Perturbation of Hamiltonian systems with Keplerian potentials (Q1107700)

The paper deals with the existence of periodic solutions for a second order Hamiltonian system of the form \[ (P_{\epsilon})\quad q''+\alpha q| q|^{-\alpha -2}+\epsilon V_ q(t,q)=0 \] for \(\epsilon\) small, where \(\alpha >0\) and V is T-periodic in t. It is shown that \((P_{\epsilon})\) has ``several'' T-periodic solutions bifurcating from the circular orbits of the unperturbed system \((P_ 0)\quad q''+\alpha q| q|^{-\alpha -2}=0\) without any assumption on V if \(\alpha\neq 1\), and under the assumption V even in q and T/2 periodic in t if \(\alpha =1\). In the case that V does not depend on time, a change of variable allows to prove the same existence result (for any given \(T>0)\) also for \(\epsilon =1\), provided V grows less than \(| q|^{-\alpha}\) in the origin.