In the preceding chapter on rigid-body motion we took a step beyond single-particle mechanics to explore the behavior of a more complex system containing many particles bonded rigidly together. Now we will explore additional sets of many-particle systems in which the individual particles are connected by linear, Hooke’s-law springs. These have some interest in themselves, but more generally they serve as a model for a large number of coupled systems that oscillate harmonically when disturbed from their natural state of equilibrium, such as elastic solids, electric circuits, and multi-atom molecules. We will begin with the oscillations of a few coupled masses and end with the behavior of a continuum of masses described by a linear mass density. The mathematical techniques required to analyze such coupled oscillators are used throughout physics, including linear algebra and matrices, normal modes, eigenvalues and eigenvectors, and Fourier series and Fourier transforms.

A review of Green's theorem, Stokes's theorem, and the divergence theorem.

A notable advance in mechanics took place nearly a century after Newton in the work of the French mathematician and physicist Joseph-Louis Lagrange (1736--1813). We already introduced variational principles in Chapter 3, and showed that they can give the equations of motion of nonrelativistic particles subject to arbitrary conservative forces. Although useful, that is not enough: now we need to see if other kinds of forces can be included in the variational approach. In this chapter we will find that forces of constraint can be included as well, which provides us with deep insights into mechanics and also enormous simplifications in problem solving. We will introduce Hamilton’s principle and the Lagrangian, concepts that are so elegant that we are encouraged to place them at the very heart of classical mechanics. We are further encouraged to do so in the following chapter, the capstone chapter to Part I of the book, where we show how they naturally emerge as we take the classical limit of the vastly more comprehensive theory of quantum mechanics.

In this second capstone chapter, we extend some of the classical mechanics from the preceding four chapters into the context of more recent developments in physics. We begin with gravitation, including some of the ideas that led Einstein to go way beyond Newton’s nonrelativistic theory to find a fully relativistic theory of gravitation. After years of strenuous effort, his work finally culminated in his stunningly original and greatest single achievement, the general theory of relativity. He was able to predict three effects that could be measured in the solar system, which he used to check his theory. We will cover all three of these. Then we will introduce so-called “magnetic gravity,” which contains the leading terms in general relativity in a form much like Maxwell’s equations for electromagnetism. Next, we delve just a bit deeper into gauge symmetry in Maxwell’s theory, partly because it deepens our understanding of electromagnetism but also because gauge symmetry has played such a large role in physical theories over the past many decades. Finally, we introduce stochastic forces, which are not fundamental forces but the result of huge numbers of small collisions.

One truism about narrative is that it is a way we have of knowing ourselves. What are we, after all, if not characters? That is, we seem to be characters, and characters are one of the two principal components in most fictional stories, the other being the action. An extreme position is that we only know ourselves insofar as we are narrativized. Whether or not this is true, it is certainly the case that, when asked to describe someone beyond what they look like or their type (she's an angel, he's a toad), we begin to narrate.

Simply put, narrative is the representation of an event or a series of events. “Event” is the key word here, though some people prefer the word “action.” Without an event or an action, you may have a “description,” an “exposition,” an “argument,” a “lyric,” some combination of these, or something else altogether, but you won't have a narrative.