A hybrid symbolic-numerical framework for artificial satellite theory and dynamics using Maxima and Julia

Abstract This work presents the development of a hybrid symbolic-numerical framework dedicated to the automation of perturbation theories in celestial mechanics, with a particular emphasis on the theory and dynamics of artificial satellites and space mission design. The system is founded upon the interoperability between two complementary tools: symcelmech, developed in Maxima, and CelestialMechanics.jl, implemented in the Julia language. symcelmech implements the Hori-Deprit recursive method based on Lie transformations, employing Poisson series, closed form and expansions in powers of eccentricity for the systematic elimination of short periodic and, where applicable, long periodic terms in high-order Hamiltonians. The resulting analytical expressions are exported via text files to CelestialMechanics.jl, a high-performance toolkit designed for high-fidelity numerical propagation and the validation of analytical results. The framework’s modular architecture focuses on managing symbolic complexity (expression swell) and creating an automated pipeline that converts complex algebraic expressions into optimized numerical functions. The system’s feasibility is demonstrated through the processing of Hamiltonians perturbed by zonal and tesseral harmonics, highlighting that the proposed algorithmic structure enables the systematic processing of high-order dynamical models with analytical rigor and computational efficiency. The framework is open-source, providing a robust platform for the development of modern perturbation theories and for the design and analysis of space missions.