  This is the first of two companion papers, in which we investigate vortex
motion on non-orientable two dimensional surfaces. We establish the
`Hamiltonian' approach to point vortex motion on non-orientable surfaces
through describing the phase space, the Hamiltonian and the local equations of
motion. This paper is primarily focused on the dynamics on the Mobius band. To
this end, we adapt some of the known notions of vortex dynamics to
non-orientable surfaces. We write Hamiltonian-type equations of vortex motion
explicitly and follow that by the description of relative equilibria and a
thorough investigation of motion of one and two vortices, with emphasis on the
periodicity of motion.
