The Kolmogorov-Arnold-Moser Theorem: “Here Comes the Surprise”
Around 1610, Galileo began the formalization of terrestrial mechanics, while Kepler did the same for celestial mechanics Sir Isaac Newton saw fit to combine these into one subject, unifying them with his calculus. Newton himself worked out the equations of motion for the two-body problem in order to demonstrate his method (1687). In 1788, the great Italian–bench mathematician Joseph Louis Lagrange, who had been working with Euler on the calculus of variations, was able to formalize his brilliant generalized method of finding the equations of motion for any mechanical system—the same method we use today. It was also Lagrange, and then Laplace, who recognized the difficulty of understanding the motion of three bodies, and so arose the question of the stability of the Solar System. In 1834, in Dublin, Sir William Rowan Hamilton placed position and momentum on equal footing as the canonical variables of dynamics, and he also showed that the variational principle beneath dynamics was the same as the principle of least time in optics, thereby uniting the two disparate formalisms at the theoretical level. Soon after Hamilton’s contribution, Joseph Liouville proved a theorem implying that if energy was conserved in a system, then any volume of initial conditions in phase space must be conserved throughout the evolution. Liouville’s theorem holds even if the system is ergodic and highly complex.
KeywordsPeriodic Orbit Canonical Transformation Irrational Number Level Curf Deterministic Chaos
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