Hénon-Heiles system

The Hénon-Heiles system is a Hamiltonian system with two degrees of freedom and Hamiltonian functionIts Hamilton's equations give $\ddot{x}=-x-2xy$ and $\ddot{y}=-y-x^2+y^2$. Its potential energy combines an isotropic quadratic minimum with a cubic interaction.

The potential energy $V(x,y)=\frac{1}{2}(x^2+y^2)+x^{2}y-y^3/3$ in the Hénon-Heiles system has the following expression in polar coordinates:so it is invariant under rotation through $2\pi /3$. Solving $\nabla V=0$ gives a strict local minimum at the origin and three saddle points, at $(0,1)$ and $(\pm \sqrt{3}/2,-1/2)$. All three saddle points have value $1/6$. Their common level set factors exactly:Thus this level set is the union of the three lines $y=-1/2$ and $y=1\pm \sqrt{3}x$. They bound an equilateral triangle containing the local minimum. The Hessian matrix is the identity at the origin; at each of the other three stationary points it has eigenvalues $3$ and $-1$. For energies $0\le E<1/6$, the central accessible region is bounded; above that value the three saddle passages allow paths from the centre to arbitrarily large radius. This follows directly from the radial potential: in each direction with $\sin (3\theta )<0$, its radial maximum occurs at $r=-1/\sin (3\theta )$, with value $1/[6{\sin }^2(3\theta )]\ge 1/6$.