# Introduction to Mathematical Physics/Introduction

The word physics comes from the Greek $\phi \upsilon \sigma \iota \varsigma$ 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 first axis is provided by the universal mathematical language. The central mathematical idea for modelling is the use of ordinary differential equations (ODE) and partial differential equations (PDE). Because of the importance of such tools in this book, they are presented in a special chapter (Chapter chapprob). But mathematical formalism goes much further. The problem of the level of mathematics used in books about physics is always a critical point, because a level too high can exclude many readers. Here, advanced mathematical tools are used each time we think they can simplify the statement of theories. For instance, statement of electromagnetism is done using distribution theory: it allows to treat globally point, surface and volume charge distributions as a whole. We also use tensors because each physical quantity is basically a tensor, and group theory because it allows to formalize easily the notion of symmetry. However, if advanced tools are often used in this book, it is at a physicist level: mathematics rigour can not be reached in such a book, so it can be seen more "digestible" to readers not familiar with those theories. Moreover, we provide in appendices some basics about mathematical theories used here. Many references to mathematics books are also provided for the reader who would like to have more information. If this book can bring people to read math books, then one of the goals of this book will be reached.
• The second axis followed along this book is the study of the N body problem. The N body problem is the fundamental problem of modern physics. It corresponds to the second step of the reductionist method of physics: phenomena are described by using entities of smaller scale. Behaviour of solids or liquids are described by using molecules. Behaviour of molecules are described using atoms. Behaviour of atoms are described using electrons and nucleons, and so on. On another hand, galaxies are described using the behaviour of stars, clusters of galaxies are described using properties of galaxies, and so on. The reductionist method is very powerful and is the foundation of modern physics with its bright results as for instance the calculation up to the tenth digit of the rays of the hydrogen spectrum. But is has also limitations. For instance, it can not be shown from basic principles of physics and the properties of H$_{2}$ O molecule implies that ice floats on water.

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 $MX=V$ just by sending the command $X=M/V$ 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.