Invariants for the time-dependent harmonic oscillator

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Abstract
Lewis showed that in the case p (t) ≡1, h=1, I (t) = (1/2) {p2(t)[ρ (t) y′ (t) −y (t) ρ′ (t)]2+h2y2(t)/ρ2(t) } is constant in time if y (t) solves (p (t) y′) ′+q (t) y =0, and ρ (t) solves p (t) ρ3(t) L[ρ]=h2 (h constant). Recently, Eliezer and Gray showed that I (t) =const is just the conservation of angular momentum in an appropriate physical interpretation. We show, using a change of variable technique, that I (t) =const reduces to sin2ϑ+cos2ϑ=1. We discuss uniqueness and extendability of solutions to the above equation in ρ.