• homoclinic orbits
• McGehee
• equilibrium region
• proof
• theorems
• Hill's region
• Hamiltonian mechanics
• Lagrangian points
• restricted three-body problem
• three-body problem
• celestial mechanics

The three-body problem is one of the central topics in celestial mechanics. Indeed,
it is very natural to ask what is the behavior of three celestial bodies moving under
their mutual gravitational attraction. Answering this question is not easy due to the
non-integrability in the Liouville sense of this problem.
To bring the study back to the integrable two-body problem, a simplification of the
original model was considered. In the restricted three-body problem, a body of negligible
mass, called third body, moves under the influence of two massive bodies, the primaries.
Sometimes in literature the more massive one is called primary and the other is the
secondary. Having negligible mass, the force that the third body exerts on the primaries
may be neglected. Furthermore, the motion of the primaries can be taken to be circular
around their common center of mass and the third body can be assumed to move in
the plane defined by the primaries’ orbits. All these properties are characteristic of the
planar circular restricted three-body problem, or briefly PCR3PB. If we assume that the
mass of the secondary is much smaller than the mass of the primary, the PCR3BP can
be considered as a perturbation of the two-body problem.
Euler and Lagrange have proved the existence of five equilibrium points
for the PCR3BP, which in the literature are commonly referred to as Lagrangian points.
Moreover, Lyapounov has shown the existence of a family of periodic orbits sur-
rounding the two Lagrangian points nearest to the secondary for energy values just above
the ones of the two points. They are called Lyapounov orbits.
The interest here is in the existence of orbits asymptotic at both ends to a fixed
Lyapounov orbit l, or equivalently homoclinc orbits relative to l. In this work we expand McGehee’s doctoral thesis in which the existence of
such kind of orbits is proven without providing many details. His work consists of an
introductory part followed by two theorems which we will refer to as the first and the
second McGehee’s theorem.
McGehee’s proof is based on the construction of a function which counts the number
of times an orbit segment with endpoints near the Lyapounov orbit winds around a solid
invariant torus. This prompted us to expand KAM theory to prove the existence of such
a torus. Indeed, as we have just noted, the PCR3BP can be viewed as a perturbation of
the integrable two-body problem, where the motion can be confined to an invariant torus.
Kolmogorov’s theorem, or its iso-energetic version, states that if
the integrable system is subjected to a weak perturbation, under suitable assumptions,
some of the invariant tori are deformed and survive. Persistent tori are called KAM tori.
This work is organized as follows:
• Chapter 1 begins with a description of the planar circular restricted three-body
problem. It is formulated by adopting the Hamiltonian formalism and considering a
reference frame rotating with the binary. The chapter continues with a discussion
on the location of the Lagrangian points and with a description of the energy
surfaces and of their projection onto the position space. In the last sections of the
chapter we study the PCR3BP in a neighborhood of the equilibrium points nearest
to the secondary. In particular, we linearize Hamilton’s equations and find their
solution in an appropriate reference frame. Of particular interest is the geometry
of the equilibrium region, namely a subset of the energy surface projecting in a
neighborhood of one of these points.
• Chapter 2 introduces the notation and preliminary notions to McGehee’s theorems.
The most important result presented in this chapter is McGehee’s representation
of the equilibrium region. Indeed in its thesis McGehee made it possible to visualize
the equilibrium region by demonstrating that it is homeomorphic to a spherical
annulus.
• Chapter 3 contains the statements and proofs of the two theorems presented by
McGehee. Before giving the proof of the first theorem, we briefly discuss
how it leads to prove the existence of a homoclinic orbit. In the last part of the
chapter we outline the proof of the second theorem retracing what we did for the
first.
• In Appendix A the iso-energietic version of Kolmogorov’s theorem is stated and is
applied to the PCR3BP. Finally, Appendix B recalls some analytical tools useful
for the development of the thesis.