The extended, circular, planar, restricted three-body problem

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Antoni Benseny[en] This thesis deals with the study of the Planar Circular Restricted Extended Three-Body problem, which is a generalisation of the regular Planar Circular Restricted Three-Body problem in which we replace the classical gravitational field $(F \sim r^{-2})$ with another one such that $F\sim r^{\alpha}$, with arbitrary values of $\alpha$, with the final objective of performing accurate simulations of the trajectory of the third body. We begin by precisely stating the problem as well as introducing some concepts that will be needed throughout this thesis for its development. We will then move onto the study of Hill Regions, which are the regions of the plane in which motion of the third body is possible. These regions are related to the value of the Jacobi constant, the only constant variable known for the restricted three-body problem. Following we will deal with the regularisation of the problem, a useful concept that becomes of utmost importance when computing the trajectory of the third body near the collisions. Finally we outline some important points that arose when computing the trajectory and how did we overcome them