= Hénon-Heiles potential
{c}
The <potential energy> $V(x,y)=\tfrac12(x^2+y^2)+x^2y-y^3/3$ in the <Hénon-Heiles system> has the following expression in <polar coordinates>:
$$
V=\frac12r^2+\frac13r^3\sin(3\theta),
$$
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 $(\pm\sqrt3/2,-1/2)$. All three <saddle points> have value $1/6$. Their common <level set> factors exactly:
$$
6V-1=(2y+1)\bigl[3x^2-(y-1)^2\bigr].
$$
Thus this <level set> is the union of the three lines $y=-1/2$ and $y=1\pm\sqrt3x$. 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 $0\leq E<1/6$, 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 $1/[6\sin^2(3\theta)]\geq1/6$.
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