A brief solution to three-body problem: Newtonian and Hamiltonian versions

The problem of the three bodies was cataloged as one of the best-positioned problems and the pinnacle of functional analysis by Poincaré himself when he discovered that the problem itself presents a chaotic behavior and that it was impossible to apply integrable methods to this system. Therefore, its analytical solution was impossible to obtain, since its solution strongly depended on the initial conditions (weak chaos). With the development of modern numerical methods, together with the immense advances in the hardware of the new computers, attempts have been made to attack this system from different schemes and numerical stencils, to describe the main physical properties of this system (the trajectory is only one of these). With this, in the present work, we will study this problem from the Newtonian and Hamiltonian versions and the restricted problem. Special interest will be devoted to the numerical analysis of this system, The work focuses on a pedagogical description of the topic (constructivist), academic clarity, and application of numerical analysis.