On the Lyapunov theory of equilibrium figures of celestial bodies

Abstract

The work of A. M. Lyapunov on the theory of equilibrium figures of celestial bodies is analyzed. The main results are mentioned, such as sufficient conditions for the existence and uniqueness of solutions to the complicated integral and integro-differential equations of the problem; the solution of the stability problem for the MacLaurin and Jacobi ellipsoids; the solution of the existence and stability problem for figures branching from the ellipsoids; the solution of the problem for slowly rotating inhomogeneous bodies in terms of series (called now the Lyapunov series) in powers of a small parameter, which is equal in the first approximation to the centrifugal-to-gravitational force ratio; and the estimation of the convergence radius of the Lyapunov series. Further development of Lyapunov’s ideas and unsolved problems is discussed.