Automorphisms of flag systems

Abstract

We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge-Ampère type equations (2nd order). Next, we describe a prolongation functor operating on the infinitesimal symmetries (automorphisms) of the Darboux flag and extending these, isomorphically, to all the symmetries of any other flag. Hence, flag systems cannot be distinguished by their symmetry algebras and the local classification of these objects is approached by considering higher order isotropies of these algebras as well as the groupoids of k − th order formal equivalences since the differential equations defining the latter provide precious information for the application of flag systems to differential equations (e.g., Cartan's criterion for non-linear Monge equations). In examining the behaviour of the isotropy algebras, that can either diminish or remain the same, when passing from a derived system Sν to the previous system S ν−1 , we obtain a full set of numerical invariants for the elementary flag systems that moreover specify the local models. flag systems and models and infinitesimal automorphisms and isotropy and Darboux prolongations and k − th order equivalence groupoids.

On the integrability problem for systems of partial differential equations in one unknown function, II

“…The above finite sequence is a prolongation space together with Pfaffian systems, at each level, naturally associated to the initially given system S. The terminating system S ℓ vanishes and S ℓ−1 is a Darboux system in 3-space. As shown in [19], every local or infinitesimal automorphism of this Darboux system extends (prolongs) canonically to an automorphism of (P ℓ−ν , S ν ), this correspondence becoming, moreover, an isomorphism of pseudo-groups as well as of pseudo-algebras. We are not to be concerned with local sections since P 1 can be considered as the total space, whereas the base space P 0 collapses to an open set, eventually to a point.…”

“…Though rather absent in the recent literature, the reader will find interesting examples in [14], [19] and [17]. As for the calculations, we try to reduce them to the strict minimum, the main concern being the determination of a fundamental set of invariants associated to a differential system that will characterise as well as describe the local equivalences.…”

“…These are systems of co-rank equal to 2 (when the characteristics vanish) hence the integral manifolds are just 1-dimensional curves, l'intégrale générale ne dépendant que de fonctions arbitraires d'une seule variable, as would claim Cartan. However, the reader might delight himself in examining the many local models exhibited in [19].…”

“…Let us now assume that the Pfaffian system P is a flag system i.e., rank P i = rank P i+1 + 1 for all i. Then, with the notations of the reference [19], the finite and infinitesimal automorphisms pseudo-groups and pseudo-algebras of any of the factored derived systems P µ (modulo their characteristic variables) are canonically equivalent (isomorphic), via a natural merihedric prolongation algorithm, to those of P and consequently all the corresponding higher order groupoids and algebroids are also canonically isomorphic.…”

“…On the other hand, the restricted system P|L is non-integrable and has rank equal to one hence is a Darboux system, [19]. The definition, per se, of the gender of this restricted system will then assert that P|L is a Darboux system of Cartan or, inasmuch, a system of Darboux class equal to 2h + 1 hence generated locally by a form…”