This is an old question but I don't see the obvious answer, so here we go.
A huge field of research in mathematical physics during the XIXe century revolved around giving explicit solutions to the equations of classical mechanics, using brute computation. At that time, finding a new first integral in the equations of motions of some physical system was a sure path to academic fame. If sufficiently many first integrals are found, the system is integrable. If moreover the motion is constrained to a bounded domain of the phase space , it is quasi-periodic and thus both regular and recurrent. The long term behavior of the system is thoroughly described.
There was certainly some hope at first to show the stability of the solar system, or at least the three body problem, by finding these first integrals. Huge efforts went into that line of research. A century later, we know that the three body problem is not integrable in general and integrability is a pretty rare property of dynamical systems. In particular it is not stable by perturbation.
But wait, we know that Earth won't suddenly fly towards Pluto and stay there forever. Actually, we are pretty sure that it will come back to its current position, and this follows from Poincaré recurrence theorem, once you note that the Liouville measure is left invariant by the motion. No need to explicitly solve the equations of motion, and the result is so general that it applies to all hamiltonian systems in restriction to a compact level of energy. And the proof is short and elegant! To be pedantic, this was a paradigm shift and the birth of a new method in the field of mathematical physics, best described by Poincaré himself in his numerous books.
We now consider the Poincaré recurrence theorem as marking the birth of a new mathematical discipline called ergodic theory, with striking applications to arithmetic, Lie groups, foliations, moduli spaces etc. To the point that people seem to have forgotten its celestial origins.