Linearization of Newton's second law

TL;DR

This paper introduces a geometric method using Eisenhart lift to linearize Newton's second law for specific potentials, revealing new ways to simplify and analyze dynamical systems.

Contribution

It presents a novel geometric approach employing Eisenhart lift to linearize Newtonian systems, expanding the toolkit for analyzing nonlinear differential equations.

Findings

Newton's second law can be globally linearized for oscillator and Morse potentials.

The Eisenhart lift relates nonlinear systems to free particle motion.

New directions are opened for geometric linearization of differential equations.

Abstract

The geometric linearization of nonlinear differential equation is a robust method for the construction of analytic solutions. The method is related to the existence of Lie symmetries which can be used to determine point transformations such that to write the given differential equation in a linear form. In this study we employ another geometric approach and we utilize the Eisenhart lift to geometric linearize the Newtonian system describing the motion of a particle in a line under the application of an autonomous force. Our findings reveal that for the oscillator, the Ermakov potential with or without the oscillator term, and the Morse potential, Newton's second law can be globally expressed in the form of that of a free particle. This study open new directions for the geometric linearization of differential equations via equivalent dynamical systems.