Newton’s equation of motion with quadratic drag force and Toda’s potential as a solvable one

Abstract

The family of exactly solvable potentials for Newton’s equation of motion in the one-dimensional system with quadratic drag force has been determined completely. The determination is based on the implicit inverse function solution valid for any potential shape, and hence exhaustive. This solvable family includes the exponential potential appearing in the Toda lattice as a special limit. The global solution is constructed by matching the solutions applicable for positive and negative velocity, yielding the piecewise analytic function with a cusp in the third-order derivative, i.e., the jerk. These procedures and features can be regarded as a generalization of Gorder’s construction (2015 Phys. Scr. 90 085208) to the energy-dissipating damped oscillators. We also derive the asymptotic formulae by solving the matching equation, and prove that the damping of the oscillation amplitude is proportional to t−1.