The global flow of the quasihomogeneous potentials of Manev–Schwarzschild type

In this paper, we study two-body problems defined by a potential of the form V(r)=a/r+b/r2+c/r3, where r is the distance between the two particles and a, b, c are arbitrarily chosen constants. The HamiltonianH(r,pr,θ,pθ)=12pr2+pθ2r2+ar+br2+cr3and the angular momentum pθ=r2θ̇ are two first integrals, independent and in involution. Let Ih (respectively Im) be the set of points on the phase space on which H (respectively pθ) takes the value h (respectively m). Since H and pθ are first integrals, the sets Ih, Im and Ihm=Ih∩Im are invariant under the systems associated to the Hamiltonian H. We characterize the global flow of the systems when a, b, and c vary, describing the foliation of the phase space by the invariant sets Ih, the foliation of Ih by the invariant sets Ihm and the movement of the flow over Ihm.