Limits of the Formal Integrals of Motion

Abstract

We consider a formal (approximate) integral of motion in Hamiltonians of the form $H=\frac{1}{2}(X^2+Y^2+{\omega }_1^2{x}^2+{\omega }_2^2{y}^2)+\epsilon (\eta x{y}^2+\alpha x^3+\beta x^{2}y+\gamma y^3)$ 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 $\beta \gamma \ne 0$. In the special case ${\omega }_1/{\omega }_2=1/1$ we find an integral when $\beta =\gamma =0$ and $\eta \alpha \ne 0$ or $\eta =\alpha =0$ and $\beta \gamma \ne 0$, but this is not possible when $\eta \alpha \beta \gamma \ne 0$. However, we find that the invariant curves and the orbits can be approximated by a non-resonant integral with ${\omega }_1/{\omega }_2=5\sqrt{2}/7=1.010\dots$.

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References in corpus (4)

• Study of chaos in hamiltonian systems via convergent normal forms
• Analytical invariant manifolds near unstable points and the structure of chaos
• Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy
• Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach