# KAM THEORY

The paper contains an answer, probably as near to complete as one could expect, to the question posed by Littlewood in the mid 1960's. The problem deals with the motion of a classical particle on the line in a potential field: $\ddot x+{\partial\over {\partial x}}V(x,t)=0$ with periodic time-dependence in $V$. This is a model problem for higher dimensional Hamiltonian systems and includes Duffing's equation with periodic forcing, the pendulum withperiodically varying torque and others. Littlewood considered the potentials $V$ with $V(x,t)\rightarrow \infty$ for $x\rightarrow\pm\infty$; with the aim to understand nonlinear resonances he asked: for which potentials are all motions bounded, i.e., no resonances persist? According to [26], for a wide class of potentials the system $\ddot x+V_x(x, t)=0$ is near--integrable for large energies -- more precisely, there are invariant tori arbitrarily far from the $t$--axis in the $(x, \dot x, t\,{\rm mod}2\pi )$--space. This near--integrability can hold despite the fact that no smallness assumptions on the time--variation of $V$ are made. The main difficulty of this result lies in reducing what may not originally seem as a small perturbation to a near--autonomous Hamiltonian system. The crucial property of $V$ that results in this near--integrability is the superquadratic growth at infinity, which physically corresponds to the fact that the large-amplitude oscillations are rapid. Some interesting identities involving singular integrals arose as a byproduct of this work; more importantly, a machinery for more explicit estimates of the Poincar\'e map in terms of the potential was developed. The thrust of the main result can be summarized by saying that

``many non--autonomous (i.e., time--dependent) Hamiltonian systems of the form $\ddot x+V_x(x,t)=0$ with $V$ growing faster than quadratic for large $x$ are well approximated by some completely integrable Hamiltonian system in the range of high energies''.

In [34] E. Zehnder and I extended these results to the case of quasiperiodic dependence of $V$ on $t$. Aubry--Mather theory in higher dimensions:

The note [38] solves, in a very special case, the problem of extending the Aubry--Mather theory to higher dimensions. The basic question is: do there exist remnants of invariant KAM tori when the problem is not a small perturbation of an integrable one? The answer is ``yes" for Hamiltonian systems with $n=2$ degrees of freedom, by the Aubry--Mather theory, while for $n= 3$ the answer is unknown. An essentially equivalent open question is: given a metric on a 3--torus and a direction vector, does there exist a locally minimal (i.e. shorter than its neighbors) geodesic with that asymptotic direction? To reinterpret this question in terms of geometric optics, it amounts to asking whether there exists a ray for any asymptotic direction in a medium in ${\bf R^3}$ with the index of refraction periodic in all three coordinates. In [38] I gave an affirmative answer for the special metric on ${\bf T}^3={\bf R}^3/{\bf Z}^3$ considered earlier by Hedlund.

The paper [44] with Juergen Moser gives a short variational proof of Moser’s twist theorem.

In our recent papers [53] and [54] with Vadim Kaloshin we gave a simple geometrical/physical picture of Arnold diffusion (a ubiquitous instability phenomenon in Hamiltonian systems) based on the variational approach. In our latest paper (yet unpublished) we describe a manifestation of Arnold diffusion in a chain of coupled pendula; despite weak coupling, the energy can ``travel” from one pendulum to another in an arbitrarily prescribed order, by an appropriate choice of initial conditions.