A Very General Hamilton Jacobi Theorem

Transcription

1 University of Salerno 9th AIMS ICDSDEA Orlando (Florida) USA, July 1 5, 2012

2 Introduction: Generalized Hamilton-Jacobi Theorem Hamilton-Jacobi (HJ) Theorem: Consider a Hamiltonian system with Hamiltonian H = H(x, p). Then S = S(x) is a solution of the HJ equation iff for any solution x(t) of the ODE dx dt = H (x, ds(x)), p (x(t), ds(x(t))) is a solution of the Hamilton equations. (Cariñena et al., ): Our formulation of the Hamilton-Jacobi problem is based on the idea of obtaining solutions of a second order differential equations by lifting solutions of an adequate first order differential equation. The main characters are differential equations!

3 Introduction: Generalized Hamilton-Jacobi Theorem The HJ formalism is a cornerstone of the calculus of variations and the theory of Hamiltonian systems. Moreover, it is a first, important step through the quantization of a mechanical system. The aim of the talk is to present a higher derivative, field theoretic version of the generalized HJ theorem. This is achieved by recasting the latter within the geometric theory of PDEs.

4 Geometric Theory of PDEs Introduction: Generalized Hamilton-Jacobi Theorem (Q, L): a Lagrangian system Generalized HJ Theorem TQ Q T Q T Q 1 Every integral curve of is a solution (,W ) of the Euler Lagrange Eqs. iff TQ Q when Leg is invertible, = Leg 1 W and 2 i (, W ) ω = (, W ) de (GHJ) with im(, W ) graph(leg) (, W ) ω = 0 and (, W ) E = 0 are locally equivalent to: W = ds and S is a solution of the standard Hamilton-Jacobi Equation! See also (V 2012), (de León, Marrero, Martín de Diego, Vaquero 2012).

5 Outline 1 Geometric Theory of PDEs 2 3

6 Outline 1 Geometric Theory of PDEs 2 3

7 Diffieties A PDE is represented by a geometric object called a diffiety. A PDE F (x,..., u I,...) = 0, I q may be understood as a submanifold E 0 in a jet manifold J q := {(x,..., u I,...) : I q}. E 0 can be prolonged to a submanifold E J E : D J F (x,..., u I,...) = 0, J 0. E is naturally endowed with the involutive distribution C generated by total derivatives D i := x i + u Ii u I. Solutions of E 0 identify with integral manifolds of (E, C ), which is the diffiety representing E.

8 Hamilton-Jacobi Subdiffieties Remark: Diffieties are generically -dimensional. The Frobenius Theorem fails for -dimensional manifolds! Compatible equations of the form v = f(x, v) x are represented by finite-dimensional diffieties, and vice-versa. Definition: An HJ subdiffiety Y of E is a finite dimensional one If the vector field is a solution of the GHJ Equation, then it is represented by an HJ subdiffiety of the EL Equations! The condition Y E can be interpreted as a GHJ Equation for E

9 Examples Consider the Burgers equation u t = u xx + uu x (B) Equation { ut = A(t, x, u) u x = B(t, x, u) is compatible and contained into Eq.(B) iff { Bt A x + AB u BA u = 0 A = B x BB u ub. For instance, A = u 2 /(x x 0 ) and B = u/(x x 0 ). Corresponding (local) solutions of Eq.(B) are u = x x 0 t t 0.

10 Examples Consider the KdV equation u t = 6uu x u xxx (KdV) Equation u tt = A(t, x, u, u t, u x ) u xt = B(t, x, u, u t, u x ) u xx = C(t, x, u, u t, u x ) is contained into Eq. (KdV) iff For instance, A = u2 t A x + u x A u + AA ux + CA ut + u t 6uu x = 0. (ut 6uux )2 ut(ut 6uux )2 12u, B = x 4 12ux 3 Corresponding (local) solutions of Eq. (KdV) are and C = u = 2 1/3 (2 1/3 (x ct c 0 ); 0, c 1 ) + c/6. (ut 6uux )2 12u. x 2

11 Outline 1 Geometric Theory of PDEs 2 3

12 Geometric Structures α : P M: a fiber bundle = J 1 P: 1st jet bundle it is an affine bundle modelled over V := T M VP Definition (multimomentum bundle J α P) Sections of J α P = linear morphisms V n T M. coordinates (x i, y) on P = coordinates (x i, y, p i ) on J α π : E M: a fiber bundle = π k : J k M: kth jet bundle coor. (x i, u) on E coor. (x i, u I ) on J k coor. (x i, u I, p I.i ) on J π k. Mechanics Field Theory Q R J k TQ R J k+1 T Q R J π k TQ Q T Q R J k+1 J k J π k

13 Ingredients and Constructions S = L : a (k + 1)th derivative Lagrangian field theory on π: L = L[x, u]d n x, L[x, u] C (J k+1 ) determines a (dynamical) Hamiltonian n + 1-form on J k+1 J k J π k : Ω = dp I.i du I d n 1 x i de d n x, E = p I.i u Ii L[x, u] Hamilton-like equations: i σ Ω σ = 0, σ : M J k+1 J k J π k (HEL) Solutions of (HEL) can only be found in the Legendre subbundle: P := {δ I Ki pk.i = L/ u I : I = k + 1} J k+1 J k J π k Mechanics Field Theory TQ Q T Q R J k+1 J k J π k ω de dt Ω i σ ω σ = de σ i σ Ω σ = 0 graph(leg) P

14 Outline 1 Geometric Theory of PDEs 2 3

15 Geometric Theory of PDEs (π, L ): a Lagrangian field theory Theorem [LV]: Let im be compatible. Then J k+1 J k J π k J k+1 (,W ) J π k J k 1 every solution of im is a solution of the Euler Lagrange Eqs. iff 2 i (, W ) Ω = 0 (GHJ) for some W such that im(, W ) P Remark: Condition 1. is the same as im is represented by a HJ subdiffiety Y of the EL Eqs. E EL Remark: Sol(E EL ) carries a (pre)symplectic structure One can search for Y such that Sol(Y ) Sol(E EL ) is a isotropic submanifold! This is problematic for n > 1.

16 An Example The Lagrangian field theory Lagrangian Field Equations { 1 2 (u2 x + vx 2 + vu t uv t + u 2 ) + 1 vt = u + u 2 + u xx 3 u3 u t = v xx For = (du Adx Bdt) u + (dv Cdx Ddt) v : { D x A + u 2 + u = 0 (GHJ) B + x C = 0 1 A solution is: A = 3 u3 + 1 c2 2 u 2, B = ca, C = cu, D = c 2 u corresponding to travelling wave solutions of the field equations [ ] u = 3(1 c2 ) 2 sech 2 1 c 2 8 (x x 0 ct)

17 A Bibliographic Reference L. V., Hamilton-Jacobi Diffieties, J. Geom. Phys., 61 (2011) ; e-print: arxiv: