Passive velocity field control (PVFC) is a control methodology for fully actuated mechanical systems, in which the motion task is specified behaviorally in terms of a velocity field (as opposed to a timed trajectory), and the closed-loop system is passive with respect to a supply rate given by the environment power input. It is intended for safety critical and coordination intensive applications where the mechanical system is required to interact with the physical environment. The control law is derived geometrically and the geometric and robustness properties of the closed-loop system are analyzed. It is shown that the closed-loop unforced trajectories are geodesics of a closed-loop connection which is compatible with an inertia metric, and that the velocity of the system converges exponentially to a scaled multiple of the desired velocity field. The robustness property of the system exhibits some strong directional preference. In particular, disturbances that push in the direction of the desired momentum do not adversely affect performance. Moreover, robustness property also improves with more energy in the system. In Part II of the paper, the application of PVFC to contour following as well as experimental results are presented.