As an example consider the study of the action of the tetrahedral group
*T*_{d} on the complex projective space CP_{2} 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
[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 CP_{2} 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 (*T*_{d} or
*O*_{h}) equilibrium configuration [Efstathiou-2004], such resonance is
exact; in some other cases, such as ozone molecule
[Kozin-2004], it is
approximate. In both cases, we consider and reduce the approximate dynamical
(or polyad) symmetry
*S*_{1} generated by the flow of the linear 3-mode system in 1:1:1
resonance.

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
*O _{h}* symmetry, see on the left.

Analysis of quantum rotational states usung symmetries of the classical limit effective rotational Hamiltonian goes back to the work of

Indeed, from a look of it, one would think of a spherical top... The spectral density (and difficulty) is due both to the small rotational constants of this relatively heavy molecule and the accidental closeness of its equilibrium configuration to that of the symmetric top. The latter fact already rang some bells...

Months of a very intense work, requiring the state-of-the-art analysis of thousands of spectroscopic lines followed. It turned out that certain skills, like riding a bicycle or skiing, stay with us longer than we think. After almost 20 years, I had to recall all what I learned and used while working in the Spectroscopy section of the Herzberg Institute of Astrophysics in Ottawa (see below). After succeeding in understanding the ν

This is how our normal form worked.
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 Z_{2}-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 Z_{2}-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.

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 H_{3}^{+}
[SFHTZ-1993] or P_{4} and other molecules with
cubic symmetry. The Hénon-Heiles system is a 1:1 resonant 2-oscillator
with the anharmonic *D*_{3} 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
*C*_{3} symmetry and share the same *(x,y)* image (blue loop)
but differ in direction. Six other *C*_{2} 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.

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.

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 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.

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.

The underlying singularity of the corresponding classical 1:-2 resonant system
is due to the presence of special “short” orbits of the oscillator
action S_{1}, whose period is half that of the regular S_{1}
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 π_{1} of the regular
fibers (tori) in the bundle while we move along Γ.
According to Nekhoroshev, we cannot continue the whole π_{1} when we
cross the singular line, but we *can* continue a complete
*subgroup* ζ of π_{1}. What exactly happens to the
basis cycles of π_{1} 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 π_{1}, 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!