Exploring the orbital evolution of planetary systems

The aim of this paper is to encourage the use of orbital integrators in the classroom to discover and understand the long term dynamical evolution of systems of orbiting bodies. We show how to perform numerical simulations and how to handle output data in order to reveal the dynamical mechanisms that dominate the evolution of arbitrary planetary systems in timescales of millions of years using a simple but efficient numerical integrator. Through some examples we reveal the fundamental properties of planetary systems: the time evolution of the orbital elements, the free and forced modes that drive oscillations in eccentricity and inclination, the fundamental frequencies of the system, the role of the angular momenta, the invariable plane, orbital resonances, and the Kozai-Lidov mechanism.