Re: [math-fun] onion of revolution?

Once you have stopped actively twirling, the equilibrium state minimizes total energy, potential energy plus kinetic energy. This is a standard problem in calculus of variations with one fixed boundary and one free boundary. It gives rise to a differential equation which may or may not have solutions expressible using well known functions, e.g. Bessel functions. This problem is typical of those on the infamous Cambridge Math Tripos examinations, given from the early 19th century to about 1905. Whittaker's _Analytical Dynamics_ will tell you more than you want to know about such problems.