Study shows magnetic billiards are algebraically non-integrable in most cases.
problem Non-integrability of magnetic billiards in convex domains.
method Analyzing polynomial integrals of motion in magnetic billiard flows.
result Magnetic billiards are algebraically non-integrable for most smooth boundary curves.
We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. Thi…
Study shows non-integrability of magnetic billiards on curved surfaces.
problem Non-existence of polynomial integrals for magnetic billiards.
method Examined billiard motion on constant curvature surfaces with magnetic fields.
result For most magnetic field values, billiard motion lacks polynomial integrals.
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 integrable then the boundary curve is necessarily a circle. This result is a manifestation of the so-called Hopf rigidity phenomenon which was recen…
Study of billiards in sub-Finsler geometry, including unusual orbits.
problem Exploring billiard dynamics in sub-Finsler spaces.
method Symplectic and variational approaches, control theory.
result Unusual orbits like gliding and creeping orbits exist.
Survey of integrable billiard models and inequalities.
problem Understanding integrable billiard dynamics.
method Analysis of various billiard models.
result Isoperimetric inequality for Mather β-function.
Lower bounds on periodic Finsler billiard trajectories in convex hypersurfaces.
problem Estimating the number of periodic Finsler billiard trajectories.
method Morse and Lusternik-Schnirelmann theories applied to extremal polygons inscribed in a smooth closed hypersurface.
result For prime r≥3, the number of r-periodic Finsler billiard trajectories is not less than (r−1)(d−2)+1. This paper connects billiards in ellipses to focal billiards in ellipsoids.
problem Proving the existence of isometric counterparts between billiards in ellipses and focal billiards in ellipsoids.
method Continuous transition via isometric focal billiards in a fixed ellipsoid.
result Established the connection between planar and spatial billiards.
New dynamical system for convex curves solves integrable billiards conjecture.
problem Algebraic Birkhoff conjecture on integrable billiards.
method Introducing Angular billiard system.
result New results on algebraic Birkhoff conjecture.
This paper proves integrability of Birkhoff billiards inside convex cones.
problem Proving integrability of Birkhoff billiards in non-traditional shapes.
method Analyzing the billiard inside a convex cone, proving integrability using a first integral of degree two.
result The Birkhoff billiard inside a convex C3 cone is integrable. Dan Reznik discovered conserved quantities for ellipses using billiard maps.
problem Conservation laws in periodic billiard trajectories.
method Non-standard generating function for the billiard ball map.
result Proved identities valid for all smooth convex billiard tables.
Billiard motion in ellipses analyzed with canonical coordinates.
problem Understanding billiard motion in ellipses.
method Canonical coordinates and kinematic analysis.
result Explicit parametrization of billiard motions using Jacobian elliptic functions.
Study polynomial integrals for Birkhoff billiards on spheres and hyperbolic planes.
problem Existence of polynomial integrals for Birkhoff billiards.
method Extension of S. Bolotin's method and use of Angular billiard.
result New obstructions on polynomial integrability.
The paper establishes uniform hyperbolicity in nonflat billiards.
problem Uniform hyperbolicity in nonflat billiards.
method Geodesic flow and bouncing behavior on nonflat billiards.
result A sufficient condition for nonflat billiards to be uniformly hyperbolic.
Abstract collects open problems in billiards and symplectic geometry.
problem Open problems in billiards and symplectic geometry.
method Compilation of open problems from discussions.
result Compilation of open problems.
Billiard trajectories and geodesics are closely related geometrically.
problem Understanding the relationship between billiard trajectories and geodesics on surfaces.
method Establishing mutual approximation results for billiard trajectories and geodesic segments on surfaces.
result For Riemannian billiard tables, there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory.
Open problems in billiards and optics from a workshop.
problem Open problems in billiards and geometric optics.
method Collection of open problems from a workshop.
result No specific key result mentioned; collection of problems.
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 plane or on the Hemisphere is circular billiard.
Method finds differential equations for integrable billiard tables.
problem Finding differential equations for integrable billiard tables.
method Introducing a method to find differential equations for functions defining tables.
result Illustrated method in three billiard systems.
Rolling systems limit to billiard models with no-slip collisions.
problem Understanding how rolling systems behave as billiard models with no-slip collisions.
method Showed that no-slip billiards arise as limits of non-holonomic rolling systems.
result Rolling systems limit to billiard models with no-slip collisions.
Proves properties of periodic billiard orbits in ellipses.
problem Understanding periodic orbits in ellipses.
method Geometric and complex analytic methods.
result Sum of cosines of angles remains constant in one-parameter family of polygons.
Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.
problem Proves conjecture for centrally-symmetric billiards.
method Uses non-standard generating function, invariant curve structure, and integral-geometry approach.
result Billiard curve is an ellipse under given conditions.
Study on billiard trajectories with fixed bounces.
problem Counting periodic trajectories with specific bounces.
method Analyzes two-dimensional dispersive billiard systems.
result Asymptotic growth of primitive periodic trajectories.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
Paper finds conditions for different norms to produce same billiard paths.
problem Conditions for different norms to define the same billiard reflection law.
method Extending previous works by Milena Radnović and Serge Tabachnikov, the paper establishes conditions for two different non-symmetric norms to define the same billiard reflection law.
result Conditions for two different norms to define the same billiard reflection law.
A simple proof shows standard billiard for certain convex domains.
problem Characterizing billiards in convex domains that are both projective and Minkowski.
method Direct simple proof in C1-smoothness, semi-local and local versions proved. result Standard Euclidean billiard in an appropriate structure.
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in Rn. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiar…
The study finds billiard trajectories with infinitely many reflections in certain cones.
problem Existence of billiard trajectories with infinitely many reflections.
method Analysis of C3 convex cones and elliptic cones in R3. result Existence of C2 convex cones with billiard trajectories having infinitely many reflections. Given a planar compact convex billiard table T, we give an algorithm to find the shortest generalised closed billiard orbits on T. (Generalised billiard orbits are usual billiard orbits if T has smooth boundary.) This algorithm is finite if T is a polygon and provides an approximation scheme in general. As an i…
Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from the boundary making the angle of incidence equal the angle of reflection. The bi…
Investigates billiard dynamics on smooth curves in higher dimensions.
problem Extending conventional billiard dynamics to higher-dimensional smooth curves.
method Area-preserving twist map, KAM theory, Mather's converse KAM, interpolating Hamiltonians.
result Uniform distribution of impact points for small chords on nice wires.
Gutkin billiard tables studied in higher dimensions, rigidity proven.
problem Characterizing billiard tables with constant angle invariants.
method New generating function for billiards, rigidity proof.
result In higher dimensions, only spheres have Gutkin billiard tables with constant angle invariants.
Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.
problem Outer billiards in complex hyperbolic plane.
method Symplectic form and geodesic foliation analysis.
result Outer billiard map is a diffeomorphism and symplectomorphism.
Study on wind-tree models yields formulas for periodic trajectories.
problem Counting periodic trajectories in wind-tree models.
method Asymptotic formulas and explicit computation of Siegel-Veech constants.
result Asymptotic formulas for closed billiard trajectories in wind-tree models.
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C2,1. In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of t…
We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points A,B such that no finite set of points can block all billiard trajectories from A to B.
Researchers calculate complexity of billiard paths in regular polygons.
problem Calculating the complexity of billiard paths in regular polygons.
method Counting saddle connections on lattice surfaces, focusing on combinatorial length.
result They answered a question about billiard language complexity in regular polygons.
Study shows only spheres satisfy special billiard properties in higher dimensions.
problem Characterizing convex hypersurfaces with specific billiard properties.
method Examined billiard chords and their angles with tangent planes.
result Only spheres satisfy the Gutkin property in higher dimensions.
Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.
problem Finding optimal shapes to minimize the average length of billiard trajectories.
method Used techniques from Teichmüller theory.
result Optimal shapes minimize average lengths of billiard trajectories in specific polygons.
This note connects tiling billiards dynamics to Novikov's problem via helicoidal construction.
problem Understanding dynamics of tiling billiards and topology of subsurface sections.
method Helicoidal construction by Ivan Dynnikov.
result Relationship between tiling billiards and Novikov's problem in higher genus.
In this paper the problem of estimating the number of periodical billiard trajectories is considered. The main result is the theorem on Morse theory for periodical billiard trajectories.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.
problem Understanding the measure of maximizing orbits in symmetric billiard tables.
method Introduced a closed invariant set of locally maximizing orbits and gave an effective bound on its measure.
result An effective bound on the measure of the invariant set in terms of the isoperimetric defect of the curve.
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.
Locally maximizing orbits studied in twist maps and billiards.
problem Characterize orbits in locally maximizing class for twist maps.
method Geometric and variational analysis of orbits in the cotangent bundle of a torus or ball bundle over a sphere.
result Two generating functions for the Birkhoff billiard map have the same class of locally maximizing orbits.
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
Geometrically interprets integrability of geodesic flow using web theory.
problem Integrability of geodesic flow by quadratic integrals.
method Geometric interpretation through web theory and integrable billiards construction.
result Constructs integrable billiards on surfaces with quadratic geodesic integrals.