Foliation of the phase space for the Kepler problem with anisotropic perturbations

We study a particular perturbation of the Kepler problem given by the potential U(r,θ)=-1/r-b/r2(1+∈cos2θ), where b and ε are the perturbation parameters. This problem has two first integrals in involution: the first one is the well known Hamiltonian H=(p2r+p2θ/r2)-1/r-b/r2(1+∈cos2θ); the second one is given by G=p2θ/2-b/(1+∈cos2θ). The sets H -1(h),G -1(g) and H-1(h){n-ary intersection}G-1(g) are invariant under the flow of the Hamiltonian system. From here we obtain a nice foliation of the phase space. In this paper we study the topology of the above foliation. © 2008 Birkhäuser Verlag Basel/Switzerland.