1,365 research outputs found
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 in simultaneously tangent to d-1
quadrics from a given confocal family. Using this tool, the related results of
Reid, Donagi and Knoerrer 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: the Cayley's theorem, the Weyr's theorem, the Griffiths-Harris
theorem and the Darboux theorem.Comment: 36 pages, 11 figures; to be published in Advances in Mathematic
The Wagner Curvature Tensor in Nonholonomic Mechanics
We present the classical Wagner construction from 1935 of the curvature
tensor for completely nonholonomic manifolds in both invariant and coordinate
way. The starting point is the Shouten curvature tensor for nonholonomic
connection introduced by Vranceanu and Shouten. We illustrate the construction
on two mechanical examples: the case of a homogeneous disc rolling without
sliding on a horizontal plane and the case of a homogeneous ball rolling
without sliding on a fixed sphere. In the second case we study the conditions
on the ratio of diameters of the ball and the sphere to obtain a flat space -
with the Wagner curvature tensor equal zero.Comment: 22 page
Geometry of Integrable Billiards and Pencils of Quadrics
We study the deep interplay between geometry of quadrics in d-dimensional
space and the dynamics of related integrable billiard systems. Various
generalizations of Poncelet theorem are reviewed. The corresponding analytic
conditions of Cayley's type are derived giving the full description of
periodical billiard trajectories; among other cases, we consider billiards in
arbitrary dimension d with the boundary consisting of arbitrary number k of
confocal quadrics. Several important examples are presented in full details
demonstrating the effectiveness of the obtained results. We give a thorough
analysis of classical ideas and results of Darboux and methodology of Lebesgue,
and prove their natural generalizations, obtaining new interesting properties
of pencils of quadrics. At the same time, we show essential connections between
these classical ideas and the modern algebro-geometric approach in the
integrable systems theory.Comment: 49 pages, 14 figure
Systems of the Kowalevski type and discriminantly separable polynomials
Starting from the notion of discriminantly separable polynomials of degree
two in each of three variables, we construct a class of integrable dynamical
systems. These systems can be integrated explicitly in genus two
theta-functions in a procedure which is similar to the classical one for the
Kowalevski top. The discriminnatly separable polynomials play the role of the
Kowalevski fundamental equation. The natural examples include the Sokolov
systems and the Jurdjevic elasticae.Comment: 29 pages, 0 figures. arXiv admin note: substantial text overlap with
arXiv:1106.577
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