## Thesis etd-11302022-141600 |

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Thesis type

Tesi di laurea magistrale

Author

BATTAGLINI, ALICE

URN

etd-11302022-141600

Thesis title

A detailed proof of two theorems on the restricted three-body problem due to McGehee

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Dott. Baù, Giulio

Keywords

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

Graduation session start date

16/12/2022

Availability

Full

Summary

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.

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.

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