1. Sophies world: geometry and integrabilityVladimir Dragovi cMathematical Institute SANU, Belgrade Math. Phys. Group, University of LisbonSSSCP2009Ni, 4 April 2009s

2. Contents1 Sophies world 2 Billiard within ellipse 3 Poncelet theorem 4 Generalization of the Darboux theorem 5 Higher-dimensional generalization of the Darboux theorem 6 A new view on the Kowalevski top 3. Sophies world , 18501891 4. Sophies world: the story of the history of integrabilityActa Mathematica, 1889 5. Addition Theoremssin(x + y ) = sin x cos y + cos x sin y cos(x + y ) = cos x cos y sin x sin ysn x cn y dn y + sn y cn x dn x sn (x + y ) = 1 k 2 sn 2 x sn 2 y cn x cn y sn x sn y dn x dn y cn (x + y ) = 1 k 2 sn 2 x sn 2 y 21 (x) (y )(x + y ) = (x) (y ) +4(x) (y ) 6. The Quantum Yang-Baxter Equation 23 R 12 (t1 t2 , h)R 13 (t1 , h)R(t2 , h) = R 23 (t2 , h)(R 13 (t1 , h)R 12 (t1 t2 , h) R ij (t, h) : V V V V V Vt spectral parameter h Planck constant 7. The Euler-Chasles correspondenceE : 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 equationdx dy =0p4 (x) p4 (y ) Poncelet theorem for triangles 9. Elliptical Billiard 10. Billiard within ellipseA trajectory of a billirad within an ellipse is a polygonal line withvertices on the ellipse, such that successive edges satisfy the billiardreection law: the edges form equal angles with the to the ellipse atthe common vertex. 11. Focal property of ellipses 12. Focal property of ellipses 13. Focal property of ellipses 14. Caustics of billiard trajectories 15. Caustics of billiard trajectories 16. Poncelet theorem (Jean Victor Poncelet, 1813.)Let C and D be two given conics in the plane. Suppose there existsa closed polygonal line inscribed in C and circumscribed about D.Then, there are innitely many such polygonal lines and all of themhave the same number of edges. Moreover, every point of the conicC is a vertex of one of these lines. 17. Mechanical interpretation of the Poncelet theoremLet us consider closed trajectory of billiard system within ellipse E.Then every billiard trajectory within E, which has the same causticas the given closed one, is also closed. Moreover, all thesetrajectories are closed with the same number of reections at E. 18. j jj 19. Generalization of the Darboux theoremheoremLet E be an ellipse in E2 and (am )mZ , (bm )mZ be two sequencesof the segments of billiard trajectories E, sharing the same caustic.Then all the points am bm (m Z) belong to one conic K,confocal with E. 20. 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. 21. For a hyperbola as a caustic, it holds: if segments am , bm intersect the long axis of E in the same direction, then K is a hyperbola, otherwise it is an ellipse. 22. Grids in arbitrary dimension heoremLet (am )mZ , (bm )mZ be two sequences of the segments of billiardtrajectories within the ellipsoid E in Ed , sharing the same d 1caustics. Suppose the pair (a0 , b0 ) is s-skew, and that by thesequence of reections on quadrics Q1 , . . . , Qs+1 the minimalbilliard trajectory connecting a0 to b0 is realized.Then, each pair (am , bm ) is s-skew, and the minimal billiardtrajectory connecting these two lines is determined by the sequenceof reections on the same quadrics Q1 , . . . , Qs+1 . 