Acetylcholine and Loewi’s Dream

In 1936, Otto Loewi was awarded the Nobel Prize in Physiology or Medicine for the discovery of the role of acetylcholine and other chemicals in nerve and muscle transmission. Russ Hobbie and I don’t mention Loewi in the 4th edition of Intermediate Physics for Medicine and Biology, but we do discuss acetylcholine. In Chapter 6, we write

At the end of a nerve cell the signal passes to another nerve cell or to a muscle cell across a synapse or junction. A few synapses in mammals are electrical; most are chemical…In electrical synapses, channels connect the interior of one cell with the next. In the chemical case a neurotransmitter chemical is secreted by the first cell. It crosses the synaptic cleft (about 50 nm) and enters the next cell.

At the neuromuscular junction the transmitter is acetylcholine (ACh). ACh increases the permeability of nearby muscle to sodium, which then enters and depolarizes the muscle membrane. The process is quantized. Packets of acetylcholine of definite size are liberated.

In Homework Problem 20 in Chapter 4, we ask the student to calculate the time required for acetylcholine to diffuse across the synaptic cleft. The release of acetylcholine at the nerve-muscle junction in discrete quanta provides a nice example of Poisson Statistics described in Appendix J. In Chapter 7, when discussing the heart, we mention how acetylcholine, released by parasympathetic nerves, decreases the heart rate.

The Left Hand of the Electron,
by Isaac Asimov.

I can’t tell you about Otto Loewi and acetylcholine without mentioning the fascinating tale of Loewi’s dream. Since Isaac Asimov is a much better storyteller than I am, I will simply quote from his essay “The Eureka Phenomenon” published in The Left Hand of the Electron.

The German physiologist Otto Loewi was working on the mechanism of nerve action, in particular, on the chemicals produced by nerve endings. He woke at 3 A.M. one night in 1921 with a perfectly clear notion of the type of experiment he would have to run to settle a key point that was puzzling him. He wrote it down and went back to sleep. When he woke in the morning, he found he couldn't remember what his inspiration had been. He remembered he had written it down, but he couldn't read his writing.

The next night, he woke again at 3 A.M. with the clear thought once more in mind. This time, he didn't fool around. He got up, dressed himself, went straight to the laboratory and began work. By 5 A.M. he had proved his point and the consequences of his findings became important enough in later years so that in 1936 he received a share in the Nobel prize in medicine and physiology.

Viral Outbreak: The Science of Emerging Disease

Textbooks such as the 4th edition of Intermediate Physics for Medicine and Biology are essential for studying and learning a new topic, but other ways of learning can be equally effective (or, sometimes, even better). Today I want to mention two examples.

Each December, the Howard Hughes Medical Institute presents its Holiday Lectures on Science. These excellent seminars will be webcast live on December 2 and 3, starting at 10 A.M. This year, the lectures are about “Viral Outbreak: The Science of Emerging Disease.” Joseph DeRisi (University of California, San Francisco) and Eva Harris (University of California, Berkeley) will explain how to detect and fight infectious agents. The lectures will answer questions such as “Why is dengue fever becoming a worldwide health threat,” “What other epidemics are on the horizon,” and “How can we detect and counter emerging infectious diseases?” If you miss the live webcast, you can download an on-demand webcast starting December 6. I have watched these holiday lectures in the past, and they are very good. They are aimed at a serious high school student, or an undergraduate science major. They are also great for a physicist looking for a general introduction to a biological or medical topic.

Of course, the best way to learn science is to do science. For undergraduates (the main readers of Intermediate Physics for Medicine and Biology), the first exposure to doing science may come during a summer research project. Now is the time to start looking for summer research opportunities. One that I recommend is the Summer Internship Program in Biomedical Science at the National Institutes of Health. I worked at NIH for seven years, and it is a wonderful place to do scientific research. My advice is to apply for this internship today. You won’t regret it.

Bionumbers

One feature of the 4th edition of Intermediate Physics for Medicine and Biology that distinguishes it from many other medical or biological textbooks is its focus on analyzing biomedical topics quantitatively. This point of view is also promoted at the BIONUMB3R5 (bionumbers) website, established by researchers in the systems biology department at Harvard. There is also a BIONUMB3R5 wiki where many researchers are coming together to provide new insights into key numbers in biology.

I particularly like the “Bionumber of the Month” feature. The March 2010 entry (“What are the Time Scales for Diffusion in Cells”) could easily be made into a homework problem for Chapter 4 of Intermediate Physics for Medicine and Biology. The January 2010 entry (“What is Faster, Transcription or Translation?”) is fascinating.

