Time transformations in the extended phase-space

Abstract

Time transformations involving momenta in addition to the coordinates are studied from the points of view of stabilization and regularization of the equations of motion. The generalization of Sundman's transformation by using the potential function to transform the time is further generalized by using the Lagrangian function for the same purpose. The possibility of the stabilization of the equations of motion is investigated similarly to Stiefel's and Baumgarte's recent results but instead of a factorial, an additive control function is introduced in all equations of motion. The relation between the original and new independent variables is integrated by a modification of Ebert's theorem and it is shown that the new independent variable is Hamilton's principal function. Numerical examples illustrate the method and seem to indicate that the computation of close approach trajectories benefit especially by the transformations discussed. The Appendix offers an analytic treatment regarding the stabilization of the constant of energy.