Small Divisors Problem -- from Eric Weisstein's World of Physics

Small Divisors Problem

Given a perturbed Hamiltonian system,

(1)

where I and are the n-D action-angle variables. Hamilton's equations then become

(2)

(3)

where is the n-D frequency vector. Now, perturbatively construct a new set of canonical variables which transform the system into integrable form

(4)

by means of a generating function satisfying

(5)

(6)

S is expanded in a power series

(7)

As shown in Tabor (1989, pp. 103-104), can be written in a Fourier series as

(8)

If the fundamental frequencies are commensurable, i.e.,

(9)

the sum will diverge.

References

Marmi, S. "An Introduction to Small Divisors Problems" 27 Sep 2000. http://xxx.lanl.gov/abs/math.DS/0009232/.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.

Yoccoz, J.-C. "An Introduction to Small Divisors Problem." In From Number Theory to Physics (Ed. M. Waldschmidt, P. Moussa, J.-M. Luck, and C. Itzykson). New York: Springer-Verlag, pp. 659-679, 1992.