Emmy Noether is recognized as one of the greatest mathematicians of the twentieth century. She was born in Germany in 1882 to an intellectual Jewish family and died in the United States in 1935. Emmy trained as a language teacher, but after passing the qualifying exams to teach, she decided to study mathematics at the University of Erlangen. At that time in Germany a university education was limited to men, although women were allowed to attend classes if given permission by the professor. (She was half of the total female student body at that university.) She spent a semester at the University of Gottingen, at that time a world leader in mathematics and physics. There she attended lectures from a number of leading mathematicians, including Hermann Minkowski (who you will run into in Chapter 20) and Karl Schwarzschild (whose theory of black holes you will encounter in Chapter 9).

This chapter and the next are a study of the general motion of a rigid body. This is a fairly complicated topic which involves mathematical concepts that you may not have encountered before.

I denoted this chapter as “optional” because it contains essentially no new physics. However, it does introduce some useful mathematical concepts and techniques. The methods introduced here are applied in several areas of physics including quantum mechanics and solid-state physics.

This book has concentrated on “integrable” problems, that is, problems whose equations of motion can be integrated yielding closed-form solutions.

As you will soon appreciate, the physics of a pendulum is much more complex than you might imagine.

In this chapter we consider the basic concepts of the statics and dynamics of fluids. As the name indicates, a fluid is any substance that flows, such as liquids and gases. The general categories of our study are fluid statics or hydrostatics concerning the behavior of fluids at rest, and fluid dynamics which is a study of the motion of fluids and of objects moving with respect to fluids. This is further subdivided into studies of the dynamics of liquids and gases or hydrodynamics and gas dynamics.

Classical field theory is primarily a study of electromagnetic and gravitational fields. This chapter is an introduction to field theory and is limited to a few aspects of the gravitational field.

This chapter treats several advanced concepts in statics, well beyond the brief summary of statics in Section 1.6. We will begin with a few definitions and two simple theorems concerning systems of forces acting on rigid bodies, then go on to analyze the statics of freely deformable bodies such as a string or cable hanging from stationary supports. This is followed by definitions of stress and strain and a generalization of Hooke’s law. The last topic is d’Alembert’s principle and the concept of virtual work. You will see how this principle can be used to derive Lagrange’s equations. An important application is an investigation of the properties of a fluid in equilibrium (hydrostatics), but we will leave that for Chapter 19 where we consider fluids in general.

The orbital motion of a planet around the Sun was one of the first important problems to be analyzed in terms of Newton’s laws of motion. The gravitational force attracting a planet to the Sun is a central force, that is, a force directed towards a fixed point. The motion of a planet is a prime example of the more general problem of the behavior of a particle acted upon by a central force.

You already know a lot of physics and quite a bit of mathematics. You have been exposed to introductory courses in Mechanics, Electromagnetism, Thermodynamics, and Optics. You are now beginning your studies of these topics in a much more profound and rigorous manner. I hate to tell you this, but you are expected to know those concepts from the introductory courses! I know how easy it is to forget definitions and mathematical relationships if you are not using them all the time. So this chapter is a brief review of some of the concepts from your introductory mechanics course that you will be using in this intermediate level course. (I have included only those concepts that are absolutely necessary.) If the brief explanations in this chapter are not sufficient, please go back to your introductory physics textbook and review the material there. The standard introductory physics texts are well written and contain many instructive figures and diagrams. It is a good idea to refer to that text whenever you are exposed to the same material on a more advanced level.

In this chapter you will learn about two different approaches for solving physics problems. ccThese approaches are associated withcca Lagrange and Hamilton, although other physicists anad mathematicians contributed to them, including Leibniz and Euler.

A wave is an oscillation of a medium, such as a water wave in the ocean or a wave propagating down a stretched slinky or a taut string. In general, the medium as a whole does not translate. For example, a wave in the ocean can be thought of as water molecules moving vertically up and down while the wave itself moves horizontally. A strange and interesting thing about waves is that although there is no transport of mass, the wave does transport energy and momentum.

In this chapter we turn our attention to problems that can be solved using the law of conservation of linear momentum. Examples of such problems include the motion of a rocket, collisions in one and two dimensions, and the behavior of a system when an impulsive force acts on it.