24. An Introduction to Celestial Mechanics

Celestial mechanics is one of the most interesting applications of classical mechanics. This topic answers one of the oldest questions of facing humankind: what forces govern the motion of celestial bodies? How do celestial bodies move under the action of these forces? In this chapter we discuss the foundations of this subject. We first recall the two-body problem and introduce the orbital elements. Then, we analyze the restricted three-body problem in which the third body has a very small mass compared with the masses of the other two bodies. In particular, we describe the Lagrange equilibrium positions and their stability. Then, we consider the N–body problem showing that the Hamiltonian of this system is obtained by adding the Hamiltonian of the two-body problem, that describes a completely integrable system, to another term that can be regarded as a perturbation term. In the remaining part of the chapter, we show how the gravitation law for mass points can be extended to continuous mass distributions, and we evaluate the asymptotic behavior of the gravitational field produced by extended bodies. Then, we define the local inertial frames and tidal forces. Finally, we pose the problem of determining the form of self-gravitating bodies.