Limits of the Formal Integrals of Motion

TL;DR

This paper investigates the existence and form of formal integrals of motion in Hamiltonian systems with specific resonances and perturbations, analyzing their properties and comparing theoretical predictions with numerical results.

Contribution

It generalizes previous results on formal integrals of motion to systems with more complex perturbations and resonances, providing explicit forms and comparisons with numerical simulations.

Findings

Explicit formal integrals for various frequency ratios.

Invariant curves and orbits are approximated and compared with numerical results.

Certain integrals exist only under specific conditions on parameters.

Abstract

We consider a formal (approximate) integral of motion in Hamiltonians of the form $H=\frac{1}{2}(X^2+Y^2+\omega_1^2x^2+\omega_2^2y^2)+\epsilon(\eta xy^2+\alpha x^3+\beta x^2y+\gamma y^3)$ generalizing previous cases with $\beta=\gamma=0$. First we give the general form of this integral when $\omega_1/\omega_2$ is irrational and then we consider the case of commensurable frequencies. In particular we study the integrals for the resonances $\omega_1/\omega_2=4/1, 5/1, 3/2, 4/3, 3/1$ and $2/1$. We also calculate the invariant curves and the orbits in the cases $\omega_1/\omega_2=2/1$ and $1/1$ (with $\beta=\gamma=0$) and we compare the exact-numerical and the theoretical results predicted by the formal integral when $\beta\gamma\neq0$. In the special case $\omega_1/\omega_2=1/1$ we find an integral when $\beta=\gamma=0$ and $\eta\alpha\neq0$ or $\eta=\alpha=0$ and $\beta\gamma\neq 0$, but this is not possible when $\eta\alpha\beta\gamma\neq 0$. However, we find that the invariant curves and the orbits can be approximated by a non-resonant integral with $\omega_1/\omega_2=5\sqrt{2}/7=1.010\dots$.