Scientific work

Systems with symmetries

For a long period of time I have been involved in the study of Hamiltonian dynamical systems with symmetries of different kinds. Such systems are very common in atomic and molecular physics. (In fact it is diffiult to come up with a molecular or atomic system which has no symmetry.) One of the problems arising in this study is the analysis of group actions on manifolds.
As an example consider the study of the action of the tetrahedral group Td on the complex projective space CP2 made by Boris Zhilinskií in 1985 [Chem. Phys. 137, 1 (1989)]
Action of Td on CP_2, a work of art (Boris Zhilinskii, 1985)
isolated C3 points Incredible as it may seem, this old abstract work by Boris has recently found direct applications in molecules [Dhont-2000, VanHecke-2001]. In fact, molecular vibrations about the equilibrium configuration can be described using a normalized system, whose principal or polyad integral n is (approximately) the linear part of the vibrational Hamiltonian. This n is quite similar to the principal quantum number of the hydrogen atom. In molecules, n is a good quantum number which labels polyads (aka n shells). The space CP2 is a reduced phase space of the 3-oscillator system, which is described in the lowest approximation as an isotropic 1:1:1 resonant harmonic oscillator. In the case of the triply degenerate vibrational F modes of molecules with cubic (Td or Oh) equilibrium configuration, such resonance is exact; in some other cases, such as ozone molecule, it is approximate. In both cases we consider and reduce the approximate dynamical (or polyad) symmetry S1 generated by the flow of the linear 3-mode system in 1:1:1 resonance.
RES for the R44A1 term Knowing the group action, we can predict qualitatively the types of Hamiltonian functions compatible with given symmetry and topology of the manifold (phase space) on which these functions are defined. A classic example of such prediction is the reduced rotational Hamiltonian of the spherical top molecule defined on a 2-sphere in the presence of the Oh symmetry, see on the left.


My PhD thesis was in part dealing with the analysis and assignment of experimental high-resolution rotation-vibration spectra of spherical top molecules. Together with Volodya Krivtsun, we were one of the first to observe the ν3 band of the rare SnH4 molecule. Gerhard Herzberg, Ottawa 1991 Later, at the Herzberg Institute in Ottawa in 1990-93, I worked with Izabel Dabrowski and late Gerhard Herzberg (right) on the spectra of Rydberg molecules ArD and KrD. A combination of Izabel's expertise in the art of line assignments, my postdoc freshness, GH's inspiration and persistence, encouragement by John W. C. Johns, and, of course, a bit of luck, led to our first successful assignment and analysis of spectra of these molecules which had been puzzling people since they were first seen more than 10 years before us. The work on ArD was probably the last research work in which GH truly participated. When I remember this work, I always imagine him the way he is shown in the photo, coming into my office with suggestions and more spectra. These suggestions turned out to be very important indeed. To have an idea of the complexity of the spectra we worked with, see the example of the 4f(complex)→4d(Δ) band below.
spectrum of KrD
Several such bands are usually needed in order to extract information on the rotational structure of the participating electronic Rydberg states.

Normal forms

As a postdoc with John Delos in 1993-94, I worked on the normal form study of periodic orbits. Although the theory of normalization near periodic orbits existed at the time, there were no serious numerical practical applications of it before us. One of the systems studied in John's group was hydrogen atom in external crossed magnetic and electric fields. John was interested in bifurcations basic periodic orbits (=relative equilibria) of this system. We computed normal forms near one particular orbit, the so-called ``perpendicular'' orbit.
This is how our normal form worked.
normal form and s.o.s. of the perpendicular orbit
Here black dots show the Poincaré surface of section computed by John Shaw, red shows constant level sets of the normalized Hamiltonian. The pair of Z2-symmetric pairs of stable-unstable satellite orbits was created just before in a fold bifurcation, and now the two unstable orbits move tovards centre as the energy increases. The perpendicular orbit in the centre is about to have a period-1 Z2-symmetric bifurcation and loose stability. The normal form describes well this ``organized'' sequence of bifurcations [SadovskiiDelos-95]. The two small stable islands correspond to the two new periodic orbits which branched off for good. Secondary high-period resonances, which mark the onset of chaos, can be seen around these islands.

Relative equilibria

My work is related in many ways to the study of relative equilibria (RE), including those of vibrating and rotating molecules, and of the perturbed Kepler problem and its quantum analogue, the hydrogen atom in external fields. RE are special periodic orbits of the system which go along the orbit of the group action of a dynamical or strict symmetry group. In all systems we studied, we have the S1 group action.
potential and RE of the Henon-Heiles system, eps=0.1, E/Ecrit=0.9 A simple comprehensive explanation of what RE are and how we use them can be given using the ``textbook'' Hénon-Heiles system [DS:SPT2001], which has close micro analogues, such as the E-mode vibrations of H3+ [SFHTZ-1993] or P4 and other molecules with cubic symmetry. The Hénon-Heiles system is a 1:1 resonant 2-oscillator with the anharmonic D3 symmetric potential V(x,y) shown left. Near the central equilibrium (0,0), we have eight RE or nonlinear normal modes, whose projections on the coordinate plane are shown on top of V(x,y). Among the eight RE, two stable equivalent circular RE retain C3 symmetry and share the same (x,y) image (blue loop) but differ in direction. Six other C2 symmetric RE bounce at the potential energy limit (black) : three stable equivalent RE lie in the symmetry planes (straight red lines), while three remaining RE are unstable.

Perturbed Keplerian systems

Perturbations of the hydrogen atom by external fields are atomic analogues of classical perturbed Kepler systems. I started to work in this field as a postdoc with John Delos. We normalized near the perpendiqular RE (one of the four basic periodic orbits or Kepler ellipses) of the crossed fields system (hydrogen atom perturbed by orthogonal electric and magnetic fields) and studied bifurcations of this periodic orbit.

