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

e a collection of m analytic second order ordinary differential equations,possibly nonlinear, of the most general form.Motivated by physical applications and by geometrical questions, someauthors (Leach [Le1980], Grissom-Thompson–Wilkens [GTW1989],González-López [GL1988], Fels [Fe1995], Crampin-Martínez-Sarlet [CMS1996], Doubrov [Do2000], Grossman [Gr2000], Mahomed-Soh [MS2001], and others) have been interested in a least characterizingthose that have the Lie symmetry group of maximal dimension. In [Le1980],based on the belief that the equivalence between (1), (2) and (3) of Theorem1.2 would persist in the case m 2, it was conjectured that thesymmetry algebra of every linear system(1.5) y j xx = G j 0(x) +m∑my l G j 1,l (x) + ∑l=1l 1 =1y l 1x H j l 1(x),j = 1, . . .,m,of m 2 second order differential equations has a Lie symmetry group locallyisomorphic to the full transformation group PGL(m+1, K) of the projectivespace P m+1 (K). However, González-López [GL1988] infirmed thisexpectation, and produced a necessary and sufficient condition (see Corollary1.8 below) for local equivalence of such linear systems to the free particlesystem5(1.6) Y j XX= 0, j = 1, . . .,m.Applying Lie’s algorithm it is easily seen ([11]) that the free particle systemhas a local symmetry algebra of dimension ≤ m 2 +4 m+3, the bound beingattained by Y j XX= 0, j = 1, . . ., m, with group PGL(m + 1, K).In 1939, for fiber-preserving transformations only, Chern [Ch1939] conductedthe É. Cartan algorithm through absorptions of torsion, normalizationsand prolongations up to the reduction to an {e}-structure. Remarkably,in 1995, Fels [Fe1995] conducted the É. Cartan algorithm for general systemsyxx j = F j (x, y, y x ) and for general point transformations. As a byproductof the uniqueness of the obtained {e}-structure for which all invarianttensors vanish, Fels deduced in [Fe1995] that the flat system Y j XX = 0,j = 1, . . .,m, is, up to equivalence, the only system of second order possessinga symmetry group of maximal dimension. Alas, the {e}-structuresobtained by Chern and by Fels are not parametric, so that the counterpartto (4) of Theorem 1.2 was lacking as soon as m 2. The extraordinaryheaviness of É. Cartan’s method is a well known obstacle.Recently, Neut [N2003] (see also [DNP2005]) wrote a Maple program tocompute (among other {e}-structures) Fels’ tensors in a parametric, explicitway. The digital computations succeeded in the case m = 2, yielding Theorem3 of [DNP2005], a statement that may be checked to be equivalent to