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

364for j 1 , j 2 = 1, . . ., n. The expression of Y j1 does not matter for us here.Specifying this formula to the the case n = 2 and taking account of thesymmetry Y 1,2 = Y 2,1 we get the following three second order coefficients:(2.15) ⎧Y 1,1 = Y x 1 x 1 + y x 1 · {2 }Y x 1 y − X 1x 1 x + 1 yx 2 · {−X } 2x 1 x + 1+ y x 1 y x 1 · {Y }yy − 2 X 1x 1 y + yx 1 y x 2 · {−2X 2x y}+ 1+ y x 1 y x 1 y x 1 · {−X }yy1 + yx 1 y x 1 y x 2 · {−Xyy} 2 ,Y 1,2 = Y x 1 x 2 + y x 1 · {Y }x 2 y − X 1x⎪⎨1 x + 2 yx 2 · {Y }x 1 y − X 2x 1 x + 2+ y x 1 y x 1 · {−X } 1x 2 y + yx 1 y x 2 · {Yyy − X 1x 1 y − X 2x y}+ 2+ y x 2 y x 2 · {−X 2x y}+ 1+ y x 1 y x 1 y x 2 · {−X 1yy}+ yx 1 y x 2 y x 2 · {−X 2yy},Y 2,2 = Y x 2 x 2 + y x 1 · {−X }x 1 2 x + 2 yx 2 · {2 }Y x 2 y − Xx 2 2 x + 2+ y x 1 y x 2 · {−2 }X 1x⎪⎩2 y + yx 2 y x 2 · {Yyy − 2 X 2x y}+ 2+ y x 1 y x 2 y x 2 · {−X }yy1 + yx 2 y x 2 y x 2 · {−Xyy} 2 .We would like to mention that the computation of Y j1 ,j 2, 1 j 1 , j 2 2,above is easier than the verification of (2.10). Based on the three formulas(2.15), we claim that we can guess the second and the third equations,which would be obtained by developing and by simplifying (2.9), namelywith y x 1 x 2 and with y x 2 x 2 instead of y x 1 x2 in (2.10). Our dictionary to translatefrom the first formula (2.15) to (2.10) may be described as follows.Begin with the Jacobian determinant(2.16)∣∣X 1 xX 1 1 xX 1 2 y ∣∣∣∣∣X 2 xX 2 1 xX 2 2 yY x 1 Y x 2 Y yof the change of coordinates (2.2). Since this change of coordinates is closeto the identity, we may consider that the following Jacobian matrix approximationholds:⎛⎞ ⎛(2.17) ⎝ X1 xX 1 1 xX 1 2 yX 2 xX 2 1 xX 2 ⎠ ∼ 2 y = ⎝ 1 0 0⎞0 1 0 ⎠ .Y x 1 Y x 2 Y y 0 0 1The jacobian matrix has three columns. There are six possible second orderderivatives with respect to the variables (x 1 , x 2 , y), namely(2.18) (·) x 1 x 1, (·) x 1 x 2, (·) x 2 x 2, (·) x 1 y, (·) x 2 y, (·) yy .In the Jacobian determinant (2.16), by replacing any one of the threecolumns of first order derivatives with a column of second order derivatives,we obtain exactly 3 × 6 = 18 possible determinants. For instance, by