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

computational level (differential-geometric conceptional background aboutgraph transformations may be found in [Ol1986], Ch. 2).First of all, we seek how the Y j dY jX:= are explicitely related to the dX yl x .It suffices to replace, in the identity(3.4) ()m∑mY j X · X x dx + X y l dy l = Y j X dX = dY j = Yx j dx + ∑Y j dy ly ll=1the differentials dy l by y l x dx and then to identify the coefficient of dx onboth sides, which rapidly yields the formulas(3.5) Y j X = Y x j + ∑ ml=1 yl x Y jy lX x + ∑ ml=1 yl x X ,y lfor j = 1, . . .,m.Next, we seek how the Y j XX := d2 Y jdX 2l=123= dY j XdX are related to the yl 1 x , y l 2 xx .It suffices to again replace each dy l by y l x dx and each dy l x by y l xx dx in theidentity(3.6) ⎧⎪⎨⎪⎩Y j XX ·(X x dx +)m∑X y l dy l = Y j XX · dX = dY j Xl=1= ∂Y j mX∂x dx + ∑l=1(∂Y j mX=∂x + ∑l=1∂Y j X∂y l dy l +∂Y j X∂y l y l x +m∑l=1m∑l=1∂Y j X∂y l x∂Y j X∂y l xBefore entering the precise combinatorics of the explicit expression of Y j XX ,let us observe that the last term of (3.6) simply writes D(Y j X) dx, where Ddenotes the total differentiation operator (of order two) defined by(3.7) D := ∂∂x + m∑l=1y l x∂∂y + ∑ m ll=1∂yxxl ∂yxlSince dX ≡ DX after replacing each dy l by yx l dx, it follows that we maycompactly rewrite (3.6) as(3.8) Y j XX DX · dx = D ( Y j X)· dxConsequently, the expressions of Y j X (obtained in (3.5)) and of Y j XX are(3.9)Y j X = DY jand Y j XXDX= D ( Y j )XDX= DDY j · DX − DDX · DY j.[DX] 3.y l xxdy l x)· dx.