Periodic perturbations of central force problems and an application to a restricted 3-body problem.

We consider a perturbation of a central force problem of the form x ¨ = V ′ (| x |) x | x | + ε ∇ x U (t , x) , x ∈ R 2 ∖ { 0 } , where ε ∈ R is a small parameter, V : (0 , + ∞) → R and U : R × (R 2 ∖ { 0 }) → R are smooth functions, and U is τ -periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε = 0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ -periodic solutions bifurcating from invariant tori at ε = 0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V (r) = κ / r α for α ∈ (− ∞ , 2) ∖ { − 2 , 0 , 1 }). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction. [ABSTRACT FROM AUTHOR]