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.

Here, we add damping to the harmonic oscillator, and explore the role of the resulting new time scale in the solutions to the equations of motion.Specifically, the ratio of damping to oscillatory time scale can be used to identify very different regimes of motion: under-, critically-, and over-damped.Then driving forces are added, we consider the effect those have on the different flavors of forcing already in place.The main physical example (beyond springs attached to masses in dashpots) is electrical, sinusoidally driven RLC circuits provide a nice, experimentally accessible test case.On the mathematical side, the chapter serves as a thinly-veiled introduction to Fourier series and the Fourier transform.

A review of the basic strategy of ODE solving in a step-by-step way --- what guesses to try, and when, with references to the motivating solutions in the text.

A review of basic vector calculus expressions, like the gradient, divergence, curl and Laplacian, in cylindrical, spherical and more general coordinate systems.

Taking the continuum limit of the chains of masses from the previous chapter, we arrive at the wave equation, the physical subject of this chapter.The connection to approximate string motion is an additional motivation.Viewed as a manifestation of a conservation law, the wave equation can be extended to other conservative, but nonlinear cases, like traffic flow.Mathematically, we are interested in turning partial differential equations (PDEs) into ODEs, making contact with some familiar examples.Making PDEs into ODEs occurs in a couple of ways --- the method of characteristics, and additive/multiplicative separation of variables are the primary tools.

We turn next to the case of additional masses.In one dimension, we can attach masses by springs to achieve collective motions that occur at a single frequency, the normal modes.Building general solutions, using superposition, from this ``basis" of solutions is physically relevant and requires a relatively formal treatment of linear algebra, the mathematical topic of the chapter.

This chapter can be used as a six week “lab" component to a mathematical methods course, one section each week.The chapter is relatively self-contained, and consists of numerical methods that complement the analytic solutions found in the rest of the book.There are methods for solving ODE problems (both in initial and boundary value form) approximating integrals, and finding roots.There is also a discussion of the eigenvalue problem in the context of approximate solutions in quantum mechanics and a section on the discrete Fourier transform.

This chapter extends the discussion of waves beyond the longitudinal oscillations with which we began.Here, we look at the wave equation as it arises in electricity and magnetism, in Euler's equation and its shallow water approximation, in ``realistic" (extensible) strings, and in the quantum mechanical setting, culminating in a quantum mechanical treatment of the book's defining problem, the harmonic oscillator.

With many physical applications already on the table, in this chapter, we return to some of the simplified ones and re-complexify them.These problems require more sophisticated, and incomplete, solutions.Instead of finding the position of the bob for the simple pendulum, we find the period of motion for the ``real" pendulum.Instead of the classical harmonic oscillator, with its familiar solution, we study the period of the relativistic harmonic oscillator, and find that in the high energy limit, a mass attached to a spring behaves very differently from its non-relativistic counterpart.