On Dirichlet, Poncelet and Abel problems

Abstract

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form θ ∈ Qwhere number θ = m/n is built by means of data for a problem to solve.We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg XY -chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.

Poncelet-Darboux, Kippenhahn, and Szegő: interactions between projective geometry, matrices and orthogonal polynomials

“…The map τ : T → T, also known as the Poncelet correspondence, see [18] (and in some particular cases is related to the John mapping, see [5]), provides a connection between Poncelet curves and discrete dynamical systems and ellipsoidal billiards, see e.g. the work of Dragović and collaborators [1,17] and Schwarz [52,57].…”

Combinatorial analysis of the period mapping: the topology of 2D fibres

“…2. Suppose now we have an equality sign in (10). Let ∆ ′ ≤ ∆ be the largest simplex in the boundary such that f intersects its interior (take the convex hull of all simplexes with this property).…”

“…For the above formula to give a single valued function on a Riemann sphere, all the periods of the abelian integral should be multiples of the periods of cosine, in other words lie in the lattice 2πin −1 Z. Deformation of the initial optimization problem brings us to the deformation of the curve M and the induced deformation of the differential dη M such that all its periods are conserved since one cannot continuously jump from one point of the lattice to the other. The above representation of polynomials in terms of Riemann surfaces also gives the solution to Pell-Abel equation which in turn is related to Poncelet problem, elliptic billiards [13] and boundary value problem for string equation [10].…”

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

“…As can be expected, a point of finite order gives rise to many applications in the theory of periodic motion, and recently in [10] it was applied in a novel way to give a condition of Fritz John type on a Dirichlet problem for a planar domain bounded by an ellipse; in the same paper, the authors give the following explicit parametrization of the conics, and show that the vertices of a Poncelet N -gon give a solution to the N -periodic Toda chain, which we reproduce below.…”