Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation

We present an algorithm for constructing analytically approximate integrals of motion in simple time-periodic Hamiltonians the form $$H=H_{0}+\varepsilon H_{i}$$ , where $$\varepsilon$$ is a perturbation parameter. apply our Hamiltonian system whose dynamics governed by Mathieu equation and examine detail orbits their stroboscopic invariant curves different values . find $$\varepsilon_{crit}$$ beyond which escape to infinity construct are expressed as series parameter converge up In absence resonances concentric ellipses approximated very well integrals. Finally, we integral describes hyperbolic curve resonant case.