In this chapter we apply the tools developed in the previous two chapters to an exploration of the orbital dynamics of bodies subjected to their mutual gravitational attractions. Many aspects of what we learned in Chapters 1 and 2 will be put to good use, and the end result will be considerable insight into the behavior of our own solar system. To be sure, the field of celestial mechanics has a rich literature that goes back centuries, and this relatively short chapter will only scratch the surface. We believe, however, that we have sampled the literature well, and selected a good collection of interesting topics. Some of the themes introduced here will be featured in later chapters, when we turn to relativistic aspects of celestial mechanics.
We begin in Sec. 3.1 with a very brief survey of celestial mechanics, from Newton to Einstein. In Sec. 3.2 we give a complete description of Kepler's problem, the specification of the motion of two spherical bodies subjected to their mutual gravity. In Sec. 3.3 we introduce a powerful formalism to treat Keplerian orbits perturbed by external bodies or deformations of the two primary bodies; in this framework of osculating Keplerian orbits, the motion is at all times described by a sequence of Keplerian orbits, with constants of the motion that evolve as a result of the perturbation. We shall apply this formalism to a number of different situations, and highlight a number of important processes that take place in the solar system and beyond.