Phase portraits of separable Hamiltonian systems

We study a generalization of potential Hamiltonian systems (H(x,y)H(x,y)=y2+F(x)=y2+F(x)) with one degree of freedom; namely, those with Hamiltonian functions of type H(x,y)=F(x)+G(y)H(x,y)=F(x)+G(y), which will be denoted by XHXH. We present an algorithm to obtain the phase portrait (including the behaviour at infinity) of XHXH when FF and GG are arbitrary polynomials. Indeed, from the graphs of the one-variable functions FF and GG, we are able to give the full description on the Poincaré 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 FF and GG 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.