Lie systems and integrability conditions for t-dependent frequency harmonic oscillators

TL;DR

This paper explores the use of Lie systems to analyze time-dependent frequency harmonic oscillators, establishing integrability conditions and deriving t-dependent constants of motion within a unifying Lie group framework.

Contribution

It introduces a novel Lie group approach to study TDFHOs, providing integrability conditions and methods to find constants of motion for these systems.

Findings

Derived integrability conditions for TDFHOs

Established a framework relating TDFHOs to SL(2,R) equations

Obtained t-dependent constants of motion for specific cases

Abstract

Time-dependent frequency harmonic oscillators (TDFHO's) are studied through the theory of Lie systems. We show that they are related to a certain kind of equations in the Lie group SL(2,R). Some integrability conditions appear as conditions to be able to transform such equations into simpler ones in a very specific way. As a particular application of our results we find t-dependent constants of the motion for certain one-dimensional TDFHO's. Our approach provides an unifying framework which allows us to apply our developments to all Lie systems associated with equations in SL(2,R) and to generalise our methods to study any Lie system.