Constant of Motion for One-Diemnsional Non Autonomous Linear Systems and Harmonic Oscillator

TL;DR

This paper derives a constant of motion for one-dimensional non-autonomous linear systems, including the harmonic oscillator with time-dependent forcing, by relating it to the autonomous system's constant of motion.

Contribution

It introduces a method to find constants of motion for non-autonomous linear systems based on their autonomous counterparts.

Findings

Derived explicit constant of motion for non-autonomous systems

Applied method to harmonic oscillator with time-dependent force

Provided insights into energy conservation in time-dependent systems

Abstract

For a one-dimensional motion, a constant of motion for non autonomous an linear system (position and velocity) is given from the constant of motion associated to its autonomous system. This approach is used in the study of the harmonic oscillator with an additional time depending force.