Lagrangian mechanics question: A wire with an unlimited length and uniform linear mass density moves freely between two pulleys at heights y1 and y2, separated by a distance d.
(a) Demonstrate that determining the cable's curve is the same as determining the minimum surface of revolution -- i.e. rotating the cable around its projection onto the earth's surface. This "Plateau problem" is also known in the context of a soap bubble that connects two rings.
(b) Write down and solve the equations of motion you found for y(x).
(c) The solutions you get in (b) may not be unique. In which parameter regime do you anticipate a qualitatively different solution? Under this regime, what happens to the cable?