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A way to find the optimal trajectory of an arbitrary manipulator following a prescribed path is described. The method takes advantage of the fact that in this case overall motion takes place with one degree of freedom (DOF) only. Transforming the equations of motion to this one DOF results in a set of equations which defines the phase space of admissible motion constrained by path geometry and joint torques in a transparent way. The time optimal solution representing the maximum mobility of the path-manipulator configuration can be easily determined by a field of extremals bound by a maximum velocity curve, which acts as a trajectory source or sink. Its properties lead to an algorithm for evaluating the time-minimum curve from a sequence of accelerating/ decelerating extremals. Additional optimizing criteria are studied by applying Bellman's principle.