Les phénomènes vibratoires jouent un rôle déterminant dans presque toutes les branches de la physique: mécanique, électricité, optique, acoustique,… etc .mais malgré leur grande diversité, ils sont étudiés au moyen du même outil mathématique.

L’utilisation du formalisme de Lagrange est clairement spécifiée dans le programme officiel. Nous avons essayé d’introduire le formalisme de Lagrange le plus simplement possible dans le cas particulier d’un système à une particule possédant un degré de liberté.

Ce cours intitulé le «Oscillations libres des systèmes à un degré de liberté» se  permet à l'étudiant de construire une interprétation scientifique à l’aide des outils  mathématiques de certains phénomènes vibratoires


LEARNING GOALS
• Three fundamental quantities of physics and the units physicists use to measure them.
• How to keep track of significant figures in your calculations.
• The difference between scalars and vectors, and how to add and subtract vectors graphically.
• What the components of a vector are, and how to use them in calculations.
• What unit vectors are, and how to use them with components to describe vectors.
• Two ways of multiplying vectors.

• How to describe straight-line motion in terms of average velocity, instantaneous velocity, average acceleration, and instantaneous acceleration.
• How to interpret graphs of position versus time, velocity versus time, and acceleration versus time for straight-line motion.
• How to solve problems involving straight-line motion with constant acceleration, including free-fall problems.
• How to analyze straight-line motion when the acceleration is not constant.

• How to represent the position of a body in two or three dimensions using vectors.
• How to determine the vector velocity of a body from a knowledge of its path.
• How to find the vector acceleration of a body, and why a body can have an acceleration even if its speed is constant.
• How to interpret the components of a body’s acceleration parallel to and perpendicular to its path.
• How to describe the curved path followed by a projectile.
• The key ideas behind motion in a circular path, with either constant speed or varying speed.
• How to relate the velocity of a moving body as seen from two different frames of reference.

• What the concept of force means in physics, and why forces are vectors.
• The significance of the net force on an object, and what happens when the net force is zero.
• The relationship among the net force on an object, the object’s mass, and its acceleration.
• How the forces that two bodies exert on each other are related.

• How to use Newton’s first law to solve problems involving the forces that act on a body in equilibrium.
• How to use Newton’s second law to solve problems involving the forces that act on an accelerating body.
• The nature of the different types of friction forces—static friction, kinetic friction, rolling friction, and fluid resistance—and how to solve problems that involve these forces.
• How to solve problems involving the forces that act on a body moving along a circular path.
• The key properties of the four fundamental forces of nature.

• What it means for a force to do work on a body, and how to calculate the amount of work done.
• The definition of the kinetic energy (energy of motion) of a body, and what it means physically.
• How the total work done on a body changes the body’s kinetic energy, and how to use this principle to solve problems in mechanics.
• How to use the relationship between total work and change in kinetic energy when the forces are not constant, the body follows a curved path, or both.
• How to solve problems involving power (the rate of doing work).

• How to use the concept of gravitational potential energy in problems that involve vertical motion.
• How to use the concept of elastic potential energy in problems that involve a moving body attached to a stretched or compressed spring.
• The distinction between conservative and nonconservative forces, and how to solve problems in which both kinds of forces act on a moving body.
• How to calculate the properties of a conservative force if you know the corresponding potential-energy function.
• How to use energy diagrams to understand the motion of an object moving in a straight line under the influence of a conservative force.

• The meaning of the momentum of a particle, and how the impulse of the net force acting on a particle causes its momentum to change.
• The conditions under which the total momentum of a system of particles is constant (conserved).
• How to solve problems in which two bodies collide with each other.
• The important distinction among elastic, inelastic, and completely inelastic collisions.
• The definition of the center of mass of a system, and what determines how the center of mass moves.
• How to analyze situations such as rocket propulsion in which the mass of a body changes as it moves.

Crystal structure

 

1.1 INTRODUCTION

The aim of solid state physics is to explain the properties of solid materials as found on Earth. For almost all purposes the properties are expected to follow from Schrödinger’s equation for a collection of atomic nuclei and electrons interacting with electrostatic forces. The fundamental laws governing the behaviour of solids are therefore known and well tested. It is nowadays only in cosmology, astrophysics and high-energy physics that the fundamental laws are still in doubt.

In this book we shall be concerned almost entirely with crystalline solids, that is solids with an atomic structure based on a regular repeated pattern, a sort of three-dimensional wallpaper. Many important solids are crystalline in this sense, although this is not always manifest in the external form of the material. Because calculations are easier, more progress has been made in understanding the behaviour of crystalline than of non-crystalline materials. Many common solids—for example, glass, plastics, wood, bone—are not so highly ordered on an atomic scale and are therefore non-crystalline. Only recently has progress been made in understanding the behaviour of non-crystalline solids at a fundamental level.

Even in the restricted field of crystalline solids the most remarkable thing is the great variety of qualitatively different behaviour that occurs. We have insulators, semiconductors, metals and superconductors—all obeying different macroscopic laws: an electric field causes an electric dipole moment in an insulator (Chapter 9), a steady current in a metal or semiconductor (Chapters 3 to 6) and a steadily accelerated current in a superconductor (Chapter 10). Solids may be transparent or opaque, hard or soft, brittle or ductile, magnetic or nonmagnetic.

In this chapter we first introduce in Section 1.2 the basic ideas of crystallography. In Section 1.3 we describe some important crystal structures and in Section 1.4 we explain how x-ray diffraction is used to determine crystal structure. In Section 1.5 we discuss quasi-crystals, ordered solids that challenge much of the conventional wisdom concerning crystalline materials. Section 1.6 contains a qualitative description of the interatomic forces responsible for binding atoms into solids.