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.

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 $\langle N_k\rangle (k-k_G)^B$ becomes 1-periodic 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.

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.

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 non-Hamiltonian 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.

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 bi-invariant Riemannian metric and has initially a homogeneous distribution of mass. By the least action principle, force-free 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.

We investigate the stability of circular point vortex arrays and their evolution when their dynamics is governed by the generalised two-dimensional Euler's equations and the three-dimensional quasi-geostrophic 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} two-dimensional point vortex arrays and the \emph{standard} quasi-geostrophic 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 long-lived. Unstable vortex arrays can, however, lead to interesting and convoluted evolutions, exhibiting quasi-periodic and chaotic motion. We briefly illustrate the evolution of a small selection of representative unstable vortex arrays.

We discuss a model describing the effects of tidal dissipation on the satellite’s orbit in the two-body 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.