I would like to obtain a general formula to verify if a certain time-$t$ map of a Hamiltonian flow is twist. I have a Hamiltonian $1$ degree of freedom system $H=H(q(t),p(t))$, such that all orbits are periodic with equations of motion
$$\dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p}=-\frac{\partial H}{\partial q}$$
Now the time-$t$ map of the flow generated by $H$ $$q(t)=f(q(0), p(0))$$
$$p(t) = g(q(0),p(0))$$ where $f$ and $g$ are some functions, is, in terms of $H$ and its derivatives,
$$q(t) = q(0) + \int_{0}^{t}\frac{\partial H}{\partial p} d\tilde{t} $$ $$ p(t) = p(0) - \int_{0}^{t}\frac{\partial H}{\partial q} d\tilde{t}$$
The twist condition for the time-$t$ map is $\frac{\partial f}{\partial p(0)} \neq 0$. How could I obtain an expression for $\frac{\partial f}{\partial p(0)}$ in terms of $H$? Some kind of differentiation under the integral sign?