A. Applied dynamics.
B. KAM theory and stability.
C. Dynamics of lattices.
D. Geometric phases in mechanics.
E. Mechanics of flexible space structures.
F. Fluid dynamics.
G. Topology of resonance zones.
H. Geometry and physics of averaging.
I. Charged particles in magnetic fields.
J. Dripping faucets and continued fractions.
K. Adiabatic invariants and symplectic geometry.
L. Mathematical amusements.
M. Physical demonstrations for teaching mathematics.
My main result of recent 3-4 years on this topic is in [39] . I had shown earlier [6] that a class of van der Pol -- type relaxation oscillators resembling an original forced van der Pol equation exhibit chaotic behavior with zero probability: Lebesque almost all initial conditions give rise to simple periodic behavior, while only exceptional (physically unobservable) solutions are ``chaotic". The note [39] describes a previously unnoticed basic stretching mechanism in the phase space of forced relaxation oscillators. This stretching takes place for most initial conditions (in the sense of Lebesgue measure), and it is somewhat surprising that this simple basic mechanism has been overlooked in the half century since the original work by Cartwright and Littlewood.
The rigorous result of [17] disproved an experiment reported in a physics journal. More generally, I showed that the Josephson equation of the form $\beta\ddot\phi+\dot\phi +\sin\phi = p(t)$ (here $p(t)$ is any periodic function) exhibits only periodic or quasiperiodic behavior, but no chaos, as long as $\beta <{1\over 4}$. The interesting point about this result is that it specifies the numerical range of the parameter, rather than asking for $ \beta $ to be ``sufficiently small". This result has answered some specific questions that some experimentalists in Josephson thermometry were interested in.
The paper [22] explains an interesting period--adding phenomenon in a nonlinear circuit which was observed initially in 1927 by van der Pol. The paper explained also some experimental findings whose mechanism was not known: for instance, why the frequency lock--in plateaus overlap in one parameter range and why they are separated by ``chaotic'' gaps in another.
In [48] it is shown that for a certain class of relaxation oscillations there is a hidden linearity when the relaxation parameter becomes small. Specifically, it is well known that the Poincar\'e map of a periodically forced van der Pol type oscillator is well approximated by a circle map. All the work on the subject starting with that of Cartwright, Littlewood and Levinson amounts to understanding this map. Now the gist of [48] is the proof of the fact that this circle map is in fact close to piecewise linear (when the relaxation parameter is small) for a certain class of oscillators. This shows that the system is as simple as it can possibly be, contrary to what may have been expected on the basis of the work of the past 50 years.
My thesis [6] is a study of the van der Pol-type relaxation oscillators with periodic forcing. The main result of this paper is a geometric description of the Poincar\'e map (this geometry was not known from the earlier papers) and the qualitative analysis using symbolic dynamics and other methods.
The paper [26] 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 with periodically 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 {\it 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.
This class of problems is motivated by the desire to understand the dynamical behavior of lattices of particles in a force field. The central model problem is a chain of coupled pendula with forcing and damping. The problem arises in studying the classical model of electrons in a crystal lattice, the motion of charge--density waves, the Josephson junction, the discretization in space of sine--Gordon equations and their analogues and in other applications. Several numerical and experimental studies of the last two decades uncovered some interesting nonlinear wave analogues in the case of just two pendula -- one could call these ``caterpillar waves'' because of the way they behave. I gave a transparent geometrical explanation of the mechanism of these nonlinear ``waves" and gave a bifurcation diagram which predicts what happens for different values of parameters in the case of an arbitrary number of particles -- this was done in [18] for $ n=2 $ pendula and in [21] for $ 3\leq n \leq \infty $.
What makes the ``caterpillar waves'' ``crawl'' in the final analysis is the well--known fact that the velocity of the vectorfield near a degenerate equilibrium (i.e., the point of coalescence of two equilibria) is small compared to the velocity of the vectorfield near a nondegenerate equilibrium. The difficulty of this work was to connect the simple fact with the phenomenon of the waves.
The paper [21] predicts the coexistence of {\it two} stable traveling waves with different speeds. This phenomenon was observed experimentally several years later in newly discovered Josephson junction crystals.
The application of dynamical systems to studying PDE's, with an eye on turbulence, has not lived up to expectations, despite the appeal of this approach. The lattice of $n\geq 2$ pendula with weak torsional coupling is one of very few examples which are on the borderline, being rich enough to inherit the interesting behavior while still simple enough to be manageable analytically, and one of few cases when the geometric approach to discretized PDE viewed as a dynamical system gives a nearly complete insight into the qualitative dynamics of the waves in a lattice and into various bifurcations.
The note [37] contains a formula which expresses a continuous product of orthogonal matrices (i.e. of solution of the matrix ODE $\dot X = M(t) X $, $M^T=-M$, $X(t)\in SO(3)$) in terms of parallel transport on the 2-sphere. It is curious that the cumulative angle of rotation appears in the formula despite the fact that $M(t)$ do not commute for different $t$.
The paper [33] gives a simple proof of the Gauss-Bonnet formula by mechanical analogy. Mathematically, it is a new proof of the theorem using dual cones. The note also relates the Hannay-Berry phase with the ``writhing number" of the curve.
The note [32] is a geometrical study of rolling of a rigid surface on the plane with some applications which include an extremely short derivation of a formula found earlier by Montgomery.
This topic is motivated by the desire to understand the dynamics of various flexible sructures in space; the work involved some combination of mechanics, dynamical systems and modeling. The following is a sample of the results.
One of the results of [19] is an interesting symmetry--seeking effect: a certain freely tumbling ``satellite" with a variable tensor of inertia approaches asymptotically a rigid shape (due to internal dissipation) whose tensor of inertia is an ellipsoid of rotation; of all possible shapes the system could take it chooses the one with a rotationally symmetric inertia tensor.
