KAM Theorem

1. KAM Theorem

2. Energetic Pendulum • As the energy of a double pendulum increases the invariant tori change. • Some tori break up into discrete segments • Some new sets of tori emerge replacing old tori. l q m l f m Jf f

3. Rational Winding • Smooth curves on the Poincare section are perturbed by the non-linear effects. • An operator T can represent one iteration of the section • For finite values there are resonances • Appears as broken tori • Islands represent rational winding numbers

4. Elliptic and Hyperbolic • As the tori break up new fixed points emerge. • Some are not strictly fixed, but are cyclic. • The points near the fixed points have different properties. • Elliptic points are stable with near points staying near • Hyperbolic points are unstable elliptic hyperbolic Jf f

5. Poincare-Birkhoff • Consider three nearby tori • Rest frame of middle tori • Map Ts is fixed • Perturbation is zero • Turn on perturbation • Curves are distorted • Angular coordinates unchanged under map • Radial may be changed except for even number of fixed points C+ C0 C- elliptic hyperbolic C+ Ce C- TsCe

6. Kolmogorov, Arnol’d, and Moser • The KAM theorem relates the strength of the perturbation to the breaking tori. • Most tori remain for a small perturbation. • The set of remaining tori occupy a finite area in the Poincare section. • The tori that break have rational winding numbers • The last to break are the most irrational

7. Fraction Expansion • For any irrational number s, there is a rational approximation. • Improves as s increases • Expansions are better as continued fractions. • The expansion is unique r, s integers

8. Irrationality • Truncating the expression after n steps gives a rational fraction. • The slowest convergence is if all an= 1. • This is f the golden mean. • It is the “most” irrational number. r, s integers

9. The tori destroyed obey a a relationship with their winding number. K is the measure of all possible winding numbers The one valued f is the last KAM Tori Jf f next