This paper proposes several extensions to the existing method of parametrizing speed along a prescribed path. The velocity is modified by isotropically stretching/shrinking the tangent space. The path in closed form is determined by substitution, without the computational cost of re-integrating the velocity function. This concept can be extended to anisotropic stretching/shrinking of the momentum space to change the direction and magnitude of the momentum vector. The physical constraints of the

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... actuators on increasing and decreasing momentum (and its differentials) are incorporated into a parametrization function that achieves maximum distance for a given input of energy and satisfies boundary conditions on the momentum. This method of time parametrization especially applies to Geometric Control, where the Hamiltonian minimizes some cost function and matches the boundary configuration constraints but not the velocity constraints. The optimal (geometric) trajectory is modified by the parametrization so that the cost function is minimized if the stretching is stopped at any time. The forces stretching the momentum space are identifiable from the formulation. An asymmetric rigid satellite illustrates the modification of angular momentum, with independent parametrizations of the linear momentum.