In the paper [27] with Stephane Laederich we studied the dynamics of simple model problems: free planar chains; using calculus of variations we showed that a free two--ended chain with $N$ joints admits periodic motions in which each of the $N$ angles between the neighboring links increases by its own arbitrarily prescribed integer multiple of $2\pi$.
In [35] (with H. Broer) we proved a curious effect: for an open set of Mathieu equations $\ddot x + (a + bp(t) ) x = 0 $ with an even $p$, the $ n $th and the $ n+1 $st stability boundaries meet at $\geq n$ points (counting multiplicity). This happens for an open set of functions $p(t)=p(t+1) \in L^1$ near the piecewise constant, as well as for all known classical examples. This shows in particular that as $b$ changes from $-\infty $ to $\infty $, the $n$th forbidden gap collapses to a point at least $n$ times!
I had discovered an interesting underlying geometry behind some systems with rapid time-dependence. This observation really illuminates some important physical phenomena in a very simple way; previous treatments of these phenomena relied on formal machinery of normal forms but did not explain what is ``really going on".
The papers [42-45] develop this observation. As an example, it turns out that the effective potentials of Kapitsa arising in averaging rapid vibrations have a geometrical significance related to curvature of family of curves. In its simplest manifestation, the stabilization of an inverted pendulum by the vibration of its suspension point is quantified by the curvature of the tractrix (the pursuit curve). This effect is behind some other counterintuitive phenomena such as laser tweezers (used to move particles inside a living cell), Paul traps, etc.
The geometric observation mentioned above led me to a simple geometric explanation of why the Paul trap works (actually it led me to rediscover the Paul trap before I learned of its existence). The classical explanation given by Paul in his 1989 Nobel Prize acceptance paper is analytic/computational.
Jointly with I. Mitkov and V. Zharnitsky we applied some of these ideas to the study of some idealized anisotropic ferromagnets. The analysis in [41] predicts the existence of traveling domain walls if a rapidly oscillating magnetic field is applied. These walls separate the opposite directions of magnetization.
The short preprint [49] contains the solution of a problem posed by Arnold on the existence of periodic homotopically trivial trajectories on a 2--torus in the presence of magnetic fields. The note proves that a certain Poincar\'e section map of the energy 3--torus preserves the center of mass in an appropriate measure induced by the Liouville measure on the energy torus - this is the main step in the solution. In order to apply the Conley-Zehnder theorem the variable density of the invariant measure has to be uniformized (a relatively minor step); I found the uniformizing transformation by using the heat flow on the torus.
A related note [47] describes the destruction of periodic orbits of a charged particle in a potential on the torus when the magnetic field is turned on. It turns out that the homotopically nontrivial orbits on the torus which survive in small magnetic fields must lose their equidistribution property, in a certain precise sense. The note contains an estimate on the disparity of the gaps in terms of a certain curvature. Geometrically, the note exploits the fact that an associated Poincar\'e map is non--exact: while preserving the area, it shifts the center of mass. I used some simple geometry to make an analytic estimate on the disparity of gaps.
The semi-expository note [46] grew out of observing a dripping faucet and led me to finding a relationship between the continued fractions and the Morse-Thue sequences.
The report [1] is a survey of the work of Krein, Gelfand, Lidskii and Moser on stability of linear Hamiltonian systems. In [5] I had established a connection between adiabatic invariants and the Gelfand--Lidskii signature of a symplectic matrix. The paper deals with linear Hamiltonian systems whose Hamiltonians have both periodic and slow time--dependence: $\dot z = JH(t, \epsilon t)z$. One question was: are there adiabatic invariants and how many? I found the answer in terms of the Gelfand--Lidskii signature: if the periodic system $\dot z= JH(t, \tau)z$ (with $\tau$ frozen) is strongly stable for all $\tau$ then the number of adiabatic invariants of the system is not less than the number of clusters of the same sign in the Gelfand--Lidskii signature of the Floquet matrix.
In a different category, I am planning to publish a problem book with a mixture of problems in physics, mechanics and mathematics, roughly on the undergraduate level. In the years since high school I came up with about 120 problems some of which seem to be original and require ingenious solutions; many ask to solve a mathematical problem using physics. For instance, here is a theorem which I rediscovered by means of a physical argument: given an arbitrary closed convex surface (a ``pebble"), consider a tetrahedron of smallest possible volume containing the ``pebble" inside. Then {\it the points of contact between the faces and the surface are the centroids of the faces.}
Among other items in my collection are:
- proofs of the Pythagorean theorem (one using potential energy of ``rubber bands"; another using kinetic energy ``$ma^2/2+mb^2/2=mc^2/2 $"; the third using geometric sequences.)
- a way to find integrals using conservation of energy
- some apparently new elementary geometrical identities on minimal polyhedra
- several bicycle problems (for instance, by analyzing the tracks left by a bicycle, can one tell the direction in which it rolled? Can one accelerate without pedaling?, and so on),
and many other topics.
Finally, I have made several mechanical "toys" which illustrate various mathematical ideas. In most of my classes, from freshman to graduate, I use some of these demonstrations. Among these devices are
- a 7.5-foot cycloid used to illustrate the remarkable properties of cycloids (a cycloid is (i) an isochrone; (ii) a tautochrone and (iii) an evolute of a congruent cycloid (evolute is the envelope of the family of normals));
- a simple mechanical illustration (using a bike wheel) of the Gauss-Bonnet theorem;
- a { \it mechanical } device which gives a formulas-free "proof" of Snell's law (for the refraction of light) ;
- an upside-down pendulum which becomes stable when its suspension point is made to vibrate (using a jigsaw);
- a demonstration of a heteroclinic motion by tossing a tennis racket,
and several others.