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Andrey E. Mironov

Publications and source records attributed to Andrey E. Mironov.

At least 19 recordsLinked to original sources
arXiv2026

Integrability of Billiards Inside Cones as a Discrete-Time Hamiltonian System

In this paper, we continue to study billiards inside cones $K\subset \mathbb{R}^n$ over strictly convex closed $C^3$ manifolds with non-degenerate second fundamental form. Recently we proved that the billiard is superintegrable, i.e., the billiard admits first integrals whose values uniquely determine all billiard trajectories. In this paper we prove that this billiard system admits $n-1$ independent first integrals in involution. Consequently, the system is completely integrable as a discrete-time Hamiltonian system. This provides an example of an integrable billiard where the billiard table is neither a quadric nor consists of pieces of quadrics.

math.DS↗
arXiv2026

Integrable Geodesic Flows on Cones over Riemannian Manifolds

In this paper we study the behavior of geodesics on cones over arbitrary $C^3$-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for radial geodesics; thus, the geodesic flow is superintegrable. Moreover, we prove that the geodesic flow restricted to the open dense subset of the cotangent bundle corresponding to all non-radial trajectories is Liouville--Arnold integrable. This investigation is inspired by our recent results on Birkhoff billiards inside cones over convex manifolds where similar results hold true.

math.DG↗
arXiv2025

Integrable Billiards and Related Topics

This paper surveys our results on integrable billiards. We consider various models of billiards, including Birkhoff, outer, magnetic, and Minkowski billiards. Also, we discuss wire billiards and billiards in cones. For four models of convex plane billiards, we also discuss an isoperimetric-type inequality for the Mather $β$-function. We conclude with a section of open questions on this subject.

math.DS↗
arXiv2025

Billiard trajectories inside Cones

Recently it was proved that every billiard trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by a $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$ submanifold of the hyperplane with nondegenerate second fundamental form. In this paper, we prove the existence of $C^2$ convex cones admitting billiard trajectories with infinitely many reflections in finite time. We also estimate the number of reflections of billiard trajectories in elliptic cones in $\mathbb{R}^3$ using two first integrals.

math.DS↗
arXiv2025

Two-dimensional discrete operators and rational functions on algebraic curves

In this paper we study a connection between finite-gap on one energy level two-dimensional Schrodinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators. These operators have eigenfunctions on zero level energy parameterized by points of algebraic spectral curves. In the case of genus one spectral curves we show that the finite-gap Schrodinger operators can be obtained as a limit of the discrete operators.

nlin.SI↗
arXiv2025

Integrable Birkhoff Billiards inside Cones

One of the most interesting problems in the theory of Birkhoff billiards is the problem of integrability. In all known examples of integrable billiards, the billiard tables are either conics, quadrics (closed ellipsoids as well as unclosed quadrics like paraboloids or cones), or specific configurations of conics or quadrics. This leads to the natural question: are there other integrable billiards? The Birkhoff conjecture states that if the billiard inside a convex, smooth, closed curve is integrable, then the curve is an ellipse or a circle. In this paper we study the Birkhoff billiard inside a cone in $\mathbb{R}^n$. We prove that the billiard always admits a first integral of degree two in the components of the velocity vector. Using this fact, we prove that every trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$ submanifold of the hyperplane with nondegenerate second fundamental form. The main result of this paper is the following. We prove that the Birkhoff billiard inside a convex $C^3$ cone is integrable. This is the first example of an integrable billiard where the billiard table is neither a quadric nor composed of pieces of quadrics.

math.DS↗
arXiv2022

On differential equations of integrable billiard tables

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ${\mathbb R}^3$ with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ${\mathbb R}^3$ homeomorphic to a torus, being a table of an integrable billiard.

math.DS↗
arXiv2022

The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $γ$ admits a $C^1$-smooth first integral with non-vanishing gradient on $\mathcal A$, then the curve $γ$ is an ellipse.} The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in \cite{BM}, \cite{B}; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in B0, B1 for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

math.DS↗
arXiv2020

Magnetic billiards: Non-integrability for strong magnetic field; Gutkin type examples

We consider magnetic billiards under a strong constant magnetic field. The purpose of this paper is two-folded. We examine the question of existence of polynomial integral of billiard magnetic flow. We succeed to reduce this question to algebraic geometry test on existence of polynomial integral, which shows polynomial non-integrability for all but finitely many values of the magnitude. In the second part of the paper we construct examples of magnetic billiards which have the so called $δ$-Gutkin property, meaning that any Larmor circle entering the domain with angle $δ$ exits the domain with the same angle $δ$. For ordinary Birkhoff billiard in the plane such examples were introduced by E. Gutkin and are very explicit. Our construction of Gutkin magnetic billiards relies on beautiful examples by F.Wegner of the so called Zindler curves, which are related to the problem of floating bodies in equilibrium, which goes back to S.Ulam. We prove that Gutkin magnetic billiard can be obtained as a parallel curve to a Wegner curve. Wegner curves can be written by elliptic functions in polar coordinates so the construction of magnetic Gutkin billiard is rather explicit but much more complicated.

math.DS↗
arXiv2018

Outer billiards with the dynamics of a standard shift on a finite number of invariant curves

We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.

math.DS↗
arXiv2018

Positive one-point commuting difference operators

In this paper we study commuting difference operators containing a shift operator with only positive degrees. We construct examples of such operators in the case of hyperelliptic spectral curves.

math.AG↗
arXiv2018

A survey on Polynomial in momenta integrals for billiard problems

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 integrable magnetic billiards. As a new application of the algebraic technique we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov.

math.DS↗
arXiv2018

Algebraic non-integrability of magnetic billiards on the Sphere and Hyperbolic plane

We consider billiard ball motion in a convex domain on a constant curvature surface influenced by the constant magnetic field. We examine the existence of integral of motion which is polynomial in velocities. We prove that if such an integral exists then the boundary curve of the domain determines an algebraic curve in $\mathbf{C}^3$ which must be nonsingular. Using this fact we deduce that for any domain different from round disc for all but finitely many values of the magnitude of the magnetic field billiard motion does not have Polynomial in velocities integral of motion.

math.DG↗
arXiv2017

Energy functional for Lagrangian tori in $\mathbb{C}P^2$

In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional periodic Schrödinger operator is associated with every Lagrangian torus in ${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of the potential of the Schrödinger operators, which has a natural geometrical meaning. We study the energy functional on two families of Lagrangian tori and propose a conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori. In particular we show that if the deformation preserves a conformal type of the torus, then it also preserves the area of the torus. Thus it follows that deformations generated by Novikov-Veselov equations preserve the area of minimal Lagrangian tori.

math.DG↗
arXiv2016

Integrable magnetic geodesic flows on 2-torus: new example via quasi-linear system of PDEs

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 is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on one energy level is considered. This problem can be reduced to a remarkable Semi-hamiltonian system of quasi-linear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville metric with zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of a quadratic in momenta integral. Thus our construction gives a new example of smooth periodic solution to the Semi-hamiltonian (Rich) quasi-linear system of PDEs.

math.DS↗