Suppose we have a necklace made of a conducting material. We join the two ends and leave it on a frictionless non-conducting table. Then we charge it negatively. What is the equilibrium shape of the necklace? The answer to this is probably a circle. I am actually looking for the differential equation governing the dynamics of this necklace. Here's a (probably) simpler question posed more mathematically: We charge a non-self-intersecting closed curve on the plane negatively. The curve can be any closed curve. Let the modulus of elasticity, length and total charge be given. I am looking for the differential equation for this problem. Other results are also welcome, such as the tension in the curve at equilibrium. The differential equation is probably too complex. Book, article etc. suggestions are also welcome.