Integrable billiards, Poncelet-Darboux grids and Kowalevski top

Transcription

1 Integrable billiards, Poncelet-Darboux grids and Kowalevski top Vladimir Dragović Math. Phys. Group, University of Lisbon Lisbon-Austin conference University of Texas, Austin, 3 April 2010

2 Contents 1 Sophie s world 2 Billiard within ellipse 3 Generalization of the Darboux theorem 4 Higher-dimensional generalization of the Darboux theorem 5 A new view on the Kowalevski top 6 Kowalevski top and Buchstaber-Novikov multi-valued groups

3 Sophie s world Софья Васильевна Ковалевская,

4 Sophie s world: the story of the history of integrability Acta Mathematica, 1889

5 Addition Theorems sin(x + y) = sinx cos y + cos x sin y cos(x + y) = cos x cos y sin x sin y sn(x + y) = cn (x + y) = snx cny dny + sny cn x dnx 1 k 2 sn 2 x sn 2 y cnx cn y snx sny dnx dn y 1 k 2 sn 2 x sn 2 y (x + y) = (x) (y) ( (x) ) (y) 2 (x) (y)

6 The Quantum Yang-Baxter Equation R 12 (t 1 t 2,h)R 13 (t 1,h)R 23 (t 2,h) = R 23 (t 2,h)(R 13 (t 1,h)R 12 (t 1 t 2,h) R ij (t,h) : V V V V V V t spectral parameter h Planck constant

7 The Euler-Chasles correspondence E : ax 2 y 2 + b(x 2 y + xy 2 ) + c(x 2 + y 2 ) + 2dxy + e(x + y) + f = 0

8 The Euler-Chasles correspondence The Euler equation dx p4 (x) ± dy p4 (y) = 0 Poncelet theorem for triangles

9 Elliptical Billiard

10 Billiard within ellipse A trajectory of a billirad within an ellipse is a polygonal line with vertices on the ellipse, such that successive edges satisfy the billiard reflection law: the edges form equal angles with the to the ellipse at the common vertex.

11 Focal property of ellipses

12 Focal property of ellipses

13 Caustics of billiard trajectories

14 Poncelet theorem (Jean Victor Poncelet, 1813.) Let C and D be two given conics in the plane. Suppose there exists a closed polygonal line inscribed in C and circumscribed about D. Then, there are infinitely many such polygonal lines and all of them have the same number of edges. Moreover, every point of the conic C is a vertex of one of these lines.

15 Mechanical interpretation of the Poncelet theorem Let us consider closed trajectory of billiard system within ellipse E. Then every billiard trajectory within E, which has the same caustic as the given closed one, is also closed. Moreover, all these trajectories are closed with the same number of reflections at E.

16 Nonperiodic billiard trajectories

17 Generalization of the Darboux theorem Тheorem [V. D., M. Radnović (2008)] Let E be an ellipse in E 2 and (a m ) m Z, (b m ) m Z be two sequences of the segments of billiard trajectories E, sharing the same caustic. Then all the points a m b m (m Z) belong to one conic K, confocal with E.

18 Moreover, under the additional assumption that the caustic is an ellipse, we have: if both trajectories are winding in the same direction about the caustic, then K is also an ellipse; if the trajectories are winding in opposite directions, then K is a hyperbola.

19 For a hyperbola as a caustic, it holds: if segments a m, b m intersect the long axis of E in the same direction, then K is a hyperbola, otherwise it is an ellipse.

20 Grids in arbitrary dimension Тheorem [V. D., M. Radnović (2008)] Let (a m ) m Z, (b m ) m Z be two sequences of the segments of billiard trajectories within the ellipsoid E in E d, sharing the same d 1 caustics. Suppose the pair (a 0,b 0 ) is s-skew, and that by the sequence of reflections on quadrics Q 1,..., Q s+1 the minimal billiard trajectory connecting a 0 to b 0 is realized. Then, each pair (a m,b m ) is s-skew, and the minimal billiard trajectory connecting these two lines is determined by the sequence of reflections on the same quadrics Q 1,...,Q s+1.

21 Кandinsky, Grid 1923.

