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

TL;DR

This paper completely characterizes the family of exactly solvable one-dimensional Newtonian systems with quadratic drag, including the Toda potential, and analyzes their global solutions and damping behavior.

Contribution

It provides a complete classification of solvable potentials with quadratic drag and constructs their global solutions, extending previous work to energy-dissipating systems.

Findings

The family of solvable potentials includes the Toda potential as a limit.

The global solution is piecewise analytic with a cusp in the jerk.

Damping of oscillation amplitude is proportional to t^{-1}.

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 [Phys. Scr. 2015, {\bf 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} $.