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# The Mathematical Mechanic: Using Physical Reasoning to Solve Problems

MARK LEVI
Pages: 200
Stable URL: http://www.jstor.org/stable/j.ctt7rjgk
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1. Front Matter (pp. i-iv)
3. 1 INTRODUCTION (pp. 1-8)

Back in the Soviet Union in the early 1970s, our undergraduate class—about forty mathematics and physics sophomores—was drafted for a summer job in the countryside. Our job included mixing concrete and constructing silos on one of the collective farms. My friend Anatole and I were detailed to shovel gravel. The finals were just behind us and we felt free (as free as one could feel in the circumstances). Anatole’s major was physics; mine was mathematics. Like the fans of two rival teams, each of us tried to convince the other that his field was superior. Anatole said bluntly...

4. 2 THE PYTHAGOREAN THEOREM (pp. 9-26)

Here is a fact seemingly not worth mentioning for its triviality: Still water in a resting container, with no disturbances, shall remain at rest. I think it is remarkable that this fact has the Pythagorean theorem as a corollary (p. 17). In addition, this seeming triviality implies the law of sines (p. 18), the Archimedian buoyancy law, and the 3D area version of the Pythagorean theorem (p. 19).

The proof of the Pythagorean theorem, described in section 2.2, suggested a kinematic proof of the Pythagorean theorem, described in section 2.6. The motion-based approach makes some other topics very transparent, including...

5. 3 MINIMA AND MAXIMA (pp. 27-75)

Max/min problems tend to be well suited for the physical approach. The reason for this is perhaps the fact many physical systems find maxima or minima automatically: a pendulum finds the minimum of potential energy; the light from a pebble on the bottom of the pool to my retina chooses the path of least time; a soap bubble chooses the shape of least volume; a chain hanging by two ends chooses the shape of lowest center of mass, and so on—the list is endless.

Here is a common pattern in finding a physical solution. Let us say we have...

6. 4 INEQUALITIES BY ELECTRIC SHORTING (pp. 76-83)

The following short outline of the necessary background should suffice for the reading of this chapter. More on the concepts described below is in the appendix.

Electrical current. The current in a copper wire is the flow of the “gas” of electrons in the ionic lattice of copper, analogous to the flow of water in a pipe. Just like the flux of water through the pipe is measured in gallons per second, the electric current is measured in units of charge per second, passing through a cross section of the wire. The current is denoted byIand is expressed...

7. 5 CENTER OF MASS: PROOFS AND SOLUTIONS (pp. 84-98)

The concept of the center of mass was used by Archimedes more than 2,400 years ago.¹ Much later Euler introduced another mass-related concept, that of the moment of inertia (see section A.9), which in turn suggested some very nice solutions of mathematical problems [BB]. Here I solve several other mathematical problems using the center of mass.

Recall that the center of mass of a body is the body’s point of balance; the body suspended on that point is in equilibrium in any orientation. Full details can be found in the appendix (section A.8).

As an interesting aside, we take it...

8. 6 GEOMETRY AND MOTION (pp. 99-108)

Most of the problems in this section rely on the idea of motion. The idea of motion was already used in the section on Pythagorean theorem. In section 2.4 we pointed out that the fundamental theorem of calculus can be thought of in kinematic terms. In this section I collected a few other problems, of which I like the bike problem the best. A beautiful application of the idea of motion, which allows to find the area under the tractrix with no formulas, due to R. Foote [Fo], is stated in section 6.6 as a problem. Another problem at the...

9. 7 COMPUTING INTEGRALS USING MECHANICS (pp. 109-114)

The first two problems in this section are easy to do with calculus and without mechanics. The point here is to illustrate how the “thinking” that the calculus machinery does for us can sometimes be done by a mechanical “analog computer.”

A weightP= 1, mounted on a frictionless vertical track, hangs on a string of length 1. The string is vertical initially. As the top end of the string is moved horizontally from its initial position, the weight slides upward along the vertical line. In changing the displacementxof the top end of the string fromx...

10. 8 THE EULER-LAGRANGE EQUATION VIA STRETCHED SPRINGS (pp. 115-119)

This short chapter contains a purely mechanical interpretation of the Euler-Lagrange functional as the potential energy of an imaginary spring. This interpretation makes for an almost immediate derivation of the Euler-Lagrange equations and gives a very transparent mechanical explanation of the conservation of energy. Moreover, each individual term in the Euler-Lagrange equation acquires a concrete mechanical meaning.

Here is some motivation for the reader not familiar with the Euler-Lagrange equations.

A basic problem of the calculus of variations is to find a functionx(t) which minimizes an integral involvingxand its derivative$\dot x$:

$\smallint _0^1 L(x(t),\dot x(t))dt, \caption{(8.1)}$

whereLis a...

11. 9 LENSES, TELESCOPES, AND HAMILTONIAN MECHANICS (pp. 120-132)

The central point of this chapter is a very simple hand-waving (in a literal sense) argument in mechanics in section 9.3. This simple mechanical argument has rather unexpected consequences in mathematics and in optics.¹ Thanks to the mechanical interpretation, some of these consequences, usually discussed only in graduate courses, become much more accessible.

Here is the plan of the chapter. Section 9.1 contains the background; sections 9.3 and 9.2 describe the mechanical system and give a mechanical proof of a geometrical theorem on area preservation. Section 9.7 connects the mechanical/geometrical problem with an optical one, and the last section (9.8)...

12. 10 A BICYCLE WHEEL AND THE GAUSS-BONNET THEOREM (pp. 133-147)

This chapter tells an interesting story on how playing with a bicycle wheel can connect to a fundamental theorem from differential geometry. The internal angles in a planar triangle add up to 180°. This fact can be restated in a more general and yet more basic way: if I walk around a closed curve in the plane, then my nose, treated as a vector, will rotate by 2π(provided that I always look straight ahead).¹

Does the same hold for an inhabitant of a curved surface? Figure 10.1 shows a triangular path on the sphere. Two of the sides lie...

13. 11 COMPLEX VARIABLES MADE SIMPLE(R) (pp. 148-160)

In this section I present some theory of complex variables, with physical insight but without rigorous proofs. No prior exposure to the theory of complex variables is assumed. One idea, used in about half of the chapter, links any complex function with an idealized fluid flow in the plane (the details are in section 11.3). With this compact idea some of the basic facts of the theory become intuitively obvious.

The first section on complex numbers requires little background. The rest of the chapter should be accessible to anyone who saw line integrals. The concepts of the divergence and curl...

14. APPENDIX. PHYSICAL BACKGROUND (pp. 161-182)
15. BIBLIOGRAPHY (pp. 183-184)
16. INDEX (pp. 185-186)