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

240bracket [D i1 , D i2 ], 1 i 1 , i 2 n, is a linear combination of the vector fieldsD 1 , . . .,D n .Because of their specific form (1.38), we must then have in fact[D i1 , D i2 ] = 0. For n = 1, the condition is of course void.§2. SUBMANIFOLD OF SOLUTIONS2.1. Fundamental foliation of the skeleton. As the vector fields D i commute,they equip the skeleton ∆ E ≃ K n+m+p with a foliation F ∆E by n-dimensional integral manifolds which are (approximately) directed alongthe x-axis. We draw a diagram (see only the left side).y xyD0abxyDM (E)0a, bxDDDexp(xD)(0,a, b)The (abstract, not numerical) integration of (E ) is thus straightforwardlycompleted: the set of solutions coincides with the set of leaves of F ∆E . Thisis the true geometric content, viewed in the appropriate jet space, of theassumption of complete integrability.2.2. General solution and submanifold of solutions. To construct the submanifoldof solutions M (E ) associated to (E ) (sketched in the right handside), we execute some elementary analytico-geometric constructions.At first, we duplicate the coordinates ( y j(q)β(q) , yj) ∈ K p × K m by introducinga new subspace of coordinates (a, b) ∈ K p × K m ; thus, on the leftdiagram, we draw a vertical plane together with a- and b-axes. The leavesof the foliation F ∆E are uniquely determined by their intersections with thisplane, consisting of points of coordinates (0, a, b) ∈ K n × K p × K m .Such points (0, a, b) correspond to the initial conditions ( y j(q)x β(q) (0), y(0) )for the general solution of (E ). In fact, the (concatenated, multiple) flow of{D 1 , . . .,D n } is given by(2.3)exp ( x n D n(· · ·(exp(x 1 D 1 (0, a, b))) · · ·)) = ( x, Π(x, a, b), Ω(x, a, b) ) ∈ K n ×K m ×K p ,