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

This yields the prolongation of the transformation to the first order jet space.For the second order prolongation, introducing the second order total differentiationoperator (which geometrically corresponds to differentiation alonggraphs {(x, y(x))}) defined by(2.5) D := ∂∂x + y x∂∂y + y xx∂∂y x,we may compute, simplify and reorder the expression of the second orderderivative in the (X, Y )-coordinates:(2.6) ⎧Y XX := d2 YdX ≡ DY X2 DX = D [(Y x + y x Y y )(X x + y x X x ) −1 ]=X x + y x X y⎪⎨⎪⎩1=[X x + y x X y ] {y 3 xx [X x Y y − Y x X y ] + X x Y xx − Y x X xx ++ y x [2(X x Y xy − Y x X xy ) − (X xx Y y − Y xx X y )] +11+ y x y x [X x Y yy − Y x X yy − 2(X xy Y y − Y xy X y )]++ y x y x y x [−(X yy Y y − Y yy X y )]} .Even if not too complicated, the internal combinatorics of this expressionhas to be analyzed and expressed thoroughly. First of all, as Y XX = 0 byassumption, we may erase the cubic factor [X x +y x X y ] −3 . Next, as the factorof y xx in the right-hand side of (2.6), we just recognize the Jacobian ∆(x|y)expressed in (2.3) above. Also, all the other factors are modifications of theJacobian ∆(x|y), whose combinatorics may be understood as follows.There exist exactly three possible distinct second order derivatives: xx, xyand yy. There are also exactly two columns in (2.3). By replacing each ofthe two columns of first order derivative in ∆(x|y) by any column of secondorder derivative (leaving X and Y unchanged), we may build exactly sixnew determinants(2.7){ ∆(xx|y) ∆(xy|y) ∆(yy|y)∆(x|xx) ∆(x|xy) ∆(x|yy)where for instance{(2.8) ∆(xx|y) :=∣ X ∣ xx X y ∣∣∣ and ∆(x|xy) :=Y xx Y y∣X xX xyY x Y xy∣ ∣∣∣.Hence, by rewriting (2.6), we see that the equation y xx = F(x, y, y x ) equivalentto Y XX = 0 may be written under the general explicit form, involving