Topics in Modern Mathematics

1.1. Nearly Integrable Hamiltonian Systems. In this work we examine the system of Hamiltonian equations i = _ iJH , ~ = iJH iJcp iJl with the Hamiltonian function H = Ho(l) + eH. (I. cp). (1.1) where E: "1 is a small parameter, the perturbation E:Hl (I ,cp) is 2n periodic in CP=CP1,"'CPS' and I is an s-dimensional vector, I = Il, I s The CPi are called angular variables, and the Ii action variables. A system with a Hamiltonian depending only on the action variables is said to be integrable, and a system with Hamiltonian (1.1) is said to be nearly integrable. The system (1.1) is also called a perturbation of the system with Hamiltonian Ho. The latter system is called un perturbed. 1.2. An Exponential Estimate of the Time of Stability for the Action Variables. Let I(t), cp(t) be an arbitrary solution of the per turbed system. We estimate the time interval during which the value I(t) differs slightly from the initial value: II(t)-I(O) I "1. The main result of the work is Theorem 4.4 (the main theorem) which is proved in [1]. This theorem asserts that the above-mentioned interval is estimated by a quantity which grows exponentially as the value of perturbation decreases linearly: 1/(t)-/(O)I 0 and b 0 are given l.n Sec. 4 [IJ.