On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom

Luca Biasco,Luigi Chierchia

doi:10.3934/dcds.2003.9.233

Abstract

Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the intermediate system depend explicitly upon the angle--variable conjugated to the non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated. Under suitable convexity assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values. In non convex cases, stability holds to a small set where, in principle, the degenerate action--variable might (in exponentially long times) drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a blow up (complex) analysis near separatrices, KAM techniques and energy conservation arguments.

Highlights

Introduction and resultsAs pointed out with particular emphasis by H
Degenerate nearly–integrable Hamiltonian systems with two degrees of freedom such that the “intermediate system” depend explicitly upon the angle–variable conjugated to the non–degenerate action–variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated
Under suitable “convexity” assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values

Summary

The Hamiltonian

Regarded as a one–degree–of–freedom system in the (I1, φ1) variables, is still integrable exhibiting, in general, the typical features of a one–degree–of–freedom dimensional system (phase space regions foliated by invariant circles of possibly different homotopy, stable/unstable equilibria, separatrices, etc.). These model problems are intended to capture the main features of “general” properly degenerate systems with two degrees of freedom and, in particular, the features of the exponential approximation (1.6) to the D’Alembert Hamiltonian This is the reason for considering both the convex and the non convex case in (1.8), corresponding, respectively to σ = 1 and σ = −1 (compare, point (i) of Remark 2). Where, in the case σ = 1, (I0, φ0) is an arbitrary point in the phase space MR, while, in the “non–convex” case σ = −1, (I0, φ0) belongs to MR\N∗, N∗ being an open region whose measure does not exceed ε2/3 This theorem will be a simple corollary of the following result, which describes the distribution and density of KAM tori. In Appendix B the (lengthy but elementary) details for the construction of the real–analytic action–angle variables for the pendulum are provided

Summarizing the following estimates hold

Substituting t

We have proved that

It is easy to see that σ dψ σ dψ ε