On the internal approach to differential equations 3. Infinitesimal symmetries

Abstract

Abstract. The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher-order infinitesimal symmetries which provide a simplifying "linear aproximation" of general groups of higher-order symmetries. The classical Lie's approach is appropriately adapted. PrefaceIf the invertible higher-order transformations of differential equations are accepted as a reasonable subject, the common Lie-Cartan's methods are insufficient for complete solution of the symmetry problem. We recall that even the structure of all higher-order symmetries of the trivial (empty) systems of differential equations (that is, of the infinite-order jet spaces without any differential constraits) is unknown [1,2,3]. The same can be said for the "linearized theory" of the higher-order infinitesimal transformations treated in this article.Let us outline the core of the subject. We start with surfaceslying in the space R m+n with coordinates x 1 , . . . , x n , w 1 , . . . , w m . The higherorder transformations are defined by formulaē, ··) (i, i ′ = 1, . . . , n; j, j ′ = 1, . . . , m) (1.1)2010 Mathematics Subject Classification. 54A17, 58J99, 35A30.

The symmetry reduction of variational integrals

“…We do not need urgently this result (see [22], [9]) at this place and delay the proof. It should be only noted that the sufficience of the condition is easy since the inclusion declares that the Lie derivative L V acts on finite-dimensional spaces where the classical theory can be applied and such V does generate a group.…”

Report on the Absolute Differential Equations I

“…Variations (37) preserving the filtration (36) are either the infinitesimal point transformations or the infinitesimal Lie's contact transformations. This is a particular case of the Lie-Bäcklund theorem [10], [24]. Variations preserving other good filtrations as in Figure(1b) and therefore generating a group of higher-order transformations are in general unknown.…”