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The results of Chandrasekhar (1969) are generalized to polytropes, using a formalism based on ellipsoidal energy variational principle to construct approximate stellar equilibrium solutions and study their stability. After reviewing the energy variational method and describing the approach, several equivalent stability conditions are established and secular vs. dynamical instabilities are discussed. Then, the equilibrium structure equations are derived for isolated, rotating polytropes, and axisymmetric configurations (compressible Maclaurin spheroids) are considered. Particular attention is given to triaxial configurations, either in a state of uniform rotation (generalizing the classical Jacobi ellipsoids) or with internal fluid motions of uniform vorticity (the compressible analogues of Riemann-S ellipsoids) and to the stability of these single star configurations. The compressible generalizations of the Roche and Roche-Riemann problems for a polytrope in orbit about a point-mass companion are solved, and the generalized Darwin problem for two identical polytropes in a binary is considered.
Lai et al. (Wed,) studied this question.