Phase portraits of separable Hamiltonian systems

We study a generalization of potential Hamiltonian systems (H(x, y) = y(2) + F(x)) with one degree of freedom; namely, those with Hamiltonian functions of type H(x, y) = F(x) + G(y), which will be denoted by X-H. We present an algorithm to obtain the phase portrait (including the behaviour at infinity) of X-H when F and G are arbitrary polynomials. Indeed, from the graphs of the one-variable functions F and G, we are able to give the full description on the Poincare disk, therefore extending the well-known method to obtain the phase portrait of potential systems in the finite plane. The fact that the phase portraits can be fully described in terms of the two one-variable real functions F and G allows, as well, a complete study of the bifurcation diagrams in complete families of vector fields. The algorithm can be applied to study separable Hamiltonian systems with one degree of freedom, which include a vast amount of examples in physical applications. (C) 2011 Elsevier Ltd. All rights reserved.