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

90In , we may rewrite the final expressions of these three terms: firstly⎧I = ∆(x| · · · | j xx| · · · |y m ) − yx j · ∆(xx|y1 | · · · |y m ),m∑II = 2 y l 1x · ∆(x| · · · | j xy l 1| · · · |y m )−(5.24)⎪⎨⎪⎩III =l 1 =1m∑− 2 y j xm∑l 1 =1 l 2 =1− y j xm∑l 1 =1y l 1x · ∆(xy l 1|y 1 | · · · |y m ),y l 1x y l 2x · ∆(x| · · · | j y l 1y l 2| · · · |y m )−m∑m∑l 1 =1 l 2 =1y l 1x y l 2x · ∆(y l 1y l 2|y 1 | · · · |y m ).Coming back to (5.17), we obtain the desired expression (5.4).The proof of the — technical, though involving only linear algebra —Lemma 3.32 is complete.REFERENCES[BK1989] BLUMAN, G.W.; KUMEI, S.: Symmetries and differential equations,Springer-Verlag, New-York, 1989.[Ca1924] CARTAN, É.: Sur les variétés à connexion projective, Bull. Soc. Math. France52 (1924), 205–241.[Ch1939] CHERN, S.-S.: Sur la géométrie d’un système d’équations différentielles dusecond ordre, Bull. Sci. Math. 63 (1939), 206–212.[CMS1996] CRAMPIN, M.; MARTÍNEZ, E.; SARLET, W.: Linear connections for systemsof second-order ordinary differential equations, Ann. Inst. H. Poincaré Phys.Théor. 65 (1996), no. 2, 223–249.[Do2000] DOUBROV, B.: Contact invariants of ordinary differential equations, RIMSKokyuroku 1150 (2000), 105–113.[DNP2005] DRIDI, R.; NEUT, S.; PETITOT, M.: Élie Cartan’s geometrical vision or howto avoid expression swell, arxiv.org/abs/math.DG/0504203.[Fe1995] FELS, M.: The equivalence problem for systems of second-order ordinarydifferential equations, Proc. London Math. Soc. 71 (1995), no. 2, 221–240.[G1989] GARDNER, R.B.: The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics 58 (SIAM,Philadelphia, 1989), 127 pp.[GM2003] GAUSSIER, H.; MERKER, J.: Symmetries of partial differential equations, J.Korean Math. Soc. 40 (2003), no. 3, 517–561.[GG1983] GONZÁLEZ GASCÓN, F.; GONZÁLEZ LÓPEZ, A.: Symmetries of differential[GL1988]equations, IV. J. Math. Phys. 24 (1983), 2006–2021.GONZÁLEZ LÓPEZ, A.: Symmetries of linear systems of second order differentialequations, J. Math. Phys. 29 (1988), 1097–1105.[GKO1992] GONZÁLEZ LÓPEZ, A.; KAMRAN, N.; OLVER, P.J.: Lie algebras of vectorfields in the real plane, Proc. London Math. Soc. 64 (1992), 339–368.