Phase portraits of separable Hamiltonian systems

Abstract

We study some generalizations of potential Hamiltonian systems $(H(x, y) = y^2 + F(x))$ with one degree of freedom. In particular, we are interested in Hamiltonian systems with Hamiltonian functions of type $H(x, y) = F(x) + G(y)$ arising in applied mechanical problems. We present an algorithm to plot the phase portrait (include the behavior at infinity) of any Hamiltonian system of type $H(x, y) = F(x)+G(y)$, where $F$ and $G$ are arbitrary polynomials. We are able to give the full description in the Poincaré disk according to the graphs of $F$ and $G$, extending the well-known method for the “finite” phase portrait of potential systems.