= Solution
Write the <exact differential form> as $\beta=dh$ and take the <cotangent fiber translation> family $\rho_t(x,\xi)=(x,\xi+t\,dh_x)$. Its generating <vector field> is vertical and in local coordinates is
$$
X=\sum_j\frac{\partial h}{\partial x_j}\frac{\partial}{\partial\xi_j}.
$$
With $\omega=\sum_jdx_j\wedge d\xi_j$ and the <Hamiltonian vector field> convention from Question 3,
$$
\boxed{\iota_X\omega=-\pi^*dh=dH,\qquad H=-h\circ\pi.}
$$
Thus $\rho_t$ is a <Hamiltonian isotopy> and $f=\rho_1$ is a <Hamiltonian diffeomorphism>. The <local flow> is explicitly defined for every real $t$, regardless of <compactness>. With the alternative convention $\iota_{X_H}\omega=-dH$, the same family is generated by $H=h\circ\pi$.
The definition in the question does not require compact support for the <Hamiltonian function>; the displayed <Hamiltonian diffeomorphism> uses that definition.
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