First order PDE's with one unknown function

Valeriy A. YUMAGUZHIN

Exercises and problems of the Joint Russian - Italian School
Pereslavl', 17 August - 30 August 1999

1. Prove that for any horizontal subspace H Ã T_qJ⁰M (H is horizontal if p_0 *|_H: HÆ T_p(q)M is an isomorphism) there exists q₁ ‘ J¹M with k_q1 = H.
2. Prove that "_{q ‘ J¹M}  dU₁|_{C q} 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 ‘ 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 à T_xM,  x ‘ M.
7. Let (V²ⁿ,w) be a symplectic vector space and W Ã V be an isotropic subspace (W is isotropic if "_{v,w ‘ 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 Ã 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 Æ J¹M in standard coordinates x¹,º,xⁿ,u,p₁,º,p_n    (n = 1,2).
11. Let A: J⁰MÆ J⁰M be a point transformation defined in standard coordinates by

    Ï Ì Ó X = X(x,y) Y = Y(x,y)

    Find explicit formulae defining the lift A⁽¹⁾.
12. Check that the Legendre transformation

    < Table align=left>
    	Ï Ô Ô Ì Ô Ô Ó		Xi = pi U = n  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)ÆL (J¹M), YÆ 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 ‘ C^€(J¹M) in standard coordinates.
15. Let E = {F(x,u,p) = 0} Ã J¹M be a 1-st order PDE, let Y_F = X_F-F·X₁ be the characteristic vector field of E, and let A_t be its flow. Prove that A_t takes the Cartan distribution C(E) on E to itself.
16. Let X = Âⁿ_{i = 1}aⁱ(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 = Âⁿ_{i = 1}aⁱ(x,u)p_i-b(x,u) = 0} Ã J¹M be a quasi-linear equation. Prove that Y_F |_E = X⁽¹⁾|_E, where X = Âⁿ_{i = 1}aⁱ�xⁱ+b�u.
18. Solve the Cauchy problem for the equation

    x1·u·p1+x2·u·p2+x1·x2 = 0

    with the Cauchy data

    Ï Ì Ó 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,º,f_k ‘ C^€(M)  k < n, and let
    M_c = {  x ‘ M  |  f₁(x) = c₁,º,f_k(x) = c_k  },   c = (c₁,º,c_k) ‘ R^k.
    Assume that M_c is compact and "_{x ‘ Mc} the 1-forms df₁|_x,º,df_k|_x are linear independent. Prove that there exists a neighborhood of M_c diffeomorphic to M_c¥B_c, where B_c à R^k is an open ball with center at c.
21. Let M²ⁿ be a symplectic manifold and f₁,º,f_2n ‘ C^€(M²ⁿ). Prove that if the function f₁,º,f_2n are functionally independent, then the 2n×2n-matrix of Poisson bracket ({f_i,f_j}) is nondegenerate.