= Solution
Order the phase-space coordinates as
$$
x=(q_1,\ldots,q_n,p_1,\ldots,p_n),
\qquad
\Omega=\begin{pmatrix}0&I\\-I&0\end{pmatrix}.
$$
Then <Hamilton's equations> are
$$
\dot x_a=\Omega_{ab}\frac{\partial H}{\partial x_b}.
$$
The <Poisson bracket> is
$$
\{f,g\}=\frac{\partial f}{\partial x_a}
\Omega_{ab}\frac{\partial g}{\partial x_b},
\qquad
\boxed{\{x_a,x_b\}=\Omega_{ab}}.
$$
For $X=X(x)$ with Jacobian $J_{ab}=\partial X_a/\partial x_b$, the chain rule gives
$$
\dot X=J\Omega J^T\nabla_XH.
$$
This has Hamiltonian form with the same canonical matrix for every $H$ exactly when
$$
\boxed{J\Omega J^T=\Omega}.
$$
Taking determinants gives $(\det J)^2=1$; the symplectic condition fixes the orientation, so $\det J=1$. Hence the change-of-variables formula shows that the phase-space volume element is invariant. This is <Liouville theorem in Hamiltonian mechanics>.
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