Regularization of central forces with damping in two and three dimensions

Abstract

Regularization of damped motion under central forces in two and three dimensions is investigated and equivalent; undamped systems are obtained. The dynamics of a particle moving in 1r potential and subjected to a damping force is shown to be regularized using Levi-Civita transformation. We then generalize this regularization mapping to the case of damped motion in the potential r-2NN+1. Further equation of motion of a damped Kepler motion in three dimensions is mapped to an oscillator with inverted sextic potential and couplings, in four dimensions using Kustaanheimo–Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping coefficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in two and three dimensions, we show that the regularized equation is nonlinear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin–Sundman transformation.