Hamiltonian Systems: Stability and Instability Theory

Abstract

The solar system has long appeared to astronomers and mathematicians as a model of stability. On the other hand, statistical mechanics relies on the assumption that large assemblies of particles form highly unstable systems (at the microscopic scale). Yet all these physical situations are described, at least to a certain degree of approximation, by Hamiltonian systems.One may hope that Hamiltonian systems can be classified in two different categories, stable and unstable ones. However, the situation is much more complicated and both stable and unstable behaviors cohabit in typical systems. Even our examples are not perfect paradigms of stability and instability. Indeed, it is now clear from numerical as well as theoretical points of views that some instability is present over long time-scales in the solar systems, so that for example future collisions between planets can not be completely ruled out in view of our present understanding. On the other hand, unexpected patterns of stability have been discovered in systems involving a large number of particles.Understanding the impact of stable and unstable effects in Hamiltonian systems has been considered since Poincaré as one of the most important questions in dynamical systems. In the present text, we will discuss model Hamiltonian systems of the formwhere (q, p) ∈ T d × U , with U a bounded open subset of R n . Recall that the equations of motion areThe textbook [1] is a good general introduction on Hamiltonian systems. We will always denote by ω(p) the frequency map ∂ p h(p), which plays a crucial role. Here, as is obvious in (2), the action variables p are preserved under the evolution in the unperturbed case ǫ = 0. We will try to explain what is known on the evolution of these action variables for the perturbed system. As we will see, in many situations, these variables are extremely stable. For example, KAM theorem implies that, for a positive measure of initial conditions (q 0 , p 0 ) the trajectory (q(t), p(t)) satisfies p(t) − p (0) Cǫ for all times. Examples show that some initial conditions may lead to unstable trajectories, that is trajectories such that p(t) − p(0) 1/C for some t (depending on ǫ) and some fixed constant C independent of ǫ. However, this is, as we will see, possible only for very large time t (meaning that t as a function of ǫ has to go to infinity very quickly when ǫ −→ 0). The main questions here are to understand in what situation instability is or is not possible, and what kind of evolutions can have the actions variable p. Another important question is to estimate the speed (as a function of the parameter ǫ) of the evolutions of p.