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

Definition 4.10. L is an infinitesimal symmetry of (E ) if for every small t,its time-t flow map ϕ t is a Lie symmetry of (E ).The restriction L (κ+1)∣ ∣∆Eis obtained by replacing every yα j by F α j in allcoefficients Y j i 1, . . .,Y j i 1 ,...,i κ+1. Then the coefficients become functions of(xi 1, y j 1, y j(q 1)β(q 1 ))only.Lemma 4.11. ([Ol1986, Ol1995, BK1989], [∗]) The following three conditionsare equivalent:(1) the vector field L is an infinitesimal Lie symmetry of (E );(2) its (κ + 1)-th prolongation L (κ+1) is tangent to the skeleton ∆ E ;(3) L (κ+1) is tangent to ∆ E and the push-forward(4.12) L ∆E := (π κ,p ) ∗(L(κ+1) ∣ ∣∆E)249is an infinitesimal symmetry of the foliation F ∆E , namely for everyi = 1, . . ., n, the Lie bracket [ L ∆E , D i]is a linear combination of{D 1 , . . .,D n }.According to [Ol1986, BK1989, Ol1995], the set of infinitesimal Lie symmetriesconstitutes a Lie algebra, with the property [ L (κ+1) , L ′ (κ+1) ] =[L , L′ ] (κ+1) . We summarize by a diagram.ϕ (κ+1)Jn,m κ+1Jn,mκ+1L (κ+1)∆ Eπ κ,pϕ ∆Eπ κ,p∆ Eπ κL ∆Eπ κπ pπ pK n x × K m yϕK n x × K m yL4.13. Sophus Lie’s algorithm. We describe the general process. Its complexitywill be exemplified in Section 5 (to be read simultaneously).The tangency of L (κ+1) to ∆ E is expressed by applying L (κ+1) to theequations 0 = −yα j + F α j , which yields:(4.14) 0 = −Y j α +n∑i=1X i ∂F j α∂x i +n∑l=1Y l ∂F j α∂y l +p∑q=1Y j(q)β(q)∂Fαj ,∂y j(q)β(q)for (j, α) ≠ (j, 0) and ≠ (j(q), β(q)). Restricting a coefficient Y j i 1 ,...,i λto∆ E , namely replacing everywhere in it each yα j by F α j , provides a specialized