22 Pencil of conics Two conics and tangential pencil C 1 : a 0 w a 2w a 4w a 3w 2 w 3 + 2a 5 w 1 w 3 + 2a 1 w 1 w 2 = 0 C 2 : w 2 2 4w 1w 3 = 0 Coordinate pencil F(s,z 1,z 2,z 3 ) := det M(s,z 1,z 2,z 3 ) = 0 M(s,z 1,z 2,z 3 ) = F := H + Ks + Ls 2 = 0 0 z 1 z 2 z 3 z 1 a 0 a 1 a 5 2s z 2 a 1 a 2 + s a 3 z 3 a 5 2s a 3 a 4

23 Darboux coordinates t C2 (l 0 ) : z 1 l 2 0 2z 2 l 0 + z 3 = 0 ẑ 1 l 2 2ẑ 2 l + ẑ 3 = 0 ẑ 1 = 1, ẑ 2 = x 1 + x 2, ẑ 3 = x 1 x 2 2 F(s, x 1,x 2 ) = L(x 1,x 2 )s 2 + K(x 1,x 2 )s + H(x 1,x 2 ) H(x 1, x 2 ) = (a 2 1 a 0a 2 )x 2 1 x2 2 + (a 0a 3 a 5 a 1 )x 1 x 2 (x 1 + x 2 ) + (a5 2 a 0 a 4 )(x1 2 + x2) 2 + (2(a 5 a 2 a 1 a 3 ) (a2 5 a 0 a 4 )x 1 x 2 + (a 1 a 4 a 3 a 5 ))(x 1 + x 2 ) + a3 2 a 2a 4 K(x 1, x 2 ) = a 0 x1 2 x a 1x 1 x 2 (x 1 + x 2 ) a 5 (x1 2 + x2 2 ) 4a 2x 1 x 2 + 2a 3 (x 1 + x 2 ) a 4 L(x 1, x 2 ) = (x 1 x 2 ) 2

24 Тheorem [V. D. (2009)] (i) There exists a polynomial P = P(x) such that the discriminant of the polynomial F in s as a polynomial in variables x 1 and x 2 separates the variables: D s (F)(x 1,x 2 ) = P(x 1 )P(x 2 ). (1) (ii) There exists a polynomial J = J(s) such that the discriminant of the polynomial F in x 2 as a polynomial in variables x 1 and s separates the variables: D x2 (F)(s,x 1 ) = J(s)P(x 1 ). (2) Due to the symmetry between x 1 and x 2 the last statement remains valid after exchanging the places of x 1 and x 2.

25 Lemma Given a polynomial S = S(x,y,z) of the second degree in each of its variables in the form: S(x,y,z) = A(y,z)x 2 + 2B(y,z)x + C(y,z). If there are polynomials P 1 and P 2 of the fourth degree such that B(y,z) 2 A(y,z)C(y,z) = P 1 (y)p 2 (z), (3) then there exists a polynomial f such that D y S(x,z) = f (x)p 2 (z), D z S(x,y) = f (x)p 1 (y).

26 Gauge equivalence Gauge transformations x a 1x + b 1 c 1 x + d 1 y a 2y + b 2 c 2 y + d 2 z a 3z + b 3 c 3 z + d 3

27 Discriminantly separable polynomials definition For a polynomial F(x 1,...,x n ) we say that it is discriminantly separable if there exist polynomials f i (x i ) such that for every i = 1,...,n D xi F(x 1,..., ˆx i,...,x n ) = j i f j (x j ). It is symmetrically discriminantly separable if f 2 = f 3 = = f n, while it is strongly discriminatly separable if f 1 = f 2 = f 3 = = f n. It is weakly discriminantly separable if there exist polynomials f j i (x i) such that for every i = 1,...,n D xi F(x 1,..., ˆx i,...,x n ) = j i f i j (x j ).

28 Тheorem [V. D. (2009)] Given a polynomial F(s,x 1,x 2 ) of the second degree in each of the variables s,x 1,x 2 of the form F = s 2 A(x 1,x 2 ) + 2B(x 1,x 2 )s + C(x 1,x 2 ). Denote by T B 2 AC a 5 5 matrix such that (B 2 AC)(x 1,x 2 ) = 5 5 j=1 i=1 T ij B 2 AC xi 1 1 x j 1 2. Then, polynomial F is discriminantly separable if and only if rank T B 2 AC = 1.

29 Geometric interpretation of the Kowalevski fundamental equation Q(w,x 1,x 2 ) := (x 1 x 2 ) 2 w 2 2R(x 1,x 2 )w R 1 (x 1,x 2 ) = 0 R(x 1,x 2 ) = x 2 1x l 1 x 1 x 2 + 2lc(x 1 + x 2 ) + c 2 k 2 R 1 (x 1,x 2 ) = 6l 1 x 2 1x 2 2 (c 2 k 2 )(x 1 + x 2 ) 2 4clx 1 x 2 (x 1 + x 2 ) + 6l 1 (c 2 k 2 ) 4c 2 l 2 a 0 = 2 a 1 = 0 a 5 = 0 a 2 = 3l 1 a 3 = 2cl a 4 = 2(c 2 k 2 )

30 Geometric interpretation of the Kowalevski fundamental equation Тhеоrem [V. D. (2009)] The Kowalevski fundamental equation represents a point pencil of conics given by their tangential equations Ĉ 1 : 2w l 1w (c2 k 2 )w 2 3 4clw 2w 3 = 0; C 2 : w 2 2 4w 1w 3 = 0. The Kowalevski variables w,x 1,x 2 in this geometric settings are the pencil parameter, and the Darboux coordinates with respect to the conic C 2 respectively.

