Pendulum separatrix approach takes infinite time
= Pendulum separatrix approach takes infinite time
{title2=$\int_0^\pi\frac{dx}{2\cos(x/2)}=\infty$}
For the nondimensional <simple pendulum> $\ddot x=-\sin x$, the <separatrix> has energy $\dot x^2/2+1-\cos x=2$. Starting at $x=0$ with velocity $2$, its time to angle $a<\pi$ is $\log(\sec(a/2)+\tan(a/2))$. The saddle at angle $\pi$ is approached only as time tends to infinity.
// Target: classical-mechanics.bigb