Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

Define then the transformed jet ϕ ( ) (κ) Jx κ ∗ to be the κ-th jet of g at thepoint x ∗ := φ(x ∗ ), namely:(1.11) ϕ ( ()) (κ) Jx κ ∂ λ g j 1jm∗:=∂x i1 · · ·∂x i (x ∗)∈ J κ ∣λ n,m x∗.1i 1 ,...,i λ n, 0λκIt may be shown that this jet does not depend on the choice of a localgraph y = g(x) representing the κ-th jet Jx κ ∗at x ∗ . Furthermore, ifπ κ := Jn,m κ → Km denotes the canonical projection onto the first factor,the following diagram commutes:Jn,mκ ϕ (κ) Jn,mκπ κ π κ.K n+m ϕ K n+m1.12. Inductive formulas for the κ-th prolongation ϕ (κ) . To present them,we change our notations. Instead of (x, y), as coordinates in the target spaceK n × K m , we shall use capital letters:((1.13)X 1 , . . .,X n , Y 1 , . . .,Y m) .In the source space K n+m equipped with the coordinates (x, y), we use thejet coordinates (1.2) on the associated κ-th jet space. In the target spaceK n+m equipped with the coordinates (X, Y ), we use the coordinates((1.14) X i , Y j , Y j , Y j X i 1 X i 1X , . . ....,Y )ji 2 X i 1X i 2...X iκon the associated κ-th jet space; to avoid confusion with y i1 , y i1 ,i 2, . . . insubsequent formulas, we do not write Y i1 , Y i1 ,i 2, . . . . In these notations, thediffeomorphism ϕ whose first order approximation is close to the identitymapping in a neighborhood of x ∗ may be written under the form:(1.15) ϕ : ( ) (x i′ , y j′ ↦→ X i , Y j) ()= X i (x i′ , y j′ ), Y j (x i′ , y j′ ) ,for some C ∞ -smooth functions X i (x i′ , y j′ ), i = 1, . . .,n, and Y j (x i′ , y j′ ),j = 1, . . ., m. The first prolongation ϕ (1) of ϕ may be written under theform:(1.16) ( ) (( ))ϕ (1) : x i′ , y j′ , y j′i↦−→ X i (x i′ , y j′ ), Y j (x i′ , y j′ ), Y j x i′ , y j′ , y j′′1X i 1 i, ′1( )for some functions Y j x i′ , y j′ , y j′X i 1 iwhich depend on the pure first jet′1variables y j′iThe way how these functions depend on the first order partialderivatives functions X i , x i′ Xi , Y j , Y j and on the pure first jet1. ′ y j′ x i′ y j′307