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

⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1 xX 2 yy1 ∣∣∣∣∣ ⎩X 2 xX 2∣1 xX 2 2 yy − 2Y x 1 Y x 2 Y yy∣⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 ·⎩ −2 xX 1 1 x 1 yXy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 x 1 yXy2 ⎭ +Y x 1 Y x 1 y Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 y x 1 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 y x 2 ·⎩ − xX 1 1 yy Xy1X 2 xX 2 1 yy Xy2X 1 x 1 yX 1 x 2 X 1 yX 2 x 1 yX 2 x 2 X 2 yY x 1 y Y x 2 Y y∣ ∣∣∣∣∣⎫⎬⎭ +Y x 1 Y yy Y y∣ ∣∣∣∣∣⎫⎬⎭ .This formula and the two next (2.22), (2.23) have been checked by SylvainNeut and Michel Petitot with the help of Maple.2.11. Comparison with the coefficients of the second prolongation ofa vector field. At present, it is useful to make an illuminating digressionwhich will help us to devise what is the general form of the development ofthe equations (2.9). Consider an arbitrary vector field of the form(2.12) L :=j 1 =1n∑k=1X k∂∂x k + Y ∂∂y ,where the coefficients X k and Y are functions of (x i , y). Accordingto [Ol1986, BK1989, Ol1995], there exists a unique prolongation L (2) ofthis vector field to the second order jet space, of the formn∑(2.13) L (2) ∂n∑ n∑ ∂:= L + Y j1 + Y j1 ,j∂y2,x j 1 ∂y x j 1x j 2j 1 =1 j 2 =1where the coefficients Y j1 , Y j1 ,j 2may be computed by means of formulas(3.4) of Section 3(II). In Part II, we obtained the following perfect formulas:(2.14) ⎧⎪⎨⎪⎩Y j1 ,j 2= Y x j 1x j 2 +++n∑n∑k 1 =1n∑k 1 =1 k 2 =1n∑n∑k 1 =1 k 2 =1 k 3 =1y x k 1 · {δ k 1j 1Y x j 2y + δ k 1j 2Y x j 1y − X k 1x j 1x j 2}+y x k 1 y x k 2 ·n∑363{}δ k 1,k 2j 1 , j 2Y yy − δ k 1j 1X k 2x j 2y − δk 1j 2X k 2+x j 1yy x k 1 y x k 2 y x k 3 ·{ }−δ k 1,k 2j 1 , j 2X k 3yy ,