On some aspects of the geometry of differential equations in physics

TL;DR

This paper reviews the geometric structures of Hamilton equations, singular differential equations, and PDEs in field theories, discussing their influence on solving these systems and outlining future research directions.

Contribution

It provides a comprehensive overview of the geometric aspects of key differential equations in physics and related fields, highlighting their significance and future research avenues.

Findings

Geometric structures underpin Hamilton, singular, and PDE systems.

Main results connect geometry with differential equation analysis.

Discussion of future research directions in geometric differential equations.

Abstract

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. Furthermore, research to be developed in these areas is also commented.