31 Multi-valued Buchstaber-Novikov groups n-valued group on X m : X X (X) n, m(x,y) = x y = [z 1,...,z n ] (X) n symmetric n-th power of X Associativity Equality of two n 2 -sets: [x (y z) 1,...,x (y z) n ] и [(x y) 1 z,...,(x y) n z] for every triplet (x,y,z) X 3. Unity e e x = x e = [x,...,x] for each x X. Inverse inv : X X e inv(x) x, e x inv(x) for each x X.

32 Multi-valued Buchstaber-Novikov groups Action of n-valued group X on the set Y Two n 2 -multi-subsets in Y : φ : X Y (Y ) n φ(x,y) = x y x 1 (x 2 y) и (x 1 x 2 ) y are equal for every triplet x 1,x 2 X, y Y. Additionally, we assume: for each y Y. e y = [y,...,y]

33 Two-valued group on CP 1 The equation of a pencil F(s,x 1,x 2 ) = 0 Isomorphic elliptic curves Γ 1 : y 2 = P(x) deg P = 4 Γ 2 : t 2 = J(s) deg J = 3 Canonical equation of the curve Γ 2 Γ 2 : t 2 = J (s) = 4s 3 g 2 s g 3 Birational morphism of curves ψ : Γ 2 Γ 1 Induced by fractional-linear mapping ˆψ which maps zeros of the polynomial J and to the four zeros of the polynomial P.

34 Two-valued group on CP 1 There is a group structure on the cubic Γ 2. Together with its subgroup Z 2, it defines the standrad two-valued group structure on CP 1 : [ ( ) t1 t 2 ( ) ] 2 t1 + t 2 2 s 1 c s 2 = s 1 s 2 +, s 1 s 2 +, 2(s 1 s 2 ) 2(s 1 s 2 ) where t i = J (s i ), i = 1,2. Тheorem [V. D. (2009)] The general pencil equation after fractional-linear transformations F(s, ˆψ 1 (x 1 ), ˆψ 1 (x 2 )) = 0 defines the two valued coset group structure (Γ 2, Z 2 ).

35 i Γ1 i Γ1 id id Γ 1 Γ 1 C C ϕ 1 ϕ 2 C C i Γ1 i Γ1 m CP 1 C CP 1 C ψ 1 ψ 1 id C 4 Γ 1 Γ 1 C Γ 2 Γ 2 C i a i a m p 1 p 1 id ˆψ 1 ˆψ 1 id p 1 p 1 id m 2 CP 2 f m c τ c CP 2 C/

36 Two-valued group CP 1 Тhеоrem [V. D. (2009)] Associativity conditions for the group structure of the two-valued coset group (Γ 2, Z 2 ) and for its action on Γ 1 are equivalent to the great Poncelet theorem for a triangle.

37 Experimental Math Other n-valued groups p 3 = s s 3 Dp 3 = y 2 x 2 (x y) 2. p 4 = s s 2 1s s s 1 s 3 Dp 4 = y 3 x 3 (x y) 2 (y + 4x) 2 (4y + x) 2. p 5 = s s 2 1s s 2 s 3 Dp 5 = y 4 x 4 (x y) 4 ( y 2 11xy + x 2 ) 2 ( y xy + x 2 ) 2.

38 References 1. V. Dragović, Geometrization and Generalization of the Ko walevski top, arxiv: , to appear in Comm. Math. Phys. 2. V. Dragović, M. Radnović Integrable Billiards and Quadrics Russian Math. Surveys, 2010, Vol. 65, no. 2 p V. Dragović, Marden theorem and Poncelet-Darboux curves arxiv: (2008) 4. V. Dragović, Multi-valued hyperelliptic continued fractions of generalized Halphen type, IMRN (2009) arxiv: V. Dragović, M. Radnović, Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms, Advances in Mathematics, 219 (2008) V. Dragović, M. Radnović, Geometry of integrable billiards and pencils of quadrics, Journal de Mathématiques Pures et Appliquées 85 (2006)