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, 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
*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
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!