23. andinsky, Grid 1923. 24. Pencil of conics Two conics and tangential pencil2 2 2C1 : a0 w1 + a2 w2 + a4 w3 + 2a3 w2 w3 + 2a5 w1 w3 + 2a1 w1 w2 = 02C2 : w2 4w1 w3 = 0 Coordinate pencil F (s, z1 , z2 , z3 ) := det M(s, z1 , z2 , z3 ) = 0 0z1z2 z3 z1a0 a1 a5 2s M(s, z1 , z2 , z3 ) = z2a1a2 + s a3 z3 a5 2s a3a4F := H + Ks + Ls 2 = 0 25. Darboux coordinatestC2 (0 ) : z1 2 + z2 0 + z3 = 0 0 z1 2 + 22 + z3 = 0 zx1 + x2 z1 = 1, z2 = , z3 = x1 x2 2F (s, x1 , x2 ) = L(x1 , x2 )s 2 + K (x1 , x2 )s + H(x1 , x2 )2 2 2H(x1 , x2 ) = (a1 a0 a2 )x1 x2 + (a0 a3 a5 a1 )x1 x2 (x1 + x2 ) 2 22 1 2 + (a5 a0 a4 )(x1 + x2 ) + (2(a5 a2 a1 a3 ) + (a5 a0 a4 )x1 x22 2 + (a1 a4 a3 a5 ))(x1 + x2 ) + a3 a2 a42 222K (x1 , x2 ) = a0 x1 x2 + 2a1 x1 x2 (x1 + x2 ) a5 (x1 + x2 ) 4a2 x1 x2+ 2a3 (x1 + x2 ) a4L(x1 , x2 ) = (x1 x2 )2 26. heorem (i) There exists a polynomial P = P(x) such that the discriminant of the polynomial F in s as a polynomial in variables x1 and x2 separates the variables: Ds (F )(x1 , x2 ) = P(x1 )P(x2 ). (1)(ii) There exists a polynomial J = J(s) such that the discriminantof the polynomial F in x2 as a polynomial in variables x1 and sseparates the variables:Dx2 (F )(s, x1 ) = J(s)P(x1 ). (2)Due to the symmetry between x1 and x2 the last statement remains valid after exchanging the places of x1 and x2 . 27. 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 P1 and P2 of the fourth degree such that B(y , z)2 A(y , z)C (y , z) = P1 (y )P2 (z),(3)then there exists a polynomial f such thatDy S(x, z) = f (x)P2 (z),Dz S(x, y ) = f (x)P1 (y ). 28. Discriminantly separable polynoiamls denitionFor a polynomial F (x1 , . . . , xn ) we say that it is discriminantlyseparable if there exist polynomials fi (xi ) such that for everyi = 1, . . . , nDxi F (x1 , . . . , xi , . . . , xn ) = fj (xj ). j=i It is symmetrically discriminantly separable if f2 = f3 = = fn , while it is strongly discriminatly separable if f1 = f2 = f3 = = fn . It is weakly discriminantly separable if there exist polynomials fi j (xi )such that for every i = 1, . . . , nDxi F (x1 , . . . , xi , . . . , xn ) = fji (xj ). j=i 29. heorem Given a polynomial F (s, x1 , x2 ) of the second degree in each of the variables s, x1 , x2 of the formF = s 2 A(x1 , x2 ) + 2B(x1 , x2 )s + C (x1 , x2 ).Denote by TB 2 AC a 5 5 matrix such that5 5 ij ij1 (B 2 AC )(x1 , x2 ) =TB 2 AC x11 x2 . j=1 i =1Then, polynomial F is discriminantly separable if and only ifrank TB 2 AC = 1. 