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

205But we may also express the duality from the first jet (x, y, y x )-space to thefirst jet (a, b, b a )-space by simply composing the following three maps, thecentral one being the duality M → M ∗ :⎛⎝⎞(a, x, y) (( ↓a, Q ∗ (a, x, y), Q ∗ a(a, x, y) ) ⎠ ◦⎛(x, a, b) → ( a, x, Q(x, a, b) )) ◦ ⎝(x, A(x, y, yx ), B(x, y, y x ) )↑(x, y, y x )which in sum gives us the map:(A(x, y, yx ), Q(x, y, y x ) ↦−→∗( A(x, y, y x ), x, Q ( x, A(x, y, y x ), B(x, y, y x ) )) )( ,A(x, y, yx ), x, Q ( x, A(x, y, y x ), B(x, y, y x ) )) .Q ∗ aWith the approximations, one checks that:(x, y, y x ) ↦−→ ( y x + · · · , −y + xy x + · · · , x + · · ·),where the remainder terms “+ · · · ” are all O(x 2 ). For the differential equationy xx (x) = 0 of affine lines, these remainders disappear completely andwe recover the classical projective duality written in inhomogeneous coordinates([5], pp. 156–157). Furthermore, one shows (see e.g. [5]) that theabove duality map within first order jet spaces is a contact transformation,namely through it, the pullback of the standard contact form db−b a da in thetarget space is a nonzero multiple of the standard contact form dy − y x dx inthe source space.But what matters more for us is the following. The two fundamentaldifferential invariants of b aa (a) = F ∗( a, b(a), b a (a) ) are functions exactlysimilar to the ones written on p. 197, namely:⎞⎠,I 1 (E ∗ ) := F ∗ b ab ab ab aI 2 (E ∗ ) := D ∗ D ∗( F ∗ b ab a)− F∗baD ∗( F ∗ b ab a)− 4 D∗ ( F ∗ bb a)++ 6 F ∗ bb − 3 F ∗ b F ∗ b ab a+ 4 F ∗ b aF ∗ bb a,where D ∗ := ∂ a + b a ∂ b + F ∗ (a, b, b a ) ∂ ba . Then according to Koppisch([10]), through the duality map, I 1 (E)is transformed to a nonzero multiple ofI 2 (E ∗ ) , and simultaneously also, I2 (E ) is transformed to a nonzero multiple22 ofI 1 (E ∗ ) , so that: 0 = I 1 (E) ⇐⇒ I 2 (E ∗ ) = 00 = I 2 (E) ⇐⇒ I 1 (E ∗ ) = 0.Consequently, the differential equation (E ): y xx (x) = F ( x, y(x), y x (x) ) isequivalent to y ′ x ′ x ′(x′ ) = 0 if and only if:F yxy xy xy x= 0 and F ∗ b ab ab ab a= 0 .22 To be precise, both factors of multiplicity ([5], p. 165) are nonvanishing in a neighboroodof the origin, but for our purposes, it suffices just that they are not identically zeropower series.