Volume 27, Number 3
Volume 27, Number 3, 2022
Alexey Borisov Memorial Volume
Albouy A., Zhao L.
Darboux Inversions of the Kepler Problem
Abstract
While extending a famous problem asked and solved by Bertrand in 1873, Darboux
found in 1877 a family of abstract surfaces of revolution, each endowed with a force function,
with the striking property that all the orbits are periodic on open sets of the phase space.
We give a description of this family which explains why they have this property: they are
the Darboux inverses of the Kepler problem on constant curvature surfaces. What we call the
Darboux inverse was briefly introduced by Darboux in 1889 as an alternative approach to the
conformal maps that Goursat had just described.

Miguel N., Simó C., Vieiro A.
Escape Times Across the Golden Cantorus of the Standard Map
Abstract
We consider the Chirikov standard map for values of the parameter
larger than but close to Greene's $k_G$. We investigate the dynamics near the
golden Cantorus and study escape rates across it.
Mackay [17, 19]
described the behaviour of the mean of the number of iterates
$\langle N_k \rangle$ to cross the Cantorus as $k\to k_G$ and showed that there
exists $B<0$ so that $\langle N_k\rangle (kk_G)^B$ becomes 1periodic in a
suitable logarithmic scale. The numerical explorations here give evidence of
the shape of this periodic function and of the relation between the escape
rates and the evolution of the stability islands close to the Cantorus.

Tsiganov A. V.
Reduction of Divisors and the Clebsch System
Abstract
There are a few Lax matrices of the Clebsch system. Poles of the Baker – Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann – Roch theorem, each class has a unique reduced representative. We discuss properties of such a reduced divisor on the spectral curve of $3\times 3$ Lax matrix having a natural generalization to $gl^*(n)$ case.

Efstathiou K., Lin B., Waalkens H.
Loops of Infinite Order and Toric Foliations
Abstract
In 2005 Dullin et al. proved that the
nonzero vector of Maslov indices is an eigenvector with eigenvalue
$1$ of the monodromy matrices of an integrable Hamiltonian system.
We take a close look at the geometry behind this result and extend
it to the more general context of possibly nonHamiltonian systems.
We construct a bundle morphism defined
on the lattice bundle of an (general) integrable system, which can
be seen as a generalization of the vector of Maslov indices. The nontriviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue $1$
of the monodromy matrices, and gives rise to a corank $1$ toric foliation
refining the original one induced by the integrable system. Furthermore,
we show that, in the case where the system has $2$ degrees of freedom,
this implies the existence of a compatible free $S^{1}$ action on the regular part of the system.

Emmanuele D., Salvai M., Vittone F.
Möbius Fluid Dynamics on the Unitary Groups
Abstract
We study the nonrigid dynamics induced by the standard birational actions of
the split unitary groups $G=O_{o}\left( n,n\right) $, $SU\left( n,n\right) $
and $Sp\left( n,n\right) $ on the compact classical Lie groups $M=SO_{n}$,
$U_{n}$ and $Sp_{n}$, respectively. More precisely, we study the geometry of
$G$ endowed with the kinetic energy metric associated with the action of $G$
on $M,$ assuming that $M$ carries its canonical biinvariant Riemannian
metric and has initially a homogeneous distribution of mass. By the least
action principle, forcefree motions (thought of as curves in $G$)
correspond to geodesics of $G$. The geodesic equation may be understood as
an inviscid Burgers equation with M\"{o}bius constraints. We prove that the
kinetic energy metric on $G$ is not complete and in particular not
invariant, find symmetries and totally geodesic submanifolds of $G$ and
address the question under which conditions geodesics of rigid motions are
geodesics of $G$. Besides, we study equivalences with the dynamics of
conformal and projective motions of the sphere in low dimensions.

Reinaud J. N.
Circular Vortex Arrays in Generalised Euler’s and Quasigeostrophic Dynamics
Abstract
We investigate the stability of circular point vortex arrays and their
evolution when their dynamics is governed by the generalised
twodimensional Euler's equations and the threedimensional
quasigeostrophic equations. These sets of equations offer a family
of dynamical models depending continuously on a single parameter
$\beta$ which sets how fast the velocity induced by a vortex falls
away from it. In this paper, we show that the differences between the
stability properties of the \emph{classical} twodimensional point
vortex arrays and the \emph{standard} quasigeostrophic vortex arrays
can be understood as a bifurcation in the family of models. For a
given $\beta$, the stability depends on the number $N$ of vortices
along the circular array and on the possible addition of a vortex at
the centre of the array. From a practical point of view, the most
important vortex arrays are the stable ones, as they are robust and
longlived. Unstable vortex arrays can, however, lead to interesting
and convoluted evolutions, exhibiting quasiperiodic and chaotic
motion. We briefly illustrate the evolution of a small selection of
representative unstable vortex arrays.

Scoppola B., Troiani A., Veglianti M.
Tides and Dumbbell Dynamics
Abstract
We discuss a model describing the effects of tidal dissipation on the satellite’s orbit
in the twobody problem. Tidal bulges are described in terms of a dumbbell, coupled to the
rotation by a dissipative interaction. The assumptions on this dissipative coupling turn out to
be crucial in the evolution of the system.
