Various analogies for the equilibrium shapes of an elastic thread on two-dimensional surfaces

In accordance with the Kirchhoff analogy, the equilibrium equations of the elastic thread on a plane are equivalent to the equations of motion of a simple pendulum. This analogy is generalized to the case when the thread lays on a smooth curved surface. We derive the equilibrium equations of the threads in the general case and in the particular cases of planar, cylindrical and spherical surfaces. For these surfaces the Kirchhoff analogy is generalized to the case of a simple pendulum in an additional force field. There are also considered the electromagnetic and nonholonomic analogies for the equilibrium equations of an elastic thread.