Integrable Billiards Research Articles

Birkhoff Conjecture for Nearly Centrally Symmetric Domains

AbstractIn this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 2018) and show that a domain close enough to a centrally symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) by proving that a notion of rational integrability is equivalent to the C0-integrability condition used in their paper.

Rotation functions of integrable billiards as orbital invariants

Rotation functions of integrable billiards as orbital invariants

Integrable Mechanical Billiards in Higher-Dimensional Space Forms

Integrable Mechanical Billiards in Higher-Dimensional Space Forms

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

The billiard flow in a planar domain Ω acts on the tangent bundle TR2|Ω as geodesic flow with reflections from the boundary. It has the trivial first integral: squared modulus of the velocity. Bolotin’s conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those billiards whose flow admits an additional integral that is polynomial in the velocity and whose restriction to the unit tangent bundle is non-constant. It states that (1) if the boundary of such a billiard is C 2-smooth, nonlinear and connected, then it is a conic; (2) if it is piecewise C 2-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of ‘admissible lines’ for the pencil; (3) the minimal degree of the additional integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, which defines a reflection acting on oriented lines and the projective billiard flow. They are common generalization of billiards on surfaces of constant curvature, but in general may have no canonical conserved quantity. In a previous paper the author classified those C 4-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational 0-homogeneous function of the velocity (with coefficients depending on the position): these billiards are called rationally 0-homogeneously integrable. It was shown that: (1) the underlying curve is a conic; (2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; (3) otherwise it can be arbitrary even number. In the present paper we classify piecewise C4-smooth rationally 0-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards introduced in the present paper.

Integrable outer billiards and rigidity

Integrable outer billiards and rigidity

On Differential Equations of Integrable Billiard Tables

On Differential Equations of Integrable Billiard Tables

Projective integrable mechanical billiards

In this paper, we use the projective dynamical approach to integrable mechanical billiards as in (Zhao 2021 Commun. Contemp. Math. 24 2150085) to establish the integrability of natural mechanical billiards with the Lagrange problem, which is the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the middle of the Kepler centers, as the underlying mechanical systems, and with any finite combinations of confocal conic sections with foci at the Kepler centers as the reflection wall, in the plane, on the sphere, and in the hyperbolic plane. This covers many previously known integrable mechanical billiards, especially the integrable Hooke, Kepler and two-center billiards in the plane, as has been investigated in (Takeuchi and Zhao 2021 arXiv:2110.03376), as subcases. The approach of (Takeuchi and Zhao 2021 arXiv:2110.03376) based on conformal correspondence has been also applied to integrable Kepler billiards in the hyperbolic plane to illustrate their equivalence with the corresponding integrable Hooke billiards on the hemisphere and in the hyperbolic plane as well.

Conformal transformations and integrable mechanical billiards

In this article we explain that several integrable mechanical billiards in the plane are connected via conformal transformations. We first remark that the free billiards in the plane are conformal equivalent to infinitely many billiard systems defined in central force problems on a particular fixed energy level. We then explain that the classical Hooke-Kepler correspondence can be carried over to a correspondence between integrable Hooke-Kepler billiards. As part of the conclusion we show that any focused conic section gives rise to integrable Kepler billiards, which continues previous works of Panov [26] and Gallavotti-Jauslin [11]. We discuss several generalizations of integrable Stark billiards. We also show that any finite combinations of confocal conic sections give rise to integrable billiard systems of Euler's two-center problems.

Modeling the Degenerate Singularities of Integrable Billiard Systems by Billiard Books

Modeling the Degenerate Singularities of Integrable Billiard Systems by Billiard Books

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

Points with rotational ellipsoids of inertia, envelopes of hyperplanes which equally fit the system of points in [formula omitted], and ellipsoidal billiards

For a given system of material points in Rk which represents a sample of a full rank, we study the points for which the ellipsoid of inertia is rotational. Using an initial information about such points with a rotational ellipsoid of inertia along the line of the major axis of the central ellipsoid of inertia, we construct a pencil of confocal quadrics with the following property: all the hyperplanes for which the hyperplanar moments of inertia for the given system of points are equal, are tangent to one of the quadrics from the pencil of quadrics. We close the loop by applying the confocal pencil of quadrics to get a full description of the points with rotational hyperplanar ellipsoids of inertia. We indicate relationships of the obtained results with integrable billiards within quadrics and PCA.

