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

TL;DR

This paper develops an algorithm to construct approximate integrals of motion for time-periodic Hamiltonian systems, specifically applied to the Mathieu equation, revealing critical perturbation thresholds and invariant structures.

Contribution

The authors introduce a new algorithm for analytically approximating integrals of motion in simple time-periodic Hamiltonians, demonstrated on the Mathieu equation.

Findings

Identified critical perturbation values where orbits escape to infinity.

Constructed convergent series for integrals of motion up to the critical perturbation.

Successfully approximated invariant curves as concentric ellipses in non-resonant cases.

Abstract

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.