Report on the absolute differential equations I

TL;DR

This paper surveys the absolute theory of general systems of differential equations, emphasizing coordinate-free concepts like symmetries, prolongations, and involutivity, providing a foundational mathematical framework.

Contribution

It introduces a coordinate-free, abstract mathematical approach to the theory of differential equations, clarifying symmetries and structures without relying on jet mechanisms.

Findings

Clarification of symmetries and prolongations in a coordinate-free setting

Development of a general, structure-focused theory of differential equations

Enhanced understanding of involutivity and controllability structures

Abstract

The article provides a modest survey of the absolute theory of general systems of (partial) differential equations. The equations are relieved of all additional structures and subject to quite arbitrary change of the variables. An abstract mathematical theory in the Bourbaki sense with its own concepts and technical tools follows. In particular the external, internal, generalized and higher-order symmetries and infinitesimal symmetries together with the E. Cartan's prolongations, various characteristics, the involutivity and the controllability structures are clarified in genuinely coordinate-free terms without any use of the common jet mechanisms.