The classical reaction path Hamiltonian formulation of Miller, Handy, and Adams; is reformulated using a linear expansion of the gradient in internal coordinates. It leads to a correspondence between the are length: s, along the intrinsic reaction coordinate, and the whole set of internal coordinates and, furthermore, to a dynamical equation for s, a second-order Bernoulli-type equation, which is analytically solvable inside the validity range of the quadratic expansion of the-potential. Therefore, by virtue of the above correspondence, the time dependence of the whole set of internal coordinates is easily recovered, by means of a few functional and overlap evaluations. It thus enhances the computational performance of the overall direct dynamics method. The unimolecular 1,2 hydrogen migration, between the (corresponding) carbene and ethyne oxide, is considered as example for illustrative purposes.