We propose a Hamiltonian dynamics formalism for the current and magnetic field driven dynamics of ferromagnetic and antiferromagnetic domain walls in one dimensional systems. To demonstrate the power of this formalism, we derive Hamilton equations of motion via Poisson brackets based on the Landau-Lifshitz-Gilbert phenomenology, and add dissipative dynamics via the evolution of the energy. We use this approach to study current induced domain wall motion and compute the drift velocity. For the antiferromagnetic case, we show that a nonzero magnetic moment is induced in the domain wall, which indicates that an additional application of a magnetic field would influence the antiferromagnetic domain-wall dynamics. We consider both cases of the magnetic field being parallel and transverse to the N{é}el field. Based on this formalism, we predict an orientation switch mechanism for antiferromagnetic domain walls which can be tested with the recently discovered N{é}el spin orbit torques. ; Comment: 7 pages, 3 figures