Construction of dynamical invariants for the time-dependent harmonic oscillator with a time-dependent driven force

TL;DR

This paper develops methods to construct dynamical invariants for classical and quantum time-dependent harmonic oscillators with external driving forces, using equations of motion, and explores their algebraic relationships.

Contribution

It introduces a systematic approach to derive linear and quadratic invariants for driven oscillators at both classical and quantum levels, highlighting their algebraic connections.

Findings

Derived explicit forms of linear and quadratic invariants

Established algebraic relationships between invariants

Applicable to both classical and quantum systems

Abstract

We construct the linear and quadratic polynomial dynamical invariants for the classical and quantum time-dependent harmonic oscillator driven by a time-dependent force. To obtain them, we use exclusively the associated equations of motion for the system. We also find an algebraic relationship between the linear and quadratic invariants at the classical and quantum level.