In this paper we study the global dynamics of the Hamiltonian systems (x) over dot = H-y (x,y), (y) over dot = H-x(x,y), where the Hamiltonian function H has the particular form H(x, y) = y(2)/2 + P(x)/Q(x), P(x), Q(x) is an element of R[x] are polynomials, in particular H is the sum of the kinetic and a rational potential energies. Firstly, we provide the normal forms by a suitable mu-symplectic change of variables. Then, the global topological classification of the phase portraits of these systems having canonical forms in the Poincare disk in the cases where degree(P) = 0,1, 2 and degree(Q) = 0, 1, 2 are studied as a function of the parameters that define each polynomial. We use a blow-up technique for finite equilibrium points and the Poincare compactification for the infinite equilibrium points. Finally, we show some applications. (C) 2016 Elsevier Inc. All rights reserved.