THE DIFFIETY INSTITUTE SCHOOLS
1-st order PDE's with one unknown function
Valeriy A. YUMAGUZHIN
Exercises of the course at the Join Russian-Italian Diffiety School,
Pereslavl-Zalessky (Russia), August 17  -  30, 1999
  1. Prove that for any horizontal subspace H Ì TqJ0M (H is horizontal if p0 *|H: H® Tp(q)M is an isomorphism) there exists q1 Î J1M with kq1 = H.
  2. Prove that "q Î J1M  dU1|Cq is a nondegenerate form.
  3. Prove that a bilinear form (· , ·) is nondegenerate iff its matrix is nondegenerate.
  4. Prove that dimension of a vector space with a symplectic structure is even.
  5. Prove that dimension of a smooth manifold with a contact structure is odd.
  6. Let w be a contact form on a manifold M and f Î C¥M be a nowhere vanishing function. Prove that:
    1) f·w is a contact form,
    2) dw differs from d(f·w) by a nonzero factor on any hyperplane,
    kerw|x Ì TxM,  x Î M.
  7. Let (V2n,w) be a symplectic vector space and W Ì V be an isotropic subspace (W is isotropic if "v,w Î W   w(v,w) = 0). Prove that dimW £ n.
  8. Let N be an integral manifold of the Cartan distribution on J1M. Prove that dimN £ dimM.
  9. Let w be a symplectic form on a vector space V and W Ì V be a hyperplane. Prove that the skew-orthogonal complementation of W is 1-dimensional and lies in W.
  10. Find the expression of a contact transformation f: J1M® J1M in standard coordinates x1,¼,xn,u,p1,¼,pn    (n = 1,2).
  11. Let A: J0M® J0M be a point transformation defined in standard coordinates by
    ì
    í
    î
    X
    =
    X(x,y)
    Y
    =
    Y(x,y)
    Find explicit formulae defining the lift A(1).
  12. Check that the Legendre transformation
    ì
    ï
    ï
    í
    ï
    ï
    î
    Xi
    =
    pi
    U
    =
    n
    å
     i = 1 
    		 pi·xi - u
    Pi
    =
    xi
    is contact and it cannot be obtained by lifting of a point transformation.
  13. Check that the mapping D(J1M)®L (J1M), Y® dU1(Y,·) defines an isomorphism between vector fields that lie in the Cartan distribution and 1-forms vanishing on X1 = u.
  14. Find an explicit formula defining the Jacobi bracket of f,g Î C¥(J1M) in standard coordinates.
  15. Let E = {F(x,u,p) = 0} Ì J1M be a 1-st order PDE, let YF = XF-F·X1 be the characteristic vector field of E, and let At be its flow. Prove that At takes the Cartan distribution C(E) on E to itself.
  16. Let X = åni = 1ai(x,u)xi be a smooth vector field on J0M. Find an explicit formula defining the lift X(1) in standard coordinates.
  17. Let E = {F = åni = 1ai(x,u)pi-b(x,u) = 0} Ì J1M be a quasi-linear equation. Prove that YF |E = X(1)|E, where X = åni = 1aixi+bu.
  18. Solve the Cauchy problem for the equation
    x1·u·p1+x2·u·p2+x1·x2 = 0
    with the Cauchy data
    ì
    í
    î
    g
    =
    {(x1,x2)  |  x2 = (x1)2}
    j(x1)
    =
    (x1)3
  19. Prove the Jacobi identity for the Poisson bracket.
  20. Let M be a smooth n-dimensional manifold, f1,¼,fk Î C¥(M)  k < n, and let
    Mc = {  x Î M  |  f1(x) = c1,¼,fk(x) = ck  },   c = (c1,¼,ck) Î Rk.
    Assume that Mc is compact and "x Î Mc the 1-forms df1|x,¼,dfk|x are linear independent. Prove that there exists a neighborhood of Mc diffeomorphic to Mc´Bc, where Bc Ì Rk is an open ball with center at c.
  21. Let M2n be a symplectic manifold and f1,¼,f2n Î C¥(M2n). Prove that if the function f1,¼,f2n are functionally independent, then the 2n×2n-matrix of Poisson bracket ({fi,fj}) is nondegenerate.