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

Next, we gather the underlined terms, ordering them according to their number.This yields 6 collections of sums of monomials in the pure jet variables:(3.18)Y i1 ,i 2= Y x i 1x i 2 + ∑ ][δ k 1i 1Y x i 2y + δ k 1i 2Y x i 1y − X k 1yx i 1x i 2 k1 +k 1+ ∑ ][δ k 1,k 2i 1 , i 2Y yy − δ k 1i 1X k 2x i 2y − δk 2i 2X k 1yx i 1y k1 y k2 +k 1 ,k 2+ ∑ [ ]−δ k 1,k 3i 1 , i 2X k 2yy y k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y y − δ k 2i 2X k 1x i 1 − δk 2i 1X k 1x i 2]y k1 ,k 2++ ∑ []−δ k 2,k 3i 1 , i 2X k 1y − δ k 1,k 3i 1 , i 2X k 2y − δ k 1,k 3i 2 , i 1X k 2y y k1 y k2 ,k 3.k 1 ,k 2 ,k 3To attain the real perfect harmony, this last expression has still to be workedout a little bit.Lemma 3.19. The final expression of Y i1 ,i 2is as follows:(3.20)⎧Y i1 ,i 2= Y x i 1x i 2 + ∑ [δk 1i 1Y x i 2y + δ k 1i 2Y x i 1y − X k 1x i 1x 2] i yk1 +k 1⎪⎨⎪⎩+ ∑ ][δ k 1,k 2i 1 , i 2Y yy − δ k 1i 1X k 2x i 2y − δk 1i 2X k 2yx i 1y k1 y k2 +k 1 ,k 2+ ∑ [ ]−δ k 1,k 2i 1 , i 2X k 3yy y k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y y − δ k 1i 1X k 2x i 2 − δk 1i 2X k 2x i 1]y k1 ,k 2++ ∑ []−δ k 1,k 2i 1 , i 2X k 3y − δ k 3,k 1i 1 , i 2X k 2y − δ k 2,k 3i 1 , i 2X k 1y y k1 y k2 ,k 3.k 1 ,k 2 ,k 3Proof. As promised, we explain every tiny detail.The first lines of (3.18) and of (3.20) are exactly the same. For the transformationsof terms in the second, in the third and in the fourth lines, weuse the following device. Let Υ k1 ,k 2be an indexed quantity which is symmetric:Υ k1 ,k 2= Υ k2 ,k 1. Let A k1 ,k 2be an arbitrary indexed quantity. Thenobviously:∑(3.21)A k1 ,k 2Υ k1 ,k 2= ∑ A k2 ,k 1Υ k1 ,k 2.k 1 ,k 2 k 1 ,k 2Similar relations hold with a quantity Υ i1 ,i 2 ,...,i λwhich is symmetric withrespect to its λ indices. Consequently, in the second, in the third and in325