Embracing Integrability in Stellar and Planetary Dynamics

Imagine the universe. Going down in astronomical size scale, focus on a galaxy containing myriads of stars, and on the other hand, look at one of these stars which happens to host a few planets. Apart from the fact that gravitation is the dominant force in both of them, the systems do not seem to have much in common. However, typical models of stellar and planetary systems are Hamiltonian, which enables confrontation in a mathematical sense. In a Euclidean setting, integrability of a Hamiltonian system means that we have an explicit global transformation which maps an orbit onto the surface of a torus in phase space. Motion on the torus is uniform, and hence, the orbit is known at all times. A well-known example, dating back to the 17th century, is the two-body problem. Its solution, the Keplerian orbit, often serves as a good approximation also in near-integrable multiplanet systems, and provides a basis for, e.g., developing efficient methods for orbit integration. This is actually the carrying theme in this monograph; using an integrable system in order to model something non-integrable