For the past seven years as book review editor I have been content to solicit and lightly edit reviews. But I succumbed to the itch to review this book myself.
Its preface promises a conversation with a recognizably human author, an invitation to share his thoughts. The conversation continues as you join him while he expounds engagingly on an abstract subject. His book is aimed at “inductive thinkers” more comfortable with concrete examples than with mathematical abstractions. Such thinkers comprise the bulk of humanity. Abstract thinkers have no reproductive advantage. Their survival in microhabitats called universities is an evolutionary puzzle.
I was hooked at the outset by Figs. 1.2 and 1.3 because I have long bridled over the widespread notion that the litmus test distinguishing scientific brains from scientific barbarians is the question of whether Earth revolves around the Sun or vice versa. To give the “incorrect” answer puts you in the flat-Earth camp. This question reminds me of Banesh Hoffman's arguments in The Tyranny of Testing, in which he lambastes the Educational Testing Service for multiple-choice questions skewed against those whose knowledge is deeper than that of the questioners.
Figure 1.2 shows trajectories of a planet and the star it orbits relative to a distant observer. Despite their complexity, not to say unfamiliarity, Bohn rightly says that this is “a perfectly reasonable and useful way to describe this motion.” He notes that this kind of description was the means for discovering the first exoplanets. Figure 1.3 shows the trajectories relative to the center of mass of the planet-star system. If you hanker after simplicity, the center of mass is the appropriate origin. But one origin of coordinates cannot be right and the other wrong. The choice is governed by taste. I find Fig. 1.2 more piquant. To observe directly the (slightly) elliptical orbit of Earth would require an observer deep within the Sun where the center of mass of the Earth-Sun system resides. Kepler had to infer the orbit from Tycho Brahe's unavoidably geocentric observations. Greatly exaggerated elliptical orbits depicted in textbooks and popularizations devalue Kepler's skill at teasing small eccentricities out of nearly circular orbits. For a refreshingly critical and erudite unsnarling of “The great Copernican cliché,” the confusion between geocentrism and anthropocentrism, see Dennis Danielson's article [Am. J. Phys. 60, 1029–1035 (2001)].
In Chapter 2 Bohn first shows how not to determine the motion of a pendulum bob. In rectangular coordinates, the result is two nonlinear coupled equations that must be solved numerically. And the trajectories and make your head spin, not the pendulum oscillate. A mathematical morass is his tactic to demonstrate the benefits of switching allegiance to polar coordinates rather than mandate them by diktat. This is the point of departure for introducing generalized coordinates and velocities and Lagrange's equations for the pendulum. This segues into generalized momentum and force. A nice touch is recognition that on one side of an equation velocity is treated as a variable and on the other it is a “quantity to be solved for via differential equations. It is a small shift of notation and emphasis, but a useful one to keep in mind.” Here we see the mark of an empathetic teacher: the ability to foresee confusion. This shift could be confusing because it may not be encountered in calculus courses. The stage is now set for Hamilton's equations, trajectories in phase space, and the action of orbits.
Chapter 2 ends with the assertion that it is meant to give students “a flavor of the kinds of things analytical mechanics deals in. If that's all you wanted, maybe you can stop reading at this point. In fact, maybe I should have just written this up as a pamphlet and handed it out at airports.” This chapter does give the flavor of analytical mechanics without force-feeding.
Bohn is a realist about the ideal (“pendulum of the mind”). Theoretical physics can almost be defined as analysis of the nonexistent: frictionless and absolutely smooth surfaces, zero thickness interfaces, materials homogeneous on all scales, plane harmonic waves, processes occurring in zero time, point masses, noninteracting gas molecules and electrons in metals, inviscid air (in which airplanes could not fly), absolutely isolated systems, infinities of all kinds—the list is long. Students need reminders that idealizations are a deal with the devil. When pushed too far they can yield spurious paradoxes. And some physicists earn their paychecks by measuring differences between the real and ideal.
In Chapter 3 Bohn bids farewell to Newton and ushers in d'Alembert, whose eponymous principle is “the cornerstone of analytical mechanics.” Switching founding fathers may be wrenching for some students because problems up to this point could be solved as or more easily à la Newton. Bohn ends with a whiff of Lagrange multipliers (absent from subsequent chapters). This requires faith that multipliers exist that will transform forbidden virtual displacements into allowed ones, but obtaining a previous result may instill confidence in apparent mathematical wizardry.
Continuing with the strategy of crawl, walk, run, Bohn segues from a “proto-Lagrangian” to the real thing by way of a mass moving without acceleration in an inertial coordinate system transformed to polar coordinates. The result is mass times radial acceleration equal to a generalized force, the radial gradient of the kinetic energy. This is the “fictitious” centrifugal force sometimes blamed on transforming to a non-inertial coordinate system. Here we cannot make this force vanish, and now it makes sense why generalized forces are derivatives of the difference in kinetic and potential energies not their sum. Even if the detailed derivation of Lagrange's equations (for conservative forces) requires a student to “take a big aspirin,” they are plausible given what was obtained without the headache. And this is not to be sniffed at. How many students remember detailed derivations? What they need are equations made plausible by physical interpretation, defended only if necessary by heavy mathematical artillery.