Later with Boris Zhilinskií we studied normal forms near the equilibrium of the regularized Kepler system. We applied invariant theory and studied the topology of the reduced spaces and orbit spaces. Together with Boris and Louis Michel we uncovered and interpreted the phenomenon of crossover.

My experience with these systems was a key to the successful collaboration with Richard Cushman on the monodromy of the hydrogen atom in crossed fields.


More recently, I worked with Richard Cushman, who introduced me and Boris to the idea of monodromy, or basic topological obstuction to the existence of global action-angle variables in classical Hamiltonian dynamical systems. In this sense, monodromy was introduced by J. J. Duistermaat in 1980.
pinched torus In the simplest case of a classical mechanical integrable Hamiltonian system with two degrees of freedom, monodromy exists if one of the singular dynamically invariant subspaces of the system is an isolated fiber, which Cushman called pinched torus (see left). It can be represented as a 2-torus with one basis cycle contracted to a point. The latter corresponds to the unstable equilibrium of the focus-focus type; the rest of this fiber is the stable and unstable manifolds of the equilibrium forming a homoclinic connection. In systems with parameters, appearance of such focus-focus points and their symmetric variations results from Hamiltonian Hopf bifurcations.
We studied the quantum analogues of systems with monodromy. Boris proposed to manifest quantum monodromy as a point defect in the lattice of quantum states in the joint energy-momentum spectrum.

One of the most fundamental systems with monodromy, which we found with Richard [CushmanSadovskii-1999, CushmanSadovskii-2000], is the hydrogen atom in orthogonal (or crossed) external electric and magnetic fields. The energy-momentum diagram of this system for two different relative field strengths is shown below.

Energy-momentum diagram of the H-atom in crossed fields Cushman and Sadovskii 1999
On the left, the system has monodromy. The absence of globally defined quantum numbers is illustrated by "Zhilinskií's defect diagram". Take the elementary cell of the lattice, move it along a countour around the isolated critical (singular) value of the energy-momentum map indicated by the red circle, and compare the initial and final cells. The cells don't match!

Monodromy of floppy triatomic molecules

geometry of HCN Finding monodromy in real systems may not be all that easy, as we learned with Marc Joyeux in 2003. We had many good reasons to believe that the bending vibrational system of the isomerizing HCN/CNH molecule should have monodromy. To understand this system, I came up with a nice model system, a , which does have monodromy. However, when we tried the same analysis for the real case of HCN/CNH, we found that the segment of singular values (the upper boundary of the green leaf) became unbounded, see figure below. So, instead, we have suggested that the HCN/CNH system has the generalized two-branch global bending quantum number . Later Marc applied the same analysis to LiNC/NCLi [JST-2003] and it worked . The reason for such difference between HCN/CNH and LiNC/NCLi is the "shape" of the system: in HCN/CNH, the H atom moves on a peanut-like surface with a waist, for as the much larger Li atom cannot get too close to the CN diatom and the shape of LiNC/NCLi remains convex.

Fractional monodromy

As Boris continued to study various defects of ``quantum lattices´´, he came to the idea of fractional monodromy, a generalization of the monodromy phenomenon. He suggested that fractional monodromy corresponds to a linear defect caused by the line of ``weak'' singular values in the base of the integrable foliation. In our joint work with Nikolaí Nekhoroshev we gave rigorous mathematical definition of fractional monodromy and computed it on the example of the classical 1:-2 resonant nonlinear oscillator [NSZ-2002]. The corresponding energy-momentum diagram is shown below.
Energy-momentum diagram of the
1:-2 resonant oscillator, Nekhoroshev, Sadovskii and Zhilinskii 2002
It can be easily seen that we can only use a double elementary cell to cross the line of weak singularities shown in red, and that the monodromy matrix of the transformation of the single cell has a half-integer element.
Curled torus fiber of the 1:-2 resonant oscillator system The underlying singularity of the corresponding classical 1:-2 resonant system is due to the presence of special “short” orbits of the oscillator action S1, whose period is half that of the regular S1 orbits. Dynamically these orbits correspond to the pure excitation of the second oscillator and due to the nonlinear resonance terms they are unstable. The stable and unstable manifolds conect and form a 2D singular variety, which we call a curled torus, shown left. In fact, points of the red line in the above energy-momentum diagram are images of such tori. To compute monodromy, we consider a bundle over the closed contour Γ, which goes around the central singular point (red circle) in the image of the energy-momentum map. Since Γ has to cross the singular (red) line, this bundle has one singular fiber---the curled torus. To compute monodromy, we identifiy continuously the fundamental groups π1 of the regular fibers (tori) in the bundle while we move along Γ. According to Nekhoroshev, we cannot continue the whole π1 when we cross the singular line, but we can continue a complete subgroup ζ of π1. What exactly happens to the basis cycles of π1 can be well represented on the . After making one tour on Γ, we obtain the final subgroup ζ(1) and compare it to the original subgroup ζ(0), with which we started our tour. The map ζ(1)→ζ(0) is nontrivial. When formally extended to the whole π1, the matrix for this map has rational coefficients (½ in the case of the 1:-2 resonance). Hence the term fractional.

It is intresting to note that as a graduate student of Vl. I. Arnold in the late 60s, Nikolaí formulated the necessary condititions for the existence of global action-angle variables, i.e., for the absence of monodromy, see his Two theorems about action-angle variables, Uspekhi Mat. Nauk 24 no 5 (1969) 237-238 (in Russian); Russian Math. Surveys 24 no 5 (1969) 237-38. So after 30 years Nikolaí comes back in the field with another fundamental contribution!