Massimiliano Guzzo

The long-term diffusion properties of the action variables in real analytic quasiintegrable Hamiltonian systems is a largely open problem. The Nekhoroshev theorem provides bounds to such a diffusion as well as a set of techniques, constituting its proof, which have been used to inspect also the instability of the action variables on times longer than the Nekhoroshev stability time. In particular, the separation of the motions in a superposition of a fast drift oscillation and an extremely slow diffusion along the resonances has been observed in several numerical experiments. Global diffusion, which occurs when the range of the slow diffusion largely exceeds the range of fast drift oscillations, needs times larger than the Nekhoroshev stability times to be observed, and despite the power of modern computers, it has been detected only in a small interval of the perturbation parameter, just below the critical threshold of application of the theorem. In this paper we show through an example how sharp this phenomenon is.

We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev’s theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4–L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degreesof-freedom Hamiltonian system obtained from the Fermi–Pasta–Ulam problem by setting the number of particles equal to four.

The long term stability of the proper rotations of the perturbed Euler rigid body was recently investigated analytically in the framework of Nekhoroshev theory. In this paper we perform a parallel numerical investigation, with the double aim of illustrating the theory and to submit it to a critical test. We focus the attention on the case of resonant motions, for which the stability is not trivial (resonant proper rotations are indeed stable in spite of the presence of local chaotic motions, with positive Lyapunov exponent, around them). The numerical results indicate that the analytic results are essentially optimal, apart from a particular resonance, actually the lowest order one, where the system turns out to be more stable than the theoretical expectation.

A perturbation of a degenerate integrable Hamiltonian system has the form $H=h(I)+\varepsilon f(I,\varphi ,p,q)$ with $(I,\varphi )\in {\bf R}^n\times {\bf T}^n$, $(p,q)\in {\cal B} \subseteq {\bf R}^{2m}$ and the two-form is $dI\wedge d\varphi + dp\wedge dq$. In the case $h$ is convex, Nekhoroshev theorem provides the usual bound to the motion of the actions $I$, but only for a time which is the smaller between the usual exponentially-long time and the escape time of $p,q$ from ${\cal B}$. Furthermore, the theorem does not provide any estimate for the "degenerate variables" $p,q$ better than the a priori one $\dot p,\dot q\sim \varepsilon$, and in the literature there are examples of systems with degenerate variables that perform large chaotic motions in short times. The problem of the motion of the degenerate variables is relevant to understand the long time stability of several systems, like the three body problem, the asteroid belt dynamical system and the fast rotations of the rigid body.
In this paper we show that if the "secular" Hamiltonian of $H$, i.e. the average of $H$ with respect to the fast angles $\varphi$, is integrable (or quasi-integrable) and if it satisfies a convexity condition, then a Nekhoroshev-like bound holds for the degenerate variables (actually for the actions of the secular integrable system) for all initial data with initial action $I(0)$ outside a small neighbourhood of the resonant manifolds of order lower than $\ln \dfrac{1}{\varepsilon}$. This paper generalizes a result proved in connection with the problem of the long-time stability in the Asteroid Main Belt [9,13].

We show that $L_4$ and $L_5$ in the spatial restricted circular three-body problem are Nekhoroshev-stable for all but a few values of the reduced mass up to the Routh critical value. This result is based on two extensions of previous results on Nekhoroshev-stability of elliptic equilibria, namely to the case of "directional quasi-convexity", a notion introduced here, and to a (non-convex) steep case. We verify that the hypotheses are satisfied for $L_4$ and $L_5$ by means of numerically constructed Birkhoff normal forms.