On the equilibrium of a fluid mass at rest

Abstract

1. The following question was brought up a considerable time ago by both Liapounofft and Poincare, t but has apparently not been answered up to the present: Consider a homogeneous incompressible fluid whose particles attract one another according to Newton's law and which is acted on by no external forces. Then, are there any positions of equilibrium for the fluid besides the sphere? It will be shown that there are no such positions, whether of stable or unstable equilibrium. 2. A necessary condition for equilibrium will be obtained by examining an approximating figure made up of elementary parallelepipeds, or parallel rods. The rods will be treated as rigid and free to move in the direction of their lengths only, so that perpendicular distances between them remain unchanged. They will be so chosen that whenever two collinear rods are moved into contact with one another their ends will fit together exactly and the rods will become merged into one. If the approximating system consists of two rods only, it can be seen by inspection that its potential energy diminishes continuously as the centers of the rods approach one another. Equilibrium can, therefore, only occur if the rods are touching end to end or if they are symmetrical about a perpendicular line through their centers. If there are more than two rods, the potential energy of the approximating system is equal to the sum of the potential energies of all sub-systems consisting of two rods only. Suppose the rods are set in motion in such a way that the center of each rod approaches a fixed perpendicular plane, ir, with a velocity equal to its instantaneous absolute distance from wr. Then, as the system moves, the distance between the centers of two rods never increases, while, on the contrary, it decreases whenever the centers are at unequal distances from ir.