A variational formula for floating bodies

Well known first order necessary conditions for a liquid mass to be in equilibrium in contact with a fixed solid surface declare that the free surface interface has mean curvature prescribed in terms of the bulk accelerations acting on the liquid and meets the solid surface in a materially dependent contact angle. We derive first order necessary conditions for capillary surfaces in equilibrium in contact with solid surfaces which may also be allowed to move. These conditions consist of the same prescribed mean curvature equation for the interface, the same prescribed contact angle condition on the boundary, and an additional integral condition which may be said to involve, somewhat surprisingly, only the wetted region. An example of the kind of system under consideration is that of a floating ball in a fixed container of liquid. We apply our first order conditions to this particular problem.