Assume we have a cable which endpoints is attached to two points at $(x,h)$ and $(x+\Delta_x,h)$.
Further assume it has some mass density distribution, $\rho(m),m \in [0,l]$ and is of some length $l > \Delta_x$.
Let us call its position at $m$ : $(f_x(m),f_y(m))$. In other words $f_x$, $f_y$ are such functions so that if only external (thanks to Andrei) force is acting downwards $(-g{\bf \hat y})$ (gravity force) everywhere and of course whatever forces needed at the endpoints. Then we seek the equilibrium.
How can we find $f_x,f_y$ given the above arrangement?
Own work: I think we can maybe try to express it as an energy minimization problem. However I don't know quite how to set it up.
(Other solutions not using energy minimization are also welcome!)
First of all, you cannot have equilibrium with only one force. You also need the tension in the cable.On any piece of the cable you have three forces: the tension to the left, the tension to the right, and the gravity. What you are describing is the caternary problem.