Adiabatic Invariance of a Simple Oscillator

J. E. Littlewood [5] has derived asymptotic expressions, as $\varepsilon \to 0 + $, for the function $r^2 = \phi (\varepsilon \tau )u^2 + \phi ^{ - 1} (\varepsilon \tau )({{du} / {d\tau }})^2 $, when u is a solution of the differential equation ${{d^2 u} / {d\tau ^2 }} + \phi ^2 (\varepsilon \tau )u = 0$. He assumes that $\phi (\tau ) > 0$, $\phi ( \pm \infty ) > 0$, $\phi ^{(n)} ( \pm \infty ) = 0$, and $\phi ^{(n)} \in L( - \infty ,\infty ) 0$, for all $n > 0$. In the present paper, Littlewood’s results are re-proved and strengthened by using the established methods for the solution of differential equations by asymptotic series. A new result is an explicit series construction in powers of $\varepsilon $ for the function $r^2 $. Littlewood’s asymptotic expression was in terms of the unknown solution of the differential equation.