30. Geometric interpretation of the Kowalevski fundamental equation Q(w , x1 , x2 ) := (x1 x2 )2 w 2 2R(x1 , x2 )w R1 (x1 , x2 ) = 0R(x1 , x2 ) = x1 x2 + 61 x1 x2 + 2c(x1 + x2 ) + c 2 k 2 2 2R1 (x1 , x2 ) = 61 x1 x2 (c 2 k 2 )(x1 + x2 )2 4cx1 x2 (x1 + x2 )2 2+ 61 (c 2 k 2 ) 4c 2 2a0 = 2a1 = 0a5 = 0a2 = 31 a3 = 2ca4 = 2(c 2 k 2 ) 31. Geometric interpretation of the Kowalevski fundamental equationhremThe Kowalevski fundamental equation represents a point pencil ofconics given by their tangential equations C1 : 2w1 + 3l1 w2 + 2(c 2 k 2 )w3 4clw2 w3 = 0;22 22 C2 : w2 4w1 w3 = 0. The Kowalevski variables w , x1 , x2 in this geometric settings are thepencil parameter, and the Darboux coordinates with respect to theconic C2 respectively. 32. Multi-valued Buchstaber-Novikov groupsn-valued group on Xm : X X (X )n ,m(x, y ) = x y = [z1 , . . . , zn ](X )n symmetric n-th power of X AssociativityEquality of two n2 -sets:[x (y z)1 , . . . , x (y z)n ] [(x y )1 z, . . . , (x y )n z] for every triplet (x, y , z) X 3 . Unity ee x = x e = [x, . . . , x] for each x X . Inverse inv : X Xe inv(x) x, e x inv(x) for each x X . 33. Multi-valued Buchstaber-Novikov groups Action of n-valued group X on the set Y : X Y (Y )n (x, y ) = x y Two n2 -multi-subsets in Y : x1 (x2 y ) (x1 x2 ) y are equal for every triplet x1 , x2 X , y Y .Additionally , we assume:e y = [y , . . . , y ] for each y Y . 34. Two-valued group on CP1The equation of a pencilF (s, x1 , x2 ) = 0 Isomorphic elliptic curves1 : y 2 = P(x)deg P = 4 2 : t 2 = J(s) deg J = 3 Canonical equation of the curve 22 : t 2 = J (s) = 4s 3 g2 s g3 Birational morphism of curves : 2 1 Induced by fractional-linear mapping which maps zeros of thepolynomial J and to the four zeros of the polynomial P. 35. Double-valued group on CP1 There is a group structure on the cubic 2 . Together with itssoubgroup Z2 , it denes the standrad double-valued groupstructure on CP1 :22 t1 t2t1 + t2s1 c s2 = s1 s2 + , s1 s2 + ,2(s1 s2 )2(s1 s2 ) where ti = J (si ), i = 1, 2. heoremThe general pencil equation after fractional-linear transformations F (s, 1 (x1 ), 1 (x2 )) = 0 denes the double valued coset group structure (2 , Z2 ). 36. i1 i1 m1 1 id C4 - 1 1 C- 2 2 C QQ ia ia mi1 i1 ididQ p1 p1 id Q?Q sQ ? 1 1 C C CP1 Cp1 p1 id 1 2 1 1 id ??+CCCP1 C m2 mc c?? 2 f 2 CPCP C/ 37. Double-valued group CP1hremAssociativity conditions for the group structure of thedouble-valued coset group (2 , Z2 ) and for its action on 1 areequivalent to the great Poncelet theorem for a triangle. 38. References 1. V. Dragovi, Geometrization and Generalization of the Kocwalevski top 2. V. Dragovi, Marden theorem and Poncelet-Darboux curvescarXiv:0812.48290 (2008) 3. V. Dragovi, M. Radnovi, Hyperelliptic Jacobians as Billiardc cAlgebra of Pencils of Quadrics: Beyond Poncelet Porisms, Adv.Math., 219 (2008) 1577-1607. 4. V. Dragovi, M. Radnovi, Geometry of integrable billiards andc cpencils of quadrics, Journal Math. Pures Appl. 85 (2006),758-790. 5. V. Dragovi, B. Gaji, Systems of Hess-Appelrot type Comm.c cMath. Phys. 265 (2006) 397-435 6. V. Dragovi, B. Gaji, Lagrange bitop on so(4) so(4) andc cgeometry of Prym varieties Amer. J. Math. 126 (2004),981-1004 7. V. I. Dragovich, Solutions of the Yang equation with rationalirreducible spectral curves, Izv. Ross. Akad. Nauk Ser. 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