  We present an algorithm for constructing analytically approximate integrals
of motion in simple time periodic Hamiltonians of the form $H=H_0+ \varepsilon
H_i$, where $\varepsilon$ is a perturbation parameter. We apply our algorithm
in a Hamiltonian system whose dynamics is governed by the Mathieu equation and
examine in detail the orbits and their stroboscopic invariant curves for
different values of $\varepsilon$. We find the values of $\varepsilon_{crit}$
beyond which the orbits escape to infinity and construct integrals which are
expressed as series in the perturbation $\varepsilon$ and converge up to
$\varepsilon_{crit}$. In the absence of resonances the invariant curves are
concentric ellipses which are approximated very well by our integrals. Finally
we construct an integral of motion which describes the hyperbolic stroboscopic
invariant curve of a resonant case.
