A new treatment of Lambertian mechanics

Lambert's formulas for the time of orbital flight between two points in space are rederived by first establishing a universal differential equation governing the time function, independent of the conic type of the trajectory, the focal characteristics of the trajectory sector, and the range angle. A unified form of Lambert's formulas is then obtained as the general solution of the differential equation, and the various forms of the classical Lambert's formulas are obtained as its particular solutions under different boundary conditions. Following this basic treatment, various hypergeometric expansions for Lambert's time function and its derivatives are developed, and the behavior of the function and its implications in the solution of Lambert's problem and the isochronous trajectories are briefly reviewed. Finally, a short comparison of the present treatment with those found in current literature on Lambertian Mechanics is made and briefly discussed.