Related papers: On the completeness of the Manakov integrals
The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold $M$, for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of…
This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian Gravity and General Relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding…
We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a…
Let $(M, g)$ be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the…
The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding…
Complications arising from the non-compact nature of the phase space of N-body systems prevent any asymptotic characterization of chaotic behaviour (since no equilibrium final states can exist). This leads us to revisit some of the old…
We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point,…
We prove that the geodesic flow of a Kupka-Smale riemannian metric on a closed surface has homoclinic orbits for all of its hyperbolic closed geodesics.
We consider the modified Navier-Stokes equations in R3 describing the motion of a fluid in the presence of a rotating rigid body. Weighted Sobolev spaces are used to describe the behavior of solutions at large distances. Under suitable…
We study the geodesic motion in a space-time describing a swirling universe. We show that the geodesic equations can be fully decoupled in the Hamilton-Jacobi formalism leading to an additional constant of motion. The analytical solutions…
We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an…
We study Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral quartic in momenta. The main results of the work are local description of such metrics in terms of…
We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of $T^2 \times…
The aim of this short note is to show how to construct a complete Lyapunov function of a semiflow by using a complete Lyapunov function of its time-one map. As a byproduct we assure the existence of complete Lyapunov functions for semiflows…
We study the dynamics of a coupled system, formed by a rigid body with a cavity entirely filled with magnetohydrodynamic compressible fluid. Our aim is to derive the global existence of the unique classical solutions and weak solutions to…
This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these…
In this paper we give a natural condition for when a volumorphism on a Riemannian manifold $(M,g)$ is actually an isometry with respect to some other, optimal, Riemannian metric $h$. We consider the natural action of volumorphisms on the…
Integrable spin systems are an important subclass of integrable (soliton) nonlinear equations. They play important role in physics and mathematics. At present, many integrable spin systems were found and studied. They are related with the…
We briefly review the Kapovich-Millson notion of Bending flows as an integrable system on the space of polygons in ${\bf R}^3$, its connection with a specific Gaudin XXX system, as well as the generalisation to $su(r), r>2$. Then we…
We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic…