 Solution

Hamilton's equations give $\dot{X}=P/m+O({\lambda }^2)$ and $\dot{P}=-m{\omega }^{2}X+O({\lambda }^2)$. Hence, with constants $A\ge 0$ and $\delta$ fixed by transformed initial data,An equivalent form uses $X(0)\cos \omega t+[P(0)/(m\omega )]\sin \omega t$ for $X(t)$. The indicated errors are a fixed-time perturbative statement; a higher-order frequency correction can accumulate phase on longer times.