= Two stationary catenary rays in an inverse-height speed profile
{title2=$y=a\cosh(x/a),\quad a\cosh(d/a)=H$}
For speed $c(y)=1/y$ in $y>0$, <Fermat principle> gives the elapsed-time integrand $y\sqrt{1+(y')^2}$. Its symmetric stationary <light rays> through $(\pm d,H)$ are the displayed <catenaries>. Set $t=d/a$. The endpoint equation is $\cosh t/t=H/d$. This function decreases and then increases, with unique minimum where $t\tanh t=1$. Thus above the minimum there are two stationary rays, at the minimum one, and below it none. The travel time on either ray is $T=ad+(a^2/2)\sinh(2d/a)$, and its minimum height is $a$.
For $d=1$ and $H=\cosh1$, one solution is $a=1$, with $T=1+\sinh2/2$. A second has $a\simeq0.7021026952$, with $T\simeq2.8224491447$. Hence these endpoints do not uniquely specify the ray. Stationarity is distinct from global minimization: paths descending almost to $y=0$ have limiting travel time $H^2$, below either stationary value in this example.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1-q9-two-catenary-rays.png]
{title=Two stationary catenary rays connect the same endpoints}
{description=For speed $c(y)=1/y$, the endpoints $(-1,\cosh1)$ and $(1,\cosh1)$ admit stationary <light rays> with minimum heights $a=1$ and $a\simeq0.702103$. Their different travel times illustrate that <Fermat principle> is a stationarity condition.}
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