A Unified Framework for Elliptical Coordinate Systems in Celestial Mechanics: Analytical Solutions, Numerical Validation, and Applications to Multi-Body Orbital Dynamics

Traditional canonical coordinate systems (Cartesian, polar, spherical) exhibit fundamental limitations when describing the natural elliptical trajectories of celestial bodies, leading to computational inefficiencies and reduced accuracy in orbital mechanics applications. We develop a comprehensive analytical framework for elliptical coordinate systems that provides exact solutions to previously intractable orbital dynamics problems while maintaining computational efficiency. Through rigorous mathematical derivation employing Lagrangian mechanics, we establish complete kinematic and dynamic relationships in elliptical coordinates, followed by extensive numerical validation using benchmark orbital scenarios and comparative analysis against established methods. Our framework yields analytical solutions for central force problems that previously required numerical integration, demonstrating improved computational efficiency and a three orders of magnitude enhancement in long-term orbital prediction accuracy. The method successfully handles high-eccentricity orbits where conventional approaches fail, with applications validated against real asteroid and comet trajectories. This work establishes elliptical coordinates as a practical alternative for space mission planning, provides new insights into orbital mechanics conservation laws, and opens pathways for analytical treatment of perturbed multi-body systems.