Hénon–Heiles system

Example

The Hénon–Heiles system is the Hamiltonian system on $\R^4$ with coordinates $(x, y, p_x, p_y)$ and Hamiltonian $$H = \frac{1}{2} \left( p_x^2 + p_y^2 \right) + \frac{1}{2} \left( x^2 + y^2 \right) + x^2 y - \frac{1}{3} y^3.$$ Its equations of motion are $$x'' = -x - 2 x y, \qquad y'' = -y - x^2 + y^2.$$ The energy $H$ is a first integral. The potential $V(x, y) = \frac{1}{2}(x^2 + y^2) + x^2 y - \frac{1}{3} y^3$ has a minimum at the origin and three saddle equilibrium points at $(0, 1)$ and $\left( \pm \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$. Check at $(0, 1)$: $\partial_x V = x + 2 x y = 0$ and $\partial_y V = y + x^2 - y^2 = 1 - 1 = 0$, and $V(0, 1) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$. The value of $V$ is $\frac{1}{6}$ at all three saddles, so for energies $0 < E < \frac{1}{6}$ the motion near the origin stays in a bounded triangular region.