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

356III: Systems of second orderTable of contents1. Explicit characterizations of flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2. Completely integrable systems of second order ordinary differential equations . . . 361.3. First and second auxiliary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .370.§1. EXPLICIT CHARACTERIZATIONS OF FLATNESSIn 1883, S. Lie obtained the following explicit characterization of thelocal equivalence of a second order ordinary differential equation (E 1 ):y xx = F(x, y, y x ) to the Newtonian free particle equation with one degreeof freedom Y XX = 0. All the functions are assumed to be analytic.Theorem 1.1. ([Lie1883], pp. 362–365) Let K = R of C. Let x ∈ Kand y ∈ K. A local second order ordinary differential equation y xx =F(x, y, y x ) is equivalent under an invertible point transformation (x, y) ↦→(X(x, y), Y (x, y)) to the free particle equation Y XX = 0 if and only if thefollowing two conditions are satisfied:(i) F yxy xy xy x= 0, or equivalently F is a degree three polynomial in y x ,namely there exist four functions G, H, L and M of (x, y) such that Fcan be written as(1.2) F(x, y, y x ) = G(x, y)+y x·H(x, y)+(y x ) 2·L(x, y)+(y x ) 3·M(x, y);(ii) the four functions G, H, L and M satisfy the following system of twosecond order quasi-linear partial differential equations:(1.3) ⎧0 = −2 G yy + 4 3 H xy − 2 3 L xx+⎪⎨+ 2 (G L)y − 2 G x M − 4 G M x + 2 3 H L x − 4 3 H H y,0 = − 2 3 H yy + 4 3 L xy − 2 M xx +⎪⎩+ 2 G M y + 4 G y M − 2 (H M) x − 2 3 H y L + 4 3 L L x.Open question 1.4. Deduce an explicit necessary and sufficient conditionfor the associated submanifold of solutions y = Π(x, a, b) to be locallyequivalent to Y = B + XA.