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.
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.
As an example consider the study of the action of the tetrahedral group
on the complex projective space CP2
made by Boris
Zhilinskií in 1985
[Chem. Phys. 137, 1 (1989)
Incredible as it may seem, this old abstract work by Boris has recently
found direct applications in molecules
In fact, molecular vibrations about the equilibrium configuration can be
described using a normalized system, whose principal or polyad
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
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
modes of molecules with cubic (Td
) 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
generated by the flow of the linear 3-mode system in 1:1:1
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
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.
Several such bands are usually needed in order to extract information on the
rotational structure of the participating electronic Rydberg states.
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.
Here black dots show the Poincaré surface of section computed by John
, 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
]. 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.
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.
A simple comprehensive explanation of what RE are and how we use them can be
given using the ``textbook'' Hénon-Heiles system
], which has close micro
analogues, such as the E
-mode vibrations of H3+
] 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
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
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.
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.
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
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.
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
Finding monodromy in real systems may not be all that easy, as we learned with
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
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.
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.
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.
The underlying singularity of the corresponding classical 1:-2 resonant system
is due to the presence of special “short” orbits of the oscillator
, 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
cross the singular line, but we can
continue a complete
ζ of π1
. What exactly happens to the
basis cycles of π1
can be well represented
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
, the matrix for this map has rational coefficients
(½ in the case of the 1:-2 resonance). Hence the term
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!