# Shape of a Hanging Rope Using Calculus of Variations (General Case)

Given that the rope with uniformly distributed mass is attached to two points (important: the points do not have to be at the same height). How to derive the shape of such rope using calculus of variations? I guess, the total potential energy has to be minimal. Just in case, potential energy of a point is defined as $$mgh$$, where $$g$$ is the gravitational acceleration, $$m$$ is the mass and $$h$$ is the height. Potential energy of an arbitrary shape is the same $$mgh$$, where $$H$$ is the height of its center of mass.

• en.wikipedia.org/wiki/Catenary Dec 28 '19 at 19:14
• @joriki I can derive this without Wikipedia. The question is how to derive this formula using variational approach. Dec 28 '19 at 19:20
• Derive it for points at the same height and then argue that every part of the rope optimizes the subproblem for that part, hence the same shape must also solve the problem with different heights. Dec 28 '19 at 19:23
• A related recent question: Euler-Lagrange equation with Lagrange multipliers. Jan 7 '20 at 5:46