Introduction to Mathematical Physics/Introduction

The word physics comes from the Greek formula_1
which means Nature. Physics is the science that studies the properties of the
matter, space and time and states the laws that describe natural phenomena.
Modern physics is based on the idea that matter is not exactly as we perceive
it. This idea emitted by Galileo and later taken by philosophers like
Descartes, 
Boyle and Locke is at the origin of the reductionist approach of physics: one
look for describing properties of matter at a given scale by
properties of matter at a more 
fundamental level. This approach has two parts: the first consists in
identifying 
fundamental elements that constitutes matter. So, ordinary matter
would be compound by atoms that are themselves a assembly 
of a nucleus and electrons. The nucleus is itself compound by nucleons, that is
particles compound by quarks and gluons. 
The second consists in studying
the combinations of those fundamental elements and their interactions to
interpret observed phenomena. So, color of objects can be interpreted by
vibrations of molecular atoms constituting those objects.
In a same way, sound will be interpreted as variations of air density that are
transmitted to ear.
Fundamental elements are known, at the present state of Nature
knowledge, to interact according four interactions:
gravitational interaction, electromagnetic interaction, weak interaction and
strong interaction. In this book, the "smallest" elements we consider are
atomic nuclei and electrons, except in the chapter N body problem and matter description introducing the matter
organization. Interactions that occur at scales larger or equal to the
atomic scale are the electromagnetic
interaction and gravitational interaction. 
The goal of this book is to propose an ensemble view of modern physics. The
coherence between various fields of physics is insured by following two axes:
A very important point: we present "methods" for solving given problems. The
reader should not think that physics is reduced to the applying of some
classical methods. Modelling Nature is a very creative activity. Creativity
and 
curiosity are two fundamental qualities to do research in physics. There is
always in physics the crucial point of how to simplify the problems. In the
description of a phenomenon, some features are essential, others are not. This
is the job of the physicist to select what features are really relevant. But
of course, there is never only one model to describe a phenomenon. So,
light emission by an atom can be described by
the classical spring model or by a quantum model. Each description
gives results. The first one is much more simple 
than the second one and can be sufficient for the problem you have to
solve. The second is more powerful and is based on the present idea of
the matter structure. But those are just different models of a
same phenomenon. The second is not more "true" than the first. It can just
describe and predict more phenomena.
If the reader is convinced after reading this book that physics is wonderful
field rich of diversity and puzzling problems, then the most important goal of
this book will be reached!
However, a physicist has to know when the creative job is finished. He has
to be able to identify classical mathematical problems (that can be hidden
behind various details) and to use mathematical and numerical tools to solve
them. But here also, this is not a blind activity. Each time several methods
exist, and modeller have to choose among them. It is perhaps also necessary
that his own problem requires some new method to be solved, because his
problem has some particular characteristics. It is this way that scientific
advances and discoveries are done.
The book is organized as follows:
The first chapter is mathematical introduction. It presents a
classification 
of a large class of problems encountered as modelling physical phenomena. It
deals with partial differential equations and minimization. It presents also
numerical methods for the approximate solving of those problems. The point of
view adopted here is to leave to specialized books the presentation of the
numerical treatment of classical problems : solution of linear algebraic
equations, eigensystems, root finding, even if they are used by solving PDE
and minimization problems. The reason of this is that those operations can be
"easily" automatized (and are now
rather well implemented in classical scientific software). For instance, many
programs propose now to solve a system of linear algebraic equations formula_3
just by sending the command formula_4 to the program!
On the other hand, scientists have to know in much more detail the methods used
to solve PDE and minimization problem because the choice of methods is large,
and because each method has different behaviour which can be adapted for the
particular problem to solve.
Numerous references to other part of the book are done in this chapter: this
chapter does give the mathematical unity about which we spoke above. Indeed,
most of the mathematical problems presented in the rest of the book can be
seen as PDE problems or minimization problems.
Second chapter is a physical introduction to modelling. It presents various
physical system for which a reductionist approach is natural. 
Geometrization of the physical space and first laws of dynamics are presented
in the next chapter devoted to relativity. The reading should be done in
parallel oft the reading of the appendix on tensors.
Next chapter deals with electromagnetism. Ideas introduced at previous chapter
are developed: the notion of electromagnetic force (and its dual description in
terms of power) is bound to the electromagnetic field.
Optics is presented as an approximation of Maxwell equations. The appendix on
distributions can be useful to the reader not familiar with this theory.
The next chapter introduces the fundamental theory of quantum mechanics which
is applied to the N body problem in the next chapter. methods presented at
chapter 1 are intensively used for solving Schrödinger equation.
Next chapter Statistical physics deals with the fundamental principle
of statistical 
physics. Statistical physics is the key theory to go back up to macroscopic
scales. It allows in the next chapter to tackle various N body problems,
where N is large.
It is also the theory underlying the kinetic and continuous description of
matter that are presented in the next chapters.
The public of this book concerns persons interested in having a global idea of
physics. Indeed, physics is usually taught at university by various
professors, each of them forgetting possible connections with other
courses. It can be read advantageously by students with a 3 or 4 years
physics university background, for instance for the preparation of finals
exams. It can also be used by researchers as a detailed mementum. References
should be in this case very useful. This book can also help mathematicians to
illustrate math courses by 
real world examples. Problems provided by other field of science are vital for
mathematics. This is unfortunately too often forgotten by mathematics
teachers. 
Some exercises are presented at the end of the chapters. Some are direct
applications of the exposed matter. Other invite the reader to more reflection
and to the reading of specialized books.