We present a novel and rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the longstanding challenge of accurately modeling the rotational deformation of celestial objects such as stars and planets. By integrating classical Newtonian potential theory with modern mathematical techniques, we develop a unified formalism that significantly enhances both the precision and generality of shape modeling in astrophysical contexts. Our methodology employs Lie group theory and exponential mapping to characterize vector flows associated with rotational deformations. We derive functional equations governing perturbations in density and gravitational potential, which are analytically resolved using the shift operator and Neumann series summation. This approach extends Clairaut’s classical linear perturbation theory into the nonlinear regime. The resulting formulation yields an exact nonlinear differential equation for the shape function, describing hydrostatic equilibrium under rotation without relying on the assumption of slow rotation. This equation generalizes Clairaut’s classical result—which applies only in the linear, slow-rotation regime—by incorporating nonlinear effects and accommodating sufficiently large rotational speeds, the upper limit of which remains subject to further investigation. We refer to this as the generalized Clairaut equation to emphasize its role as a natural extension of Clairaut’s original theory. We validate this formulation by deriving exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and the unit-index polytrope. Furthermore, we introduce spectral decomposition techniques to analyze the radial harmonics of the shape function and associated gravitational perturbations. Utilizing Wigner’s formalism for angular momentum addition, we compute higher-order spectral corrections, thereby enhancing both the accuracy and computational efficiency of existing models. Our framework also derives rigorously defined boundary conditions for the spectral radial harmonics, enabling the systematic computation of Love numbers and gravitational multipole moments with high accuracy. This work represents a substantial advancement in the mathematical theory of rotating fluid figures, offering a comprehensive, nonperturbative methodology for modeling rotational deformations in astrophysical and planetary systems.

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• Continuum mechanics