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

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.

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Cited by in corpus (5)

• Explicit solutions of the ${\mathfrak{a}}_1$-type Lie-Scheffers system and a general Riccati equation
• A geometric approach to integrability of Abel differential equations
• Integrability of Lie systems through Riccati equations
• Reduction of symplectic principal $\mathbb{R}$-bundles
• A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$