Perturbation theory

Abstract

Perturbation theory is an efficient tool for investigating the dynamics of nearlyintegrable Hamiltonian systems. The restricted three-body problem is the prototype of a nearly-integrable mechanical system (Section 6.1); the integrable part is given by the two-body approximation, while the perturbation is due to the gravitational influence of the other primary. A typical example is represented by the motion of an asteroid under the gravitational attraction of the Sun and Jupiter. The mass of the asteroid is so small, that one can assume that the primaries move on Keplerian orbits. The dynamics of the asteroid is essentially driven by the Sun and it is perturbed by Jupiter, where the Jupiter-Sun mass-ratio is observed to be about 10⁻³. The solution of the restricted three-body problem can be investigated through perturbation theories, which were developed in the 18th and 19th centuries; they are used nowadays in many contexts of Celestial Mechanics, from ephemeris computations to astrodynamics.