Volume 29, Number 2
Volume 29, Number 2, 2024
GrottaRagazzo C., Gustafsson B., Koiller J.
On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d
Abstract
Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler equation in terms of 1forms yields a coupled PDEODE system. The $L^2$orthogonal components are a ``pure'' vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on $N$ point vortices on a compact Riemann surface without boundary of genus $g$, with a metric chosen in the conformal class. The phase space has finite dimension $2N+ 2g$. We compute a surface of section for the motion of a single vortex ($N=1$) on a torus ($g=1$) with a nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ($ g \geqslant 2$) having constant curvature $1$, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally, we consider multiply connected planar domains. The image method due to Green and Thomson is
viewed in the Schottky double. The Kirchhoff – Routh Hamiltonian
given in C.C. Lin's celebrated theorem is recovered by
Marsden – Weinstein reduction from $2N+2g$ to $2N$.
The relation between the electrostatic Green function and the
hydrodynamic Green function is clarified.
A number of questions are suggested.

Lin Z., Zelenko I.
Abstract
The classical result of Eisenhart states that, if a Riemannian metric $g$ admits a Riemannian metric that is not constantly proportional to $g$ and has the same (parameterized) geodesics as $g$ in a neighborhood of a given point, then $g$ is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step $2$ graded nilpotent Lie algebras, called $\mathrm{ad}$surjective, and extend the Eisenhart theorem to subRiemannian metrics on step $2$ distributions with $\mathrm{ad}$surjective Tanaka symbols. The class of adsurjective step $2$ nilpotent Lie algebras contains a wellknown class of algebras of Htype as a very particular case.

de Neeling D., Roest D., Seri M., Waalkens H.
Abstract
The recent detection of gravitational waves emanating from inspiralling black hole
binaries has triggered a renewed interest in the dynamics of relativistic twobody systems. The
conservative part of the latter are given by Hamiltonian systems obtained from socalled post
Newtonian expansions of the general relativistic description of black hole binaries. In this paper
we study the general question of whether there exist relativistic binaries that display Keplerlike
dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem
indeed exists for relativistic systems with a Hamiltonian of a Keplerlike form. This form is
realised by extremal black holes with electric charge and scalar hair to at least first order in
the postNewtonian expansion for arbitrary mass ratios and to all orders in the postNewtonian
expansion in the testmass limit of the binary. Moreover, to fifth postNewtonian order, we
show that Hamiltonians of the Keplerlike form can be related explicitly through a canonical
transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians
conserving a Laplace – Runge – Lenzlike vector are related in this way to Kepler.

Barinova M. K., Grines V. Z., Pochinka O. V., Zhuzhoma E. V.
Abstract
We consider a topologically mixing hyperbolic attractor $\Lambda\subset M^n$ for a diffeomorphism $f:M^n\to M^n$ of a compact orientable $n$manifold $M^n$, $n>3$. Such an attractor $\Lambda$ is called an Anosov torus provided the restriction $f_{\Lambda}$ is conjugate to Anosov algebraic automorphism of $k$dimensional torus $\mathbb T^k$.
We prove that $\Lambda$ is an Anosov torus for two cases:
1) $\dim{\Lambda}=n1$, $\dim{W^u_x}=1$, $x\in\Lambda$;
2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$invariant closed $k$manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.

Gelfreikh N. G., Ivanov A. V.
Abstract
We study a slowfast system with two slow and one fast variables. We assume that
the slow manifold of the system possesses a fold and there is an equilibrium of the system in
a small neighborhood of the fold. We derive a normal form for the system in a neighborhood
of the pair “equilibriumfold” and study the dynamics of the normal form. In particular, as
the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincaré
map and calculate the parameter values for the first perioddoubling bifurcation. The theory is
applied to a generalization of the FitzHugh – Nagumo system.
