The theory of differential equations is one of the largest fields within mathematics
and probably most graduates in mathematics have attended at least one course on
differential equations. But differential equations are also of fundamental importance
in most applied sciences; whenever a continuous process is modelled mathematically,
chances are high that differential equations appear. So it does not surprise
that many textbooks exist on both ordinary and partial differential equations. But
the huge majority of these books makes an implicit assumption on the structure of
the equations: either one deals with scalar equations or with normal systems, i. e.
with systems in Cauchy–Kovalevskaya form. The main topic of this book is what
happens, if this popular assumption is dropped.
This is not just an academic exercise; non-normal systems are ubiquitous in applications.
Classical examples include the incompressible Navier–Stokes equations
of fluid dynamics, Maxwell’s equations of electrodynamics, the Yang–Mills equations
of the fundamental gauge theories in modern particle physics or Einstein’s
equations of general relativity. But also the simulation and control of multibody
systems, electrical circuits or chemical reactions lead to non-normal systems of ordinary
differential equations, often called differential algebraic equations. In fact,
most of the differential equations nowadays encountered by engineers and scientists
are probably not normal.
In view of this great importance of non-normal systems, the relative lack of literature
on their general theory is all the more surprising. Specific (classes of) systems
like the Navier–Stokes equations have been studied in great depth, but the existence
of general approaches to non-normal systems seems to be hardly known, although
some of them were developed about a century ago! In fact, again and again new
attempts have been started for such general theories, in particular for ordinary differential
equations where the situation is comparatively straightforward. Classical
examples are the Dirac theory of mechanical systems with constraints and the currently
fairly popular numerical analysis of differential algebraic equations. However,
in both cases researchers have had to learn (sometimes the hard way) that the
generalisation to partial differential equations is far from trivial, as new phenomena
emerge requiring new techniques.