This is an introduction to the mathematical theory of celestial mechanics.
Since the foundamental work of Sir Issac Newton, the Newtonian N-body problem were important topics of science for centuries, both for theoretical interests and for practical purposes. While the two-body problem is a highly symmetric integrable system, the geometry and dynamics of the three-body problem are already highly complicated, and the phenomenon of deterministic chaos already appear: an extremely small difference of initial data will lead to a drastic difference of orbits after a sufficiently long time. These systems also serve as natural examples of symmetric Lagrangian/Hamiltonian systems.
In this course, we shall present some fundamental theory of the N-body problem with in particular a detailed study of the two-body problem, its solutions, symmetries and some ways to eliminate the singularities of two-body collisions. We shall then focus on the applications of variational methods to this problem which allow to identify interesting periodic orbits.
