Example 6.5.4.
The system
\begin{align*}
\frac{dx}{dt} \amp = -2x - 3y^2\\
\frac{dy}{dt} \amp = -3x^2 + 2y
\end{align*}
is Hamiltonian since for \(H(x, y) = x^3 - 2xy - y^3\)
\begin{align*}
\frac{\partial H}{\partial y} \amp = -2x - 3y^2\\
\frac{\partial H}{\partial x} \amp = -(-3x^2 + 2y).
\end{align*}