Transcription, the synthesis of mRNA from DNA, and translation, the synthesis of protein from mRNA, are the main pillars of the central dogma of molecular biology. How do the speeds of these two processes compare? …

Transcription of RNA by RNA polymerase in E. coli cells proceeds at a maximal speed of about 40–80 bp/sec… Translation by the ribosome in E. coli proceeds at a maximal speed of about 20 aa/sec… Interestingly, since every 3 base pairs code for one amino acid, the rates of the two processes are quite similar…

The “collection of fundamental numbers in molecular biology” found at the bionumbers website has the same tone as the first section of Chapter 1 in Intermediate Physics for Medicine and Biology, in which Russ Hobbie and I look at the relative size of biological objects. The collection contains this gem: “concentration of 1 nM in a cell the volume of E. coli is ~ 1 molecule/cell.”

The bionumbers website arose from an article by Rob Phillips and Ron Milo in the Proceedings of the National Academy of Sciences (Volume 106, pages 21465–21471, 2009), “A Feeling for the Numbers in Biology.” The abstract of their paper is given below:

Although the quantitative description of biological systems has been going on for centuries, recent advances in the measurement of phenomena ranging from metabolism to gene expression to signal transduction have resulted in a new emphasis on biological numeracy. This article describes the confluence of two different approaches to biological numbers. First, an impressive array of quantitative measurements make it possible to develop intuition about biological numbers ranging from how many gigatons of atmospheric carbon are fixed every year in the process of photosynthesis to the number of membrane transporters needed to provide sugars to rapidly dividing Escherichia coli cells. As a result of the vast array of such quantitative data, the BioNumbers web site has recently been developed as a repository for biology by the numbers. Second, a complementary and powerful tradition of numerical estimates familiar from the physical sciences and canonized in the so-called “Fermi problems” calls for efforts to estimate key biological quantities on the basis of a few foundational facts and simple ideas from physics and chemistry. In this article, we describe these two approaches and illustrate their synergism in several particularly appealing case studies. These case studies reveal the impact that an emphasis on numbers can have on important biological questions.

Russ and I introduce similar order-of-magnitude estimates (Fermi problems) in Chapter 1 of our book (for example, see homework problems 1–4, which are new in the 4th edition). One of my favorite Fermi problems, which I first encountered in the book Air and Water by Mark Denny, is to calculate the concentration of oxygen molecules in blood and in air, and compare them. Not too surprisingly, they are nearly the same (about 8 mM). I suspect the bionumbers folks would enjoy Air and Water. (I hope they would enjoy Intermediate Physics for Medicine and Biology, too.)

For those of you who find all of this interesting but prefer video over text, see the bionumbers video on YouTube.

Bionumbers: The data base of useful biological numbers. 

https://www.youtube.com/embed/psE0H1ejxKE

Seeing the Natural World with a Physicist’s Lens

One theme of this blog—and indeed, one theme of the 4^th edition of Intermediate Physics for Medicine and Biology—is the role of physics in the biological sciences. So imagine my delight when Russ Hobbie sent me a similarly themed article from the November 1 issue of the New York Times (a publication that, alas, has more readers than does my blog). Natalie Angier, who studied for two years at that little college down the road in Ann Arbor, wrote an article titled “Seeing the Natural World With a Physicist’s Lens.” Its thesis is that many biological systems have evolved to perfection, in the sense that physical laws don’t let them get any better. Angier writes

Yet for all these apparent flaws, the basic building blocks of human eyesight turn out to be practically perfect. Scientists have learned that the fundamental units of vision, the photoreceptor cells that carpet the retinal tissue of the eye and respond to light, are not just good or great or phabulous at their job. They are not merely exceptionally impressive by the standards of biology, with whatever slop and wiggle room the animate category implies. Photoreceptors operate at the outermost boundary allowed by the laws of physics, which means they are as good as they can be, period. Each one is designed to detect and respond to single photons of light—the smallest possible packages in which light comes wrapped…

Photoreceptors exemplify the principle of optimization, an idea, gaining ever wider traction among researchers, that certain key features of the natural world have been honed by evolution to the highest possible peaks of performance, the legal limits of what Newton, Maxwell, Pauli, Planck et Albert will allow. Scientists have identified and mathematically anatomized an array of cases where optimization has left its fastidious mark… In each instance, biophysicists have calculated, the system couldn’t get faster, more sensitive or more efficient without first relocating to an alternate universe with alternate physical constants.

Angier has written a lot of articles for the NYT, and has published several books, that will be of interest to readers of Intermediate Physics for Medicine and Biology. Enjoy!

