Geometry and Integrable Billiards Vladimir Dragovi c GFM - PowerPoint PPT Presentation

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Geometry and Integrable Billiards Vladimir Dragovi´ c GFM University of Lisbon / Mathematical Institute SANU, Belgrade Geometry and Integrability 08 Obergurgl, 13–20 December 2008

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation V. Dragovi´ c, M. Radnovi´ c, Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms , Advances in Mathematics 219 (2008) // arXiv:0710.3656 V. Dragovi´ c, M. Radnovi´ c, Geometry of integrable billiards and pencils of quadrics , Journal de Math´ ematiques Pures et Appliqu´ ees 85 (2006) V. Dragovi´ c, Multi-valued hyperelliptic continued fractions of generalized Halphen type , arXiv: 0809.4931

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation V. Dragovi´ c, M. Radnovi´ c, Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms , Advances in Mathematics 219 (2008) // arXiv:0710.3656 V. Dragovi´ c, M. Radnovi´ c, Geometry of integrable billiards and pencils of quadrics , Journal de Math´ ematiques Pures et Appliqu´ ees 85 (2006) V. Dragovi´ c, Multi-valued hyperelliptic continued fractions of generalized Halphen type , arXiv: 0809.4931

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Preliminaries 1 Poncelet Theorem and Elliptic Billiards Confocal Families of Quadrics and Billiards in Euclidean Space Poncelet Theorem in Projective Space over an Arbitrary Field Billiard Law and Algebraic Structure on the Abelian Variety A ℓ 2 Billiard Algebra and Theorems of Poncelet Type 3 Weak Poncelet Trajectories Generalizations of Theorems of Weyr and Griffiths-Harris Poncelet-Darboux Grid and Higher Dimensional Generalizations Continued Fractions 4 Basic Algebraic Lemma Hyperelliptic Halphen-Type Continued Fractions Periodicity and Symmetry Invariant Approach Multi-valued divisor dynamics Remainders, Continuants and Approximation 5

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards The Poncelet Theorem Let two conics be given in the plane. If there is a closed polygonal line inscribed in one of them and circumscribed about another one, then there is infinitely many such lines and they all have the same number of edges.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Cayley’s Condition C : ( Cx , x ) = 0, D : ( Dx , x ) = 0 – two conics in the projective plane Cayley’s Condition for Even n There is a polygon with n vertices inscribed in C and circumscribed about D if and only if: � . . . � C 3 C 4 C p +1 � � � � C 4 C 5 . . . C p +2 � � = 0 , for n = 2 p , � � . . . � � � � . . . C p +1 C p +2 C 2 p − 1 � � det( C + xD ) = C 0 + C 1 x + C 2 x 2 + . . . is the Taylor � where expansion around x = 0.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Cayley’s Condition C : ( Cx , x ) = 0, D : ( Dx , x ) = 0 – two conics in the projective plane Cayley’s Condition for Odd n There is a polygon with n vertices inscribed in C and circumscribed about D if and only if: � . . . � C 2 C 3 C p +1 � � � � C 3 C 4 . . . C p +2 � � = 0 for n = 2 p + 1 , � � . . . � � � � . . . C p +1 C p +2 C 2 p � � det( C + xD ) = C 0 + C 1 x + C 2 x 2 + . . . is the Taylor � where expansion around x = 0.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Billiard within Ellipse

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Focal Property of Elliptical Billiard

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Focal Property of Elliptical Billiard

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Caustics of Elliptical Billiard

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Caustics of Elliptical Billiard

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem and Elliptic Billiards Periodical Trajectories of Elliptical Billiard Applied to a pair of confocal conics C , D , the Cayley’s condition gives an analytical condition for periodicity of a billiard trajectory within C with D as a caustic.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Confocal Families of Quadrics and Billiards in Euclidean Space Definition of Confocal Family A family of confocal quadrics in the d -dimensional Euclidean space E d is a family of the form: x 2 x 2 1 d Q λ : a 1 − λ + · · · + a d − λ = 1 ( λ ∈ R ) , where a 1 , . . . , a d are real constants.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Confocal Families of Quadrics and Billiards in Euclidean Space Chasles Theorem Chasles Theorem Any line in E d is tangent to exactly d − 1 quadrics from a given confocal family. Tangent hyper-planes to these quadrics, constructed at the points of tangency with the line, are orthogonal to each other. Theorem Two lines that satisfy the reflection law on a quadric Q in E d are tangent to the same d − 1 quadrics confocal with Q .

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Confocal Families of Quadrics and Billiards in Euclidean Space Generalized Poncelet Theorem Consider a closed billiard trajectory within quadric Q in E d . Then all other billiard trajectories within Q , that share the same d − 1 caustics, are also closed. Moreover, all these closed trajectories have the same number of vertices.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Confocal Families of Quadrics and Billiards in Euclidean Space Generalized Cayley Condition The condition on a billiard trajectory inside ellipsoid Q 0 in E d , with nondegenerate caustics Q α 1 , . . . , Q α d − 1 , to be perodic with period n ≥ d is:   B n +1 B n . . . B d +1 . . . B n +2 B n +1 B d +2     rank . . . < n − d + 1 ,     . . .   B 2 n − 1 B 2 n − 2 . . . B n + d − 1 where � ( x − a 1 ) . . . ( x − a d )( x − α 1 )( x − α d − 1 ) = B 0 + B 1 x + B 2 x 2 + . . . and all a 1 , . . . , a d are distinct and positive.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem in Projective Space over an Arbitrary Field Reflection Law in Projective Space Let Q 1 and Q 2 be two quadrics that meet transversely. Denote by u the tangent plane to Q 1 at point x and by z the pole of u with respect to Q 2 . Suppose lines ℓ 1 and ℓ 2 intersect at x , and the plane containing these two lines meet u along ℓ . If lines ℓ 1 , ℓ 2 , xz , ℓ are coplanar and harmonically conjugated, we say that rays ℓ 1 and ℓ 2 obey the reflection law at the point x of the quadric Q 1 with respect to the confocal system which contains Q 1 and Q 2 . If we introduce a coordinate system in which quadrics Q 1 and Q 2 are confocal in the usual sense, reflection defined in this way is same as the standard one.

The Articles Outline Preliminaries Billiard Algebra on A ℓ Theorems of Poncelet Type Continued Fractions Approximation Poncelet Theorem in Projective Space over an Arbitrary Field One Reflection Theorem Suppose rays ℓ 1 and ℓ 2 obey the reflection law at x of Q 1 with respect to the confocal system determined by quadrics Q 1 and Q 2 . Let ℓ 1 intersects Q 2 at y ′ 1 and y 1 , u is a tangent plane to Q 1 at x , and z its pole with respect to Q 2 . Then lines y ′ 1 z and y 1 z respectively contain intersecting points y ′ 2 and y 2 of ray ℓ 2 with Q 2 . Converse is also true. Corollary Let rays ℓ 1 and ℓ 2 obey the reflection law of Q 1 with respect to the confocal system determined by quadrics Q 1 and Q 2 . Then ℓ 1 is tangent to Q 2 if and only if is tangent ℓ 2 to Q 2 ; ℓ 1 intersects Q 2 at two points if and only if ℓ 2 intersects Q 2 at two points.