It is physically intuitive that the catenary is unique?

Question

The catenary minimizes the potential energy of a cable and has equation $y - y_0 = A \cosh (\frac{x-x_0}{A})$. It is physically intuitive that the catenary is unique, but is there a mathematical (rigorous) proof that this is so? Thanks.

Answer 1

Assume the cable is spanned form $(p_0,q_0)$ to $(p_1,q_1)$, $p_1>p_0$, and has a certain length $L$. Then its potential energy is minimal not only globally but also locally: If this cable passes through the points $(x_0,y_0)$ and $(x_1,y_1)$ and has length $s$ in between, then in the interval $[x_0,x_1]$ it assumes the exact shape that a cable of this length suspended from $(x_0,y_0)$ and $(x_1,y_1)$ would have. It follows that minimizing the potential energy globally enforces a certain local condition which translates into a second order differential equation for the curve $x\mapsto y(x)$. The derivation of this equation happens in the first pages of any book on variational calculus, and its solutions are curves of the form $y(x)=c \cosh(a x + b)$. It turns out that the constants $a$, $b$, $c$ are uniquely determined by $(p_0,q_0)$, $(p_1,q_1)$ and the length $L$ of the cable, as long as $L\geq\sqrt{(p_1-p_0)^2+(q_1-q_0)^2}$.