Mathematics for Dynamicists - Lecture Notes
Résumé
These notes are the essence of the lecture series
"Mathematics of Dynamicists" started in November 2004 and held
regularly at the University of Sheffield as a MSc course module.
The module "Mathematics for Dynamicists" addresses to any students
of Engineering and Natural Sciences who are interested in
mathematical-analytical solution methods to tackle problems in
physics and engineering dynamics.
The lecture notes consist of two parts. Part I is a warmup of the
basic mathematics which is prerequisite for the understanding of
part II. Some students will be familiar with the basics reviewed in
part I, however the experience showed that it is useful to repeat
these basics before starting with dynamical problems of engineering
which are described by Recurrences and Ordinary Differential
Equations in part II. To ensure the appropriate maths level needed
for part II, the reader is highly recommended to answer the
questions to part I before going to part II.
Part II is the essential part and gives a brief introduction to the
theory and solution methods for linear recurrences, Ordinary
Differential Equations (ODEs) and Initial Value Problems (IVPs)
which are also referred to as Cauchy Problems. The main goal is less
to give proofs but more on applications of the methods on dynamical
problems relevant to physical and engineering sciences.
Instead of strict proofs we rather confine to short derivations with
emphasis on applications of the analytical techniques. The reader is
referred to the standard literature for any further proofs which are
beyond the scope of these lecture notes.
The selection of materials for this short-term module (2 weeks time)
can only be an incomplete and sometimes arbitrary selection of
materials. To improve the syllabus, suggestions from anybody are
highly appreciated. The enclosed lecture notes are continuously
updated and extended. Any communicated discrepancies and errors will
be corrected steadily.
I am especially grateful to my wife Dr. Jicun Wang for proof-reading
this manuscript, to Mr. Shahram Derogar (Dept. Civil \& Struct.
Eng., Univ. of Sheffield) and to Dr. Andrzej F. Nowakowski (Dept. of
Mech. Eng., Univ. Sheffield) for their helpful critical comments and
valuable inputs which significantly improved the contents of this
lecture.
Sheffield, November 2006 Thomas Michael Michelitsch