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 parameter \(\varepsilon\) 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.

我们提出了一种算法，用于构造简单时间周期哈密顿量的解析近似运动积分，形式为\（H = H_ {0} + \ varepsilon H_ {i} \），其中\（\ varepsilon \）是一个摄动参数。我们将算法应用在动力学受Mathieu方程支配的哈密顿系统中，并详细研究了不同\（\ varepsilon \）值的轨道及其频闪观测不变曲线。我们找到\（\ varepsilon_ {crit} \）的值，超出该值，轨道逃逸到无穷大，并构造积分，这些积分在扰动参数\（\ varepsilon \）中表示为序列，并收敛到\（\ varepsilon_ {crit} \）。在没有共振的情况下，不变曲线是同心椭圆，它们的积分非常好。最后，我们构造了一个运动积分，它描述了共振情况下的双曲频闪频闪不变曲线。