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

238We denote by Jn,m κ the space of κ-th jets of maps K n ∋ x ↦→ y(x) ∈ K m .Let(1.31)(x i , y j , y j i 1, y j i 1 ,i 2, . . .. . .,y j i 1 ,i 2 ,...,i κ)∈ Kn+m+mn+mn 2 +···+mn κ ,denote the natural coordinates on Jn,m κ ≃ Kn+m(1+n+···+nκ) . For instance,(x, y, y 1 ) ∈ J1,1 1 . We shall sometimes write them shortly:(1.32)(x i , y j , y j )β ∈ Kn+m+m(n+···+n κ) ,where β ∈ N n varies and satisfies |β| κ. Sometimes also, weconsider these jet coordinates only up to their symmetries y j i 1 ,i 2 ,...,i λ=y j i σ(1) ,i σ(2) ,...,i σ(λ), where σ is a permutation of {1, 2, . . ., λ}, so that J κ n,m ≃K n+m Cκ n+κ , with Cκn+κ := (n+κ)! .κ! n!Having these notations at hand, we may develope the canonical system ofcontact forms on Jn,m κ ([Ol1995], [Stk2000]):⎧n∑θ j := dy j − y j k dxk ,(1.33)⎪⎨θ j i 1:= dy j i 1−k=1n∑y j i 1 ,k dxk ,k=1· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·n∑⎪⎩ θ j i 1 ,...,i κ−1:= dy j i 1 ,...,i κ−1− y j i 1 ,...,i κ−1 ,k dxk .For instance, with n = m = 1 and κ = 2, we have θ 1 = dy −y 1 dx and θ1 1 =dy 1 − y 2 dx. These (linearly independent) one-forms generate a subspaceC T κ n,m of the cotangent T ∗ Jn,m κ whose dimension equals m Cκ−1 n+κ−1. Forthe duality between forms and vectors, the orthogonal (C T κ n,m )⊥ in TJn,mκis spanned by the n + m Cn+κ−1 κ vector fields:(1.34) ⎧⎪⎨⎪⎩D i := ∂∂x i + m∑T j 1i 1 ,...,i κ:=j 1 =1∂∂y j ,1i 1 ,...,i κy j 1ik=1∂∂y + · · · + ∑ m j 1n∑j 1 =1 k 1 ,...,k κ−1 =1y j 1i,k 1 ,...,k κ−1∂∂y j ,1k 1 ,...,k κ−1the first n ones being the total differentiation operators, considered in Part II.For n = m = 1, κ = 2, we get ∂ + y ∂x 1 ∂ + y ∂y 2 ∂∂y 1and ∂∂y 2.Classically ([Ol1986, BK1989, Ol1995]), one associates to (E ) its skeleton∆ E , namely the (n + m + p)-dimensional submanifold of Jn,m κ+1 simply