Iatrogenic Problems in End-Stage Renal Failure

In Section 5.7 of the 4th edition of Intermediate Physics for Medicine and Biology, where Russ Hobbie and I discuss the artificial kidney, we say

The artificial kidney provides an example of the use of the transport equations to solve an engineering problem… The reader should also be aware that this “high-technology” solution to the problem of chronic renal disease is not an entirely satisfactory one. It is expensive and uncomfortable and leads to degenerative changes in the skeleton and severe atherosclerosis…

The alternative treatment, a transplant, has it own problems, related primarily to the immunosuppressive therapy. Anyone who is going to be involved in biomedical engineering or in the treatment of patients with chronic disease should read the account by Calland (1972), a physician with chronic renal failure who had both chronic dialysis and several transplants.

The paper by Chad Calland, in the New England Journal of Medicine (“Iatrogenic Problems in End-Stage Renal Failure,” Volume 287, Pages 334–336, 1972), was published on the same day that Calland took his own life. Wikipedia defines “iatrogenic” as “inadvertent adverse effects or complications caused by or resulting from medical treatment or advice.” It is a problem we must constantly be aware of as we seek to improve medical care through technology. Calland wrote

The physician is more often a voyeur than a partaker in human suffering. I am a physician who has undergone chronic renal failure, dialysis and multiple transplants. As a physician-partaker, I am distressed by the controversial dialogue that separates the nephrologist from the transplant surgeon, so that, in the end, it is the patient who is given short shrift. I have observed that both nephrologist and transplant surgeon work alone in their own separate fields, and that the patient becomes lost in a morass of professional role playing and physician self-justification. As legitimate as their altruistic but differing opinions may be, the nephrologist and the transplant surgeon must work together for the patient, so that therapy is tailored to suit the individual patient, his circumstances, his needs and the quality of his life.

Glimpses of Creatures in Their Physical Worlds

Glimpses of Creatures
in their Physical Worlds,
by Steven Vogel.

I am a loyal member of Sigma Xi, the Scientific Research Society, and am a regular reader of its marvelous magazine American Scientist. One of the best parts of this bimonthly periodical is its book reviews. In the November-December 2010 issue of American Scientist, Mark Denny (author of Air and Water) reviews the new book by Steven Vogel: Glimpses of Creatures in Their Physical Worlds (Princeton University Press, 2009). Both Denny and Vogel appear in the 4th edition of Intermediate Physics for Medicine and Biology. Denny writes

Vogel’s contributions to biomechanics have had two admirable objectives. In Life in Moving Fluids (1981), Life’s Devices (1988), Vital Circuits (1992), Prime Mover (2001) and Comparative Biomechanics (2003), his goal is to explain the mechanics of biology to a general audience. If you want to know how fish swim, fleas jump and bats fly, or why hardening of your arteries is a bad thing, them dip into these sources; you will come away both informed and amused…

All too often, biologists observe only what they are prepared to see. Vogel’s second objective is therefore to expand their perspectives by conjuring up and carefully analyzing systems that might be… For example, dogs don’t sweat as humans do. Instead, they pant, evaporating water from their respirator tracts and expelling the resulting warm, moist air with each breath. But panting requires the repeated contraction of chest muscles, which adds to the heat the animal desires to loss. Could there be a better way?...

To find out, read Glimpses of Creatures in Their Physical Worlds. Here, as in Cats’ Paws and Catapults (1998), Vogel takes a decidedly nontraditional look at biology, unleashing his talent for unbridled speculation. The 12 chapters of Glimpses, which began as a series of essays in the Journal of Biosciences, have been revised and updated. They cover topics that range from the ballistics of seeds (plants use both catapults and cannons to launch their propagules) to the breathing apparatus of diving spiders (tiny hairs on the body take advantage of surface tension to maintain an airspace into which oxygen can flow), with stops along the way to explore the efficiency of man-made and natural pumps, the twist-to-bend ratios of daffodils in the breeze, and the physics of cow tipping…

If what you desire in a readable science book is food for thought, Glimpses of Creatures in Their Physical Worlds provides a feast. Biologists, engineers and physicists—indeed, anyone with curiosity about the natural world—will revel in this smorgasbord of biomechanical ideas.

I’ll put reading Glimpses on my to do list, maybe during the semester break.

If you get a copy of American Scientist so you can read Denny’s entire review, don’t miss another review in the same issue about a new edition (with notes and commentary) of the classic Flatland by Edwin Abbott. Flatland is a favorite of mine, and I agree with Colin Adams who says in his review: “In the pantheon of popular books about mathematics, one would be hard-pressed to name another that has lasted so long in popularity or had such a dramatic impact.”

Michael Faraday, Biological Physicist?

Last week in this blog I discussed the greatest physicist of all time, Isaac Newton. However, if we narrow consideration to only experimental physicists, I would argue that the greatest is Michael Faraday (with apologies to Ernest Rutherford, who is a close second). In Section 8.6 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Faraday’s greatest discovery: electromagnetic induction.

In 1831 Faraday discovered that a changing magnetic field causes an electric current to flow in a circuit. It does not matter whