Solving the Three-Body Problem via Hierarchical Differential Algebraic Closure: A Rigorous Symbolic-Numeric Framework

This paper presents a comprehensive framework for solving the three-body problem using hierarchical differential algebraic closure. We rigorously transform the classical Newtonian gravitational equations into a multivariate polynomial system through systematic introduction of auxiliary variables and constraint equations. The extended methodology from [25] is carefully adapted to celestial mechanics, providing a recursive solution structure that fully exploits the inherent symmetries of three-body configurations. Our approach achieves machineprecision accuracy while exactly preserving fundamental conservation laws at the algebraic level. Theoretical analysis demonstrates that all classical solutions, including Lagrange points and periodic orbits, admit analytic expressions within the differential algebraic closure. Extensive numerical validation confirms unprecedented accuracy and long-term stability compared to traditional numerical integration methods, with detailed error bounds and convergence proofs.