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

320Define the differential operators(2.35) ( )∂ ∂F 2 := g 2 + g 1 f 2 ,∂g 1 ∂f 1(∂ ∂ ∂F 3 := g 2 + g 3 + g 1 f 2 + f 3∂g 1 ∂g 2 ∂f 1)∂,∂f 2· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(∂ ∂ ∂F λ := g 2 + g 3 + · · · + g λ + g 1 f 2∂g 1 ∂g 2 ∂g λ−1Then we have∂∂f 1+ f 3∂∂f 2+ · · · + f λ)∂.∂f λ−1(2.36)h 2 = F 2 (h 1 ),h 3 = F 3 (h 2 ),· · · · · · · · · · · · · · ·h λ = F λ (h λ−1 ).§3. SEVERAL INDEPENDENT VARIABLES AND ONE DEPENDENTVARIABLE3.1. Simplified adapted notations. As announced after the statement ofTheorem 2.24, it is only after we have treated the case of several independentvariables that we will understand perfectly the general formula (2.25), validin the case of one independent variable and one dependent variable. We willdiscover massive formal computations, exciting our computational intuition.Thus, assume n 1 and m = 1, let κ ∈ N with κ 1 and simply denote(instead of (1.2)) the jet variables by:(3.2)(x i , y, y i1 , y i1 ,i 2, . . .,y i1 ,i 2 ,...,i κ).Also, instead of (1.30), denote the κ-th prolongation of a vector field by:(3.3) ⎧L ⎪⎨(κ) =⎪⎩n∑i=1X i+ · · · +∂∂x + Y ∂ n∑i ∂y +n∑i 1 ,i 2 ,...,i κ=1i 1 =1Y i1 ,i 2 ,...,i κY i1∂∂y i1+∂∂y i1 ,i 2 ,...,i κ.n∑i 1 ,i 2 =1Y i1 ,i 2∂∂y i1 ,i 2+