Flag systems and ordinary differential equations

Abstract

During many decades, it was a general belief that a flag system could always be represented locally by the von Weber model. However, this is not the case and, in 1978, Giaro, Ruiz and the author brought to evidence a non-homogeneous model ([9]). More recently, the same model was found independently by Mormul ([20]). The presence of such non-homogeneous models sheds some doubts on Cartan's criterion since statements such as in [4], p. 89, [11], p. 328, and [22], p. 227, become inaccurate. It is our purpose here to prove the Cartan criterion with the added hypothesis of homogeneity and show as well that this criterion can be applied directly to higher order ordinary differential equations without the necessity of reducing them to first order equations. Finally, we examine first and second order single equations, these involving flag systems of length two and three respectively.(*) Entrata in Redazione il 1 agosto 1998. Indirizzo dell'A.: Rua Campos Sales 225,

Lie Theorem via Rank 2 Distributions (Integration of PDE of Class ω = 1)

“…Indeed, from internal viewpoint the rank 2 distribution ∆, obtained from E via reduction by Cauchy characteristic, is a Goursat distribution (or Goursat in the leaves of the first integrals if the distribution is not totally non-holonomic). Since the Goursat distribution has the canonical normal form (see [18,22], we neglect singularities) the claim follows.…”

“…1.2 E has closed form of the general solution if and only if the same is true for the reduced underdetermined ODE (encoded by ∆). For rank 2 distributions the criterion for closed form description of the integral curves (without constants) is known since Cartan [5,18]: this is equivalent to ∆ being Goursat, i.e. the canonical distribution C on the jet-space…”

Automorphisms of flag systems

“…Élie Cartan also gave extraordinary contributions to this subject and no doubt went fathoms deeper. Among the several papers he wrote, perhaps the deepest and above all the most inspiring are [4] and [5] (see also the forewords in [23], [22] and [24]. Knowing the structure and, whenever possible, a local model for the Pfaffian system S associated to the set of equations, we shall be able [26] to determine the algebra of all infinitesimal automorphisms (dwelling in the base space) of the given k − th order equations, lift this algebra to a sub-algebra of infinitesimal k − th order contact transformations and subsequently prolong the later to an order where the equations become formally integrable (e.g., involutive).The prolongued algebra is then equal to the set of all infinitesimal contact automorphisms of the prolongued equations and a fortiori leaves invariant the characteristic system of the prolongation of S. The knowledge of such an algebra entails the possibility of integrating, via Lie and Cartan's methods, the (integrable) characteristic system and thereafter the initially given equations [23,24].…”

“…We infer that the above associated Pfaffian system is the Engel flag. Another noteworthy example is given by the second order Goursat equations [15,23,22] whose algebra of all infinitesimal second order contact automorphisms is isomorphic to the exceptional complex simple Lie algebra g 2 with real form g 2(2) .…”

An introduction to Lie groupoids

“…δ(f s) = f δ(s) + [ϑ(θ)f ]s where ϑ( ) denotes the Lie derivative. These operators are clearly of order ≤ 1 and, since any first order linear differential operator D : E −→ F satisfies the relation D(f s) = f D(s) + σ(D, df )s, where σ(D, df ) : E −→ F is the symbol of D evaluated on the co-vector df, we infer that (29) is equivalent to the relation(30) σ(δ, df )s = [ϑ(θ)f ]s =< θ, df > s.…”