## Perturbation of Hamiltonian systems with Keplerian potentials.(English)Zbl 0653.34032

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.
Reviewer: A.Ambrosetti

### MSC:

 34C25 Periodic solutions to ordinary differential equations 70H05 Hamilton’s equations
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### References:

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