Problema de N Corpos e sua Relação com o Caos

Abstract

Daily experience shows that nature is made up of countless bodies interacting with each other, macroscopic, like planets and stars, or microscopic, such as atoms and molecules. The laws of motion established by Isaac Newton in the 17th century were developed to explain how forces act on bodies to produce motion and how bodies interact with each other. Newton was the first to formally solve the problem of N = 2 massive bodies moving in three-dimensional space, with their initial positions and velocities known, and interacting with each other under the exclusive action of gravity. The universal gravitational law was successfully applied to solve this problem, prooving the three laws of Kepler, which were established for the description of the motion of the planets in the solar system. From this result, Newton naturally extended his analysis to the three-body problem, in order to describe the Earth-Moon-Sun system. However, the problem for N ≥ 3 bodies has become a challenge for humanity, remaining even today without a complete analytical solution, that can provide the trajectories of the system. Nevertheless, restricted solutions of this problem showed that extremely complex dynamics could be observed in simple deterministic systems. These results marked the beginning of the study of chaotic systems. In this work we present how the fundamental properties of these systems can be introduced and discussed from the two-body problem in the context of celestial mechanics. We start with this problem to discuss the basic principles and concepts of Classical Mechanics such as the initial state of a system, its center of mass and the principles of conservation of energy and of angular and linear momenta. These were essential to demonstrate analytically that the trajectories of the system are described by stable and periodic orbits, as established by Kepler’s laws. The emergence of chaos in this system is observed by considering the general three-body problem. The addition of one more body caused the stability and regularity of the trajectories observed in the two-body system to disappear, giving way to irregular, non-periodic and extremely sensitive dependence on the initial conditions of the system. All analysis of the chaotic system was done numerically. By comparing the results obtained for the two- and three-body systems, it was possible to show that the chaotic systems are characterized not only for being non-linear and showing sensitive dependence on the initial conditions, but also for having a deterministic dynamics.