Bohn states Hamilton's Principle according to which the action, the path integral of the Lagrangian, is stationary, usually a minimum, and notes that it expresses a philosophical stance about the “elegance in which the world works.” But he stays clear of the teleological stance that Nature performs its daily chores as efficiently as possible. Tell that to the 500 × 106 bipeds who suffer from lower back pain.
Another nice touch is a footnote explaining the origin of the term reduced mass for a two-body system: it is less than either mass. A trifle you might say. Maybe, but attaching meaning to what would otherwise be obscure jargon is an aid to learning. Imparting meaning is a motif of this book.
The extension of Lagrange's equations to include the Lorentz magnetic force is clever and simple. Having previously derived the Coriolis force for a mass in a planar rotating coordinate system, Bohn notes that these forces have the same form, and by analogy makes a plausible guess as to the magnetic addition to the Lagrangian. This does “what it is supposed to do, namely, generate [correct] equations of motion,” and its form derived for a uniform magnetic field holds more generally.
After a previous cameo appearance of Hamilton's equations of motion for a simple pendulum, Bohn eventually switches allegiance from the difference between kinetic and potential energies to their sum. Because he touches on the spinning ice skater problem I digress on what has been treated superficially in countless textbooks. Although the skater's angular momentum is constant its rotational kinetic energy is not. Rarely mentioned is that the energy increase must originate from energy transformations within the skater. For a world record rotational speed of 308 rpm, the estimated energy cost to a skater is about the caloric value of one-tenth of a peanut.
Bohn adopts the ratchet and pawl approach to Hamilton's equations describing the “restless interconversion between potential and kinetic energy.” Students who find the general derivation a bit daunting still will have learned something that need not be unlearned. He asserts that “Knowing which variables you are talking about is the whole thing when it comes to getting Hamilton's equations from Lagrange's.” Amen!
His treatment of Hamilton-Jacobi theory “is the apex of mathematical abstraction in this book.” Despite his admirable efforts to simplify, students may find it hard going, as I did, but I belong to the benighted majority with a limited tolerance for abstraction. For a nearly lethal dose see Aurel Wintner's Analytical Foundations of Celestial Mechanics. All I could tolerate from it without wooziness was that “traditional references to the origin of the fundamental mathematical notions in analytical dynamics are almost always incorrect…the name ‘Hamiltonian equations’ is not correct…‘the Hamilton-Jacobi theory’ is only a particular case of Cauchy’s theory of characteristics, which is of an older date” (p. 413).
As another example of Bohn‘s wry humor, he gives the origin of libration, from the Latin libra for balance scales, and counsels readers to not confuse it with libation. Here is an example in which a dash of humor can cement the meaning of jargon in the minds of readers, a device shunned by the purveyors of rigor (mortis?).
He points out in Chapter 8 that a bounded oscillator acted on by a nonlinear restoring force is periodic with a frequency depending on its energy. But this is the fundamental frequency. The uniqueness of the simple harmonic oscillator is that all its Fourier components vanish except one with a frequency independent of its energy, unlike the complex harmonic oscillator, which is not anharmonic. The motion of any one-dimensional system with two turning points connected by a continuous set of allowed positions is periodic and hence harmonic in that the frequencies of its Fourier components are integral multiples of a fundamental frequency. For more about this see Neville H. Fletcher, “Harmonic? Anharmonic? Inharmonic?,” Am. J. Phys. 70, 1205–1207; Erratum: 71, 492 (2002).
A minor point of confusion is that the Lennard-Jones potential in the text becomes the van der Waals potential in the caption to Fig. 8.3. Absent short range repulsion the former reduces to the latter.
The final chapter on applications begins with “nobody reads all the way to the end of a physics book, so it doesn't really matter what's in here.” “Oh ye of little faith.” This nobody did. Previous theory is applied to standard textbook problems: the trajectory of a projectile in a uniform gravitation field, the orbit of a comet, the normal modes of two coupled oscillators, and finally the double pendulum as a foretaste of chaotic motion, “which requires a whole new set of concepts.”
The list of further reading includes Jerry Ginsberg's (misspelled Ginsburg) Engineering Dynamics. These days some engineering students are taught Lagrangian and Hamiltonian mechanics, unlike in my day as a mechanical engineering student more than half a century ago.
Robert Graves and Alan Hodge in The Reader Over Your Shoulder “suggest that whenever anyone sits down to write he should imagine a crowd of his prospective readers…looking over his shoulder.” The imagined readers looking over Bohn's shoulder are students, not those who might sniff at his rough-and-ready derivations, sense of humor, and conversational style.
Craig Bohren is the author or co-author of several books ranging from popular science, to textbooks, to a technical monograph. He also trains dogs for obedience and agility competition, mobile autonomous trace gas detectors for tracking, and in a former life for Search and Rescue.
