Perturbative Methods in Studying the Secular Evolution of Many Body Problem with Variable Masses

The classical many body problem with constant masses is non-integrable, therefore, to investigate the dynamics of planetary systems, perturbation theory based on the exact solution of the two-body problem is usually applied. However, in the case of variable masses, the two-body problem is generally non-integrable as well. Consequently, to apply perturbation theory, another integrable model problem is used as the unperturbed motion, the solutions of which are close to the solution of the two-body problem with variable masses (in particular, when the mass changes according to Meshchersky's first law, the solutions coincide exactly). In this talk, perturbation theory is used to derive the differential equations determining the orbital elements of the relative motion of one body around another. Here, the models of perturbed aperiodic motion along a conic section and along a quasi-conic section are considered and compared. It is shown that the model based on aperiodic motion along a quasi-conic section is more suitable for generalization to the many body problem with variable masses. N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).