Abstract

The long-term evolution of planetary systems is a classical problem in celestial mechanics, commonly studied through secular theories obtained by averaging out the fast orbital motion. The remarkable diversity of extrasolar planetary systems often pushes classical secular models beyond their traditional domains of applicability, motivating the development of more general secular theories. In this talk, we present two complementary approaches toward this goal. The first is a numerical implementation of second-order canonical perturbation theory for the planetary three-body problem, based on the Lie transform formalism. We derive the secular Hamiltonian up to second order in the planetary mass ratio without any expansion in the orbital elements, and apply the resulting model to planetary systems near or away from MMRs. We show that second-order terms substantially improve the accuracy of the predicted orbital precession frequencies. The second approach revisits the analytical expansion of the disturbing function, which encodes the gravitational interactions between planets. We introduce a hybrid formulation combining elements of the classical Laplace and Legendre series, whose terms preserve an exact dependence on both eccentricity and semi-major axis ratio. We show that the resulting Laplace-Legendre expansion bridges the domains of applicability of the two classical approaches, providing a single framework applicable across a wide range of orbital configurations.

Mathematical Physics and Dynamical Systems Seminar