Poincaré, la mecánica clásica y el teorema de la recurrencia

Abstract

This work commemorates the $101^{}{{}^{\hbox{th}}}$ anniversary of Henri Poincaré's death. We pinpoint his main contributions to classical mechanics while enlivening the discussion with a brief remembrance of his academic career. We employ a physical pendulum for illustrating his techniques for analysing properties of solutions to differential equations without actually solving them. We next use an elastic pendulum for exhibiting how Poincaré maps allow us to distiguish the periodic from the chaotic solutions, a type of solutions which Poincaré himself discovered while studying the famous three body problem. We also give a heuristic proof of his extraordinary recurrence theorem according to which every bound solution of a conservative dynamical system should return, after a time, $T_r$, to be as close as we like to its initial conditions. We regard this as a very important result that ought to be known by all physics students.