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

3101.25. Prolongation of a vector field to the κ-th jet space. Consider a vectorfield(1.26) L =n∑i=1X i (x, y) ∂∂x i + m∑j=1Y j (x, y) ∂∂y j ,defined in K n+m . Its flow:(1.27) ϕ t (x, y) := exp (tL )(x, y)constitutes a one-parameter family of diffeomorphisms of K n+m close tothe identity. The lift (ϕ t ) (κ) to the κ-th jet space constitutes a one-parameterfamily of diffeomorphisms of J κ n,m. By definition, the κ-th prolongationL (κ) of L to the jet space J κ n,m is the infinitesimal generator of (ϕ t) (κ) ,namely:(1.28) L (κ) := d [ dt∣ (ϕt ) (κ)] .t=01.29. Inductive formulas for the κ-th prolongation L (κ) . As a vector fielddefined in K n+m+m(n+m)! n! m! , the κ-th prolongation L (κ) may be written underthe general form:(1.30) ⎧ n∑L (κ) = X i ∂∂x + ∑ mY j ∂i ∂y + j⎪⎨⎪⎩i=1++m∑j=1m∑j=1n∑i 1 =1n∑Y j i 1i 1 ,...,i κ=1j=1∂∂y j i 1+Y j i 1 ,...,i κm∑j=1n∑i 1 ,i 2 =1∂∂y j i 1 ,...,i κ.Y j i 1 ,i 2∂∂y j i 1 ,i 2+ · · ·+Here, the coefficients Y j i 1, Y j i 1 ,i 2, . . . , Y j i 1 ,i 2 ,...,i κare uniquely determinedin terms of partial derivatives of the coefficients X i and Y j of the originalvector field L , together with the pure jet variables ( y j i 1, . . .,y j i 1 ,...,i κ),by means of the following fundamental inductive formulas ([OL1979],