2017/12/31 by Daisuke Takahashi, Daisuke A. Takahashi
Mathematics · Physics and Astronomy · #Acceleration #Classical mechanics #Drag #Equations of motion #Experimental and Theoretical Physics Studies #Exponential function #Jerk #Mathematical analysis #Mathematical physics #Mathematics #Mechanical and Optical Resonators #Model Reduction and Neural Networks #Physics #Piecewise #Quadratic equation #Toda lattice #nlin.SI #physics.class-ph
paper · pdf · doi:10.1088/1402-4896/aac969
published as Phys. Scr. 93, 075204 (2018) · 6 pages, 2 figures, final version published in Phys. Scr
openalex publication_date 2018/06/01 · arxiv created 2018/06/19 · arxiv updated 2018/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 .