87}$$
[When we try to isolate the two velocities $\dot{q}^i$ in the equation for momenta $p_i=f_i(q,\dot{q},t)$, we discover that the two momenta $p_i$ are not independent.] OP has ...
1
Well, given
$$
\dot{p} = - \frac{\partial H}{\partial x}
\quad
\text{and}
\quad
\dot{q} = \frac{\partial H}{\partial p}
$$
we have
$$
H = - \int dx_{i} \, g_{i}(x,p,t)
$$
and
$$
H = \int dp_i \, f_{i} (x, p, t).
$$
You deal with the constants of integration using the obvious constraint:
$$
- \int dx_{i} \, g_{i} (x, p, t)
=
\int d p_{i} \, f_{i} (x,...
1
The way I inverted the symplectic form was to write it out in coordinate notation. Not keeping track of overall constants (homework exercise!) we can write
$$
\Omega \sim \int d^2 z du \partial_u \delta A_z \wedge \delta A_{\bar z} \sim \int du du' d^2 z d^2 z' \partial_u \delta(u-u') \delta^2(z-z') \delta A_z(u) \wedge \delta A_{{\bar z}'} (u')
$$
In matrix ...
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