Volume 29, Number 2

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler equation in terms of 1-forms yields a coupled PDE-ODE 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.

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 sub-Riemannian metrics on step $2$ distributions with $\mathrm{ad}$-surjective Tanaka symbols. The class of ad-surjective step $2$ nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called 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 Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler.

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) $\dim{\Lambda}=n-1$, $\dim{W^u_x}=1$, $x\in\Lambda$; 2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k-1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$-invariant closed $k$-manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.

We study a slow-fast 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 “equilibrium-fold” 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 period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh – Nagumo system.