Integrable Systems: Do They Exist?

NOTICE

• DO NOT TRUST THE FOLLOWING

History

• Kepler found integrable systems and elliptic orbits
• Priority for the principle of least action is not so clear but Leibniz found early
• Hamilton investigated the principle of least action, or rather he called it the principle of stationery action, and analogy between lights and planets in the form of calculus of variations
• Poincare investigated three-body problem and found that no analytic solution other than the Lagrange points. With his recurrent theorem, energy-conserving orbits go arbitrary near the starting point
• (Supposedly) von Neumann might have asked himself: There's no such thing like thermodynamically isolated system, other than possibly the whole universe. The second law is solid and Boltzmann's "heat death" look so inevitable. We can see that Haar invariant measure exists iff the universe is uniform and eternal. But it does not seem the general case. Why can we calculate the time-evolution of orbits as if they were ergodic?
• Kuramoto says that pull-in synchronization is observed perversely in chemistry and biology

Answer?

• Even blackholes are predicted to evaporate someday, but gravity works against dissipation, radiation for a while
• Lucretius' mystic clinamen works for both directions: heavy mass point might be somewhat self-stabilizing in its orbit, bend everything, including radiation from itself, slightly toward itself and steal momentum even from the (vacuum?). Orbits of lighter ones only emit photons, loss energy, get smaller, more uncertain and disappear earlier

TODO

• Isaac, after all, what is gravity?