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=CP1CPS 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.