High order normal form construction near the elliptic orbit of the Sitnikov problem

Abstract

We consider the Sitnikov problem; from the equations of motion we derive the approximate Hamiltonian flow. Then, we introduce suitable action-angle variables in order to construct a high order normal form of the Hamiltonian. We introduce Birkhoff Cartesian coordinates near the elliptic orbit and we analyze the behavior of the remainder of the normal form. Finally, we derive a kind of local stability estimate in the vicinity of the periodic orbit for exponentially long times using the normal form up to 40th order in Cartesian coordinates.