Nonintegrability of Hamiltonian Systems Close to Integrable Ones

Abstract

We begin the consideration of the analytic obstacles to integrability with the analysis of Poincaré’s “basic problem of dynamics”. This problem deals with the Hamiltonian system whose Hamiltonian has the form

$$ H=H_0 (y_1 , \ldots ,y_n )+\varepsilon H_1(x_1,\ldots ,x_n ,y_1\ldots ,y_n )+\cdots.$$