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

244vanish at the origin, since Π = b+xa+O 3 . By elementary Cramer formulas,we get:⎧A x = −Π b Π xx + Π x Π xb, B x = −Π x Π xa + Π a Π xx,Π b Π xa − Π a Π xb Π b Π xa − Π a Π xb(2.33)⎪⎨⎪⎩ A y1 =−Π xbA y =,Π b Π xa − Π a Π xbB y =Π b,Π b Π xa − Π a Π xbB y1 =Π xaΠ b Π xa − Π a Π xb,−Π aΠ b Π xa − Π a Π xb.Replacing in (2.30), no simplification occurs and we get what we wanted:(2.34)⎧F x = Π xxx + Π [ ] [ ]xxa − Πb Π xx + Π x Π xb + Πxxb − Πx Π xa + Π a Π xx,Π ⎪⎨b Π xa − Π a Π xbF y = −Π xxa Π xb + Π xxb Π xa,Π b Π xa − Π a Π xb⎪⎩ F y1 = Π xxa Π b − Π xxb Π a,Π b Π xa − Π a Π xbOne sees D F = F x + Π x F y + Π xx F y1 = Π xxx simply, as predicted byLemma 2.22.Second order derivatives F xx , F xy , F xy1 , F yy , F yy1 , F y1 y 1have still reasonablecomplexity, when expressed in terms of Jx,a,b 4 Π. Beyond, the computationsexplode.Open question 2.35. A second order ordinary differential equation y xx =F(x, y, y x ) has two fundamental differential invariants, namely ([Tr1896,Ca1924, GTW1989, Ol1995]):(2.36)I 1 (E 1 ) := ∂4 Fand∂y14I 2 (E 1 ) := DD ( )F y1 y 1 − Fy1 D ( ) ( )F y1 y 1 − 4 D Fyy1 + 6 Fyy − 3 F y F y1 y 1+ 4 F y1 F yy1 .Compute I 1 M 1and I 2 M 1.Although the notion of diffeomorphism is clear and apparently obviousfrom the intuitive, geometric and conceptual viewpoints, in concrete applicationsand in explicit computations, it almost never straightforward totransfer algebrico-differential objects.Open problem 2.37. For general (E ) and M , build closed combinatorialformulas executing the double translation (2.27).2.38. Plan for the sequel. We will endeavour a general theory showing thatthe study of systems (E ) and the study of submanifolds of solutions M gives