A well-known problem in mechanics is the three-body problem, in which we have three masses interacting only by their mutual gravitational force. Between the methods employed in its solution, from analytical to numerical, we present, in this work, a computational approach using dynamical systems. To do this, we show the fundamental concepts about this mathematical tool, highlighting the stability and the Lyapunov exponents, using several examples built with the mathematical softwares wxMaxima, Maple e Mathematica. To treat the three-body problem, we did a brief historic and formulated it beginning with the universal law of gravitation and the second Newton's law, restricting the movement to the x − y plane, to, thus, reduce from 18 to 12 the number of equations that are necessary in the associated dynamical system. By using the softwares, thus, we plot the trajectories of the bodies for some known solutions and we have used the Lyapunov exponents to study their stability. As expected, in all the cases these exponents indicated chaos in this system, which implies in the instability of the studied solutions, as well a great sensibility to the initial conditions. This work shows, then, a method to the combination of analytical and numerical processes, aiming in obtaining important qualitative informations for dynamical systems of interest, by using computational softwares.
|
