Μελέτη περιοδικών και ασυμπτωτικών λύσεων στο περιορισμένο πρόβλημα των τεσσάρων σωμάτων

Abstract

In this thesis we are concerned with the periodic and asymptotic solutions of the restricted four - body problem. In chapter 1 we describe the three - body and four - body problem, starting with historical information. We also present the needed equations of motion and integrals of the problem. In chapter 2 we study numerically the problem of four - bodies, according to the Lagrangian equilateral triangle configuration. We find the equilibrium points and the allowed regions of motion. In chapter 3 we study the stability of the relative equibrium solutions. We also illustrate the regions of the basins of attraction for the equilibrium points of the present dynamical model. In chapter 4 we present families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem. Series of critical periodic orbits of each family and in any case of the mass parameters are also calculated. In chapter 5 we study, numerically, families of simple non-symmetric periodic orbits of the restricted four-body problem, where we consider the three primary bodies as Sun, Jupiter and a Trojan Asteroid and as a massless fourth body, a spacecraft. The primary bodies are set in the stable Lagrangian equilateral triangle configuration. We also study analytically the solutions in the neighborhood of the stable equilibrium points and the linear stability of each periodic solution. The results of the chapters 2,3,4 and 5 have been published in three journals and a part of chapter 5 has been presented in an international conference. Chapters 2 and 3 have been published in “International Journal of Bifurcation and Chaos, 21, 2011, pp. 2179-2193” under the title of “Equilibrium Points and their stability in the restricted four-body problem”. Chapter 4 has been titled “Families of periodic orbits in the restricted four- body problem” and published in “Astrophysics and Space Science, 336, 2011, pp. 357-367”. Chapter 5 has been titled “Periodic solutions in the Sun - Jupiter - Trojan Asteroid - Spacecraft system,” and published in “Planetary and Space Science, 75, 2013, pp. 148-157”. The conference was the “10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina, Greece, 5-8 September 2011, pp. 23-24” and part of the chapter 5 was presented under the title of “Families of periodic orbits in the Sun - Jupiter - Trojan Asteroid system”. This thesis was compiled while the author was in receipt of “K.Karatheodory” research grant.