Publications (16)
New Semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces
Michael, Bialy, Andrey Mironov
We consider magnetic geodesic flows on the 2-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a Semi-Hamiltonian…
Integrable geodesic flows on 2-torus: formal solutions and variational principle
Michael, Bialy, Andrey E. Mironov
In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic fl…
Smooth solutions for a -system of mixed type
Michael, Bialy
In this note we analyze smooth solutions of a -system of the \textit{mixed} type. Motivating example for this is a 2-components reduction of the Benney moments chain which appea…
In search of periodic solutions for a reduction of the Benney chain
Michael, Bialy, Andrey Mironov
We search for smooth periodic solutions for the system of quasi-linear PDEs known as the Lax dispersionless reduction of the Benney moments chain. It is naturally related to the ex…
Richness or Semi-Hamiltonicity of quasi-linear systems which are not in evolution form
Michael, Bialy
The aim of this paper is to consider quasi-linear systems which are not in the form of evolution equations. We propose new condition of Richness or Semi-Hamiltonicity for such a sy…
On Periodic solutions for a reduction of Benney chain
Michael, Bialy
We study periodic solutions for a quasi-linear system, which is the so called dispersionless Lax reduction of the Benney moments chain. This question naturally arises in search of…
Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane
Michael, Bialy, Andrey E. Mironov
We consider a convex curve lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the do…
Effective bounds in E.Hopf rigidity for billiards and geodesic flows
Michael, Bialy
In this paper we show that in some cases the E.Hopf rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set $\mathcal{M…
On Newton equations which are totally integrable at infinity
Michael, Bialy
In this paper Hamiltonian system of time dependent periodic Newton equations is studied. It is shown that for dimensions and higher the following rigidity results holds true: I…
A remark on the number of invisible directions for a smooth Riemannian metric
Michael, Bialy
In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible direction…
A survey on Polynomial in momenta integrals for billiard problems
Michael, Bialy, Andrey E. Mironov
In this paper we give a short survey of recent results on algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss i…
Integrable magnetic geodesic flows on 2-torus: new example via quasi-linear system of PDEs
Sergey V. Agapov, Michael, Bialy +1
The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral…
Cubic and Quartic integrals for geodesic flow on 2-torus via system of Hydrodynamic type
Michael, Bialy, Andrey Mironov
In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of…
Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane
Michael, Bialy
This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic p…
On Totally integrable magnetic billiards on constant curvature surface
Michael, Bialy
We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrab…
Algebraic non-integrability of magnetic billiards
Michael, Bialy, Andrey E. Mironov
We consider billiard ball motion in a convex domain of the Euclidean plane bounded by a piece-wise smooth curve influenced by the constant magnetic field. We show that if there exi…