In this paper we consider the time evolution of vortices simulated by the method of contour dynamics. Special attention is being paid to the Hamiltonian character of the governing equations and in particular to the conservational properties of numerical time integration for them. We assess symplectic and nonsymplectic schemes. For the former methods, we give an implementation which is both efficient and yet effectively explicit. A number of numerical examples sustain the analysis and demonstrate the usefulness of the approach.