Related papers: Integrable Billiards and Related Topics
In this article we discuss pointwise spectral rigidity results for several billiard systems (e.g., Birkhoff billiards, symplectic billiards and $4$-th billiards), showing that a single value of Mather's $\beta$-function can determine…
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…
This article is concerned with the study of Mather's \beta-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it…
For strictly convex billiard maps of smooth boundaries, we get a Birkhoff normal form via a list of constructive generating functions. Based on this, we get an explicit formula for the beta function (locally), and explored the relation…
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…
In this article we explain that several integrable mechanical billiards in the plane are connected via conformal transformations. We first remark that the free billiard in the plane are conformal equivalent to infinitely many billiard…
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…
The three-dimensional Kepler problem is related to the four-dimensional isotropic harmonic oscillators by the Kustaanheimo-Stiefel Transformations. In the first part of this paper, we study how certain integrable mechanical billiards are…
Consider a strictly convex set $\Omega$ in the plane, and a homogeneous, stationary magnetic field orthogonal to the plane whose strength is $B$ on the complement of $\Omega$ and $0$ inside $\Omega$. The trajectories of a charged particle…
We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those…
We focus on the outer length billiard dynamics, acting on the exterior of a strictly-convex planar domain. We first show that ellipses are totally integrable. We then provide an explicit representation of first order terms for the formal…
In this paper, we use the projective dynamical approach to integrable mechanical billiards as in [Zhao, 2021] to establish the integrability of natural mechanical billiards with the Lagrange problem, which is the superposition of two Kepler…
We study billiards in plane domains, with a perpendicular magnetic field and a potential. We give some results on periodic orbits, KAM tori and adiabatic invariants. We also prove the existence of bound states in a related scattering…
We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result…
In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff…
We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if,…
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…
Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The…
The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a…
In this article, we consider mechanical billiard systems defined with Lagrange's integrable extension of Euler's two-center problems in the Euclidean space, on the sphere, and in the hyperbolic space of arbitrary dimension $n \ge 3$. In the…