| Abstract: | The three-body problem, dealing with the dynamics and evolution of three gravitating bodies is one of the oldest, generally still open problems in modern science, from the days of Newton to these days. Its study frustrated Newton, explored by Euler and Lagrange, and lead Poincare to introduce the field of chaos. Indeed, it is an active and extensively studied field to this day both in physics and mathematics. The evolution of three-body systems plays a key role in the evolution of stars and planets in the universe and serves as a major player in the production of various cosmic explosions and the production of gravitational wave sources. It thereby, indirectly, also affects the production of (mostly) heavy elements in the universe, produced in such explosions. I will briefly introduce the history of this problem and its importance, and then explore the dynamics of three-body systems in different regimes. I will first describe the evolution of secular hierarchical triple systems, which have some similarities to coupled oscillators, and pinpoint various important implications for such evolution, studied in my group, from the scales of Solar system asteroids and moons, through mergers of compact objects such as white dwarfs and neutron stars, to the production of gravitational-wave sources near massive black holes. In the second part of the talk, I will focus on the solution to the chaotic, non-hierarchical regime of the three-body problem considered a major challenge and the holy grail in this field. It could effectively be analyzed only through brute-force few-body numerical simulations available with sufficient computational power only over the last few decades. Due to the chaotic nature of the problem, predicting the evolution of a single chaotic three-body system is effectively impossible, however, as I'll show, a solution to the distribution of the final outcomes, i.e. the cross-sections and branching ratios for the various outcomes can be tackled using detailed-balance approach and coupled with the use of a random-walk approach. Our novel solution provides, for the first time, a statistical-analytical description of the chaotic three-body problem throughout its dynamical evolution and can directly reproduce the results of the decades of numerical studies, without even requiring any fitting parameter, solving the problem and effectively making decades of numerical simulations obsolete. Furthermore, our approach allows a robust method to include additional dissipative processes to the problem and/or additional external potentials and perturbations, aspects that are critical for realistic astrophysical systems. |