Resonance of ellipsoidal billiard trajectories and extremal rational functions

We study resonant billiard trajectories within quadrics in the d-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of d intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to s-weak billiard trajectories, we come to a functional-polynomial relation of the same generalized Pell type.

Synthetic fuzzballs: a linear ramp from black hole normal modes

We consider a black hole with a stretched horizon as a toy model for a fuzzball microstate. The stretched horizon provides a cut-off, and therefore one can determine the normal (as opposed to quasi-normal) modes of a probe scalar in this geometry. For the BTZ black hole, we compute these as a function of the level n and the angular quantum number J. Conventional level repulsion is absent in this system, and yet we find that the Spectral Form Factor (SFF) shows clear evidence for a dip-ramp-plateau structure with a linear ramp of slope ~ 1 on a log-log plot, with or without ensemble averaging. We show that this is a robust feature of stretched horizons by repeating our calculations on the Rindler wedge (times a compact space). We also observe that this is not a generic feature of integrable systems, as illustrated by standard examples like integrable billiards and random 2-site coupled SYK model, among others. The origins of the ramp can be traced to the hierarchically weaker dependence of the normal mode spectrum on the quantum numbers of the compact directions, and the resulting quasi-degeneracy. We conclude by noting an analogy between the 4-site coupled SYK model and the quartic coupling responsible for the non-linear instability of capped geometries. Based on this, we speculate that incorporating probe self-interactions will lead to stronger connections to random matrix behavior.

Classification of Liouville foliations of integrable topological billiards in magnetic fields

The topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In particular, peculiar saddle singularities are discovered in which singular circles have different orientations - such systems were also previously encountered in mechanical systems in a magnetic field on surfaces of revolution homeomorphic to a 2-sphere. Bibliography: 13 titles.

Topology of Liouville foliations of integrable billiards on table-complexes

Topology of Liouville foliations of integrable billiards on table-complexes

Billiard Ordered Games and Books

The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko's school, in particular of V. Vedyushkina. These more complex billiard domains are obtained by gluing planar sets bounded by arcs of confocal conics along common edges. Such domains are used in this paper to construct the configuration space for billiard ordered games. We analyse dynamical and topological properties of the systems obtained in that way.

Topology of Integrable Billiard in an Ellipse in the Minkowski Plane with the Hooke Potential

Topology of Integrable Billiard in an Ellipse in the Minkowski Plane with the Hooke Potential

Evolutionary force billiards

A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are "billiard-equivalent", despite the fact that the former one is square integrable, and the latter one allows a linear integral.

Topological type of isoenergy surfaces of billiard books

The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products . The Fomenko-Zieschang invariants, which classify Liouville foliations on isoenergy surfaces up to fibrewise homeomorphisms — that is, up to Liouville equivalence of the corresponding integrable Hamiltonian systems — are calculated for several integrable billiards of this type. Bibliography: 14 titles.

Fluctuating Number of Energy Levels in Mixed-Type Lemon Billiards

In this paper, the fluctuation properties of the number of energy levels (mode fluctuation) are studied in the mixed-type lemon billiards at high lying energies. The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by B=0.1953,0.083, is studied. It is shown that the fluctuation of the number of energy levels follows the Gaussian distribution quite accurately, even though the relative fraction of the chaotic part of the phase space is only 0.28 and 0.16, respectively. The theoretical description of spectral fluctuations in the Berry–Robnik picture is discussed. Also, the (golden mean) integrable rectangular billiard is studied and an almost Gaussian distribution is obtained, in contrast to theory expectations. However, the variance as a function of energy, E, behaves as E, in agreement with the theoretical prediction by Steiner.