Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: Beyond Poncelet porisms

The thirty years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper. Starting with the observation of the billiard nature of some classical constructions and configurations, we construct the billiard algebra, that is a group structure on the set T of lines simultaneously tangent to d−1 quadrics from a given confocal family in the d-dimensional Euclidean space. Using this tool, the related results of Reid, Donagi and Knörrer are further developed, realized and simplified. We derive a fundamental property of T: any two lines from this set can be obtained from each other by at most d−1 billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: s-skew lines in T and s-weak Poncelet trajectories, s=−1,0,…,d−2. The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in this paper enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results: Cayley's theorem, Weyr's theorem, Griffiths-Harris theorem and Darboux theorem.     

An Ellipsoidal Billiard with a Quadratic Potential

There exists an infinite hierarchy of integrable generalizations of the geodesic flow on an n-dimensional ellipsoid. These generalizations describe the motion of a point in the force fields of certain polynomial potentials. In the limit as one of semiaxes of the ellipsoid tends to zero, one obtains integrable mappings corresponding to billiards with polynomial potentials inside an (n-1)-dimensional ellipsoid.In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke) potential, its representation in Lax form, and a theta function solution. We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The method we use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.     

A survey of the analytical description of periodic elliptical billiard trajectories

The complete Poncelét theorem and its generalization related to billiard systems within several confocal quadrics in ℝ^d is considered. Starting from algebro-geometric conditions for periodic elliptic billiard trajectories, obtaining explicit Cayley-type conditions is demonstrated. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

Shortest billiard trajectories

In this paper we prove that any convex body of the d-dimensional Euclidean space (d ≥ 2) possesses at least one shortest generalized billiard trajectory moreover, any of its shortest generalized billiard trajectories is of period at most d + 1. Actually, in the Euclidean plane we improve this theorem as follows. A disk-polygon with parameter r > 0 is simply the intersection of finitely many (closed) circular disks of radii r, called generating disks, having some interior point in common in the Euclidean plane. Also, we say that a disk-polygon with parameter r > 0 is a fat disk-polygon if the pairwise distances between the centers of its generating disks are at most r. We prove that any of the shortest generalized billiard trajectories of an arbitrary fat disk-polygon is a 2-periodic one. Also, we give a proof of the analogue result for ε-rounded disk-polygons obtained from fat disk-polygons by rounding them off using circular disks of radii ε > 0. Our theorems give partial answers to the very recent question raised by S. Zelditch on characterizing convex bodies whose shortest periodic billiard trajectories are of period 2. K. Bezdek partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Hirota equation and Bethe ansatz

Recent analyses of classical integrable structures in quantum integrable models solved by various versions of the Bethe ansatz are reviewed. Similarities between elements of quantum and classical theories of integrable systems are discussed. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton theory. Functional relations for quantum transfer matrices become the classical Hirota bilinear difference equation; solving this classical equation gives all the basic results for the spectral properties of quantum systems. Vice versa, typical Bethe ansatz formulas under certain boundary conditions yield solutions of this classical equation. The Baxter T-Q relation and its generalizations arise as auxiliary linear problems for the Hirota equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 1, pp. 54–100, July, 1998.     

Integrable billiards model important integrable cases of rigid body dynamics

Abstract—A generalized billiard is considered, in which a point moves on a locally flat surface obtained by isometrically gluing together several plane domains along boundaries being arcs of confocal quadrics. Under this motion, a point moves from one domain to another, passing through the glued boundaries. Many integrable cases of rigid body dynamics with appropriate parameter values at certain levels of integrals are modeled by classical or generalized billiards; in the paper, Liouville equivalence is proved by comparing Fomenko–Zieschang invariants.     

Generalized Calogero–Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals that are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero–Moser type which we explicitly specify. In the case of classical Coxeter groups, we also obtain generalized Calogero–Moser systems with added quadratic potential.     

Lower estimates for the number of closed trajectories of generalized billiards

The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let k be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length k in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let M be a smooth closed m-dimensional submanifold of a Euclidean space, and let p > 2 be a prime integer. Then M has at least

Integrable elliptic pseudopotentials

We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are multiparameter deformations of ones constructed by Krichever in studying the Whitham-averaged solutions of the KP equation and yield new integrable (2+1)-dimensional systems of hydrodynamic type. Moreover, an interesting class of integrable (1+1)-dimensional systems described in terms of solutions of an elliptic generalization of the Gibbons-Tsarev system is related to these pseudopotentials.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras II: A nonlinear system arising from the group G 6.1 and its exact integration

In the first paper of this series a correspondence was established between coupled systems of two-dimensional nonlinear wave equations and the six-dimensional simply transitive Lie algebras. In the present paper we make use of this result to construct a Darboux integrable and exactly integrable nonlinear system associated with the six-parameter nilpotent Lie group G _6,1 and we give its exact general solution in terms of four arbitrary functions. The procedure is shown to be an exact linearization of the nonlinear problem.     

Generalizations of the neumann system—a curve theoretical approach part III: Order n systems

The Neumann system is a well-known algebraically completely integrable Hamiltonian system. Its geometry has roots in hyperelliptic curve theory and the isospectral deformation theory of Hill's operator. In this paper generalizations of the Neumann system are found for n-sheeted Riemann surfaces and the isospectral deformation theory of operators of order n. Trace formulas, Lax pairs, and constants of motion are found. The new systems are shown to be algebraically completely integrable.     

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

We consider natural complex Hamiltonian systems with n degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential V of degree k > 2. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each k there exists an explicitly known infinite set ⊂ ℚ such that if the system is integrable, then all eigenvalues of the Hessian matrix V″(d) calculated at a non-zero d ∈ ℂ ⁿ satisfying V′(d) = d, belong to . The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning V we prove the following fact. For each k and n there exists a finite set such that if the system is integrable, then all eigenvalues of the Hessian matrix V″(d) belong to . We give an algorithm which allows to find sets . We applied this results for the case n = k = 3 and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.      

Classical Integrable Systems and Billiards Related to Generalized Jacobians

We study some classical integrable systems of dynamics (the Euler top in space, the asymptotic geodesic motion on an ellipsoid) which are linearized on unramified coverings of generalized Jacobian varieties. We find explicit expressions for so called root functions living on such coverings which enable us to solve the problems in terms of generalized theta-functions. In addition, general and asymptotic solutions for ellipsoidal billiards and the billiard in an ellipsoidal layer are obtained.     

The dressing method and nonlocal Riemann-Hilbert problems

Summary We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.     

Classical solutions of quasilinear parabolic systems on two dimensional domains

Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system.     

Abstract Lebesgue-Radon-Nikodym theorems

Summary In this paper generalizations of the classical Lebesgue-Radon-Nikodym type decomposition of additive set functions are obtained for pairs of vector measures when both measures take values in possibly different Banach spaces. Some applications of these results are made to (i) the representation of wearly compact operators on the spaces of integrable scalar functions relative to a vector measure to an arbitrary Banach space, and (ii) a problem of comparison of measures in inference theory. The abstract conditional expectations of operator valued strongly measurable and integrable random variables on a σ-finite space are briefly treated. Supported, in part, under the NSF Grants GP-1349 and GP-5921.     

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical ? ? ? ^n?1 structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.     

Higher Weights of Anticodes and the Generalized Griesmer Bound

We show that the minimum r-weight d_r of an anticode can be expressed in terms of the maximum r-weight of the corresponding code. As examples, we consider anticodes from homogeneous hypersurfaces (quadrics and Hermitian varieties). In a number of cases, all differences (except for one) of the weight hierarchy of such an anticode meet an analog of the generalized Griesmer bound.