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

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In this paper, we study two-body problems defined by a potential of the form V( r)= a/ r+ b/ r 2+ c/ r 3, where r is the distance between the two particles and a, b, c are arbitrarily chosen constants. The Hamiltonian H(r,p r,θ,p θ)= 1 2 p r 2+ p θ 2 r 2 + a r + b r 2 + c r 3 and the angular momentum p θ=r 2 θ ̇ are two first integrals, independent and in involution. Let I h (respectively I m ) be the set of points on the phase space on which H (respectively p θ ) takes the value h (respectively m). Since H and p θ are first integrals, the sets I h , I m and I hm = I h ∩ I m 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 I h , the foliation of I h by the invariant sets I hm and the movement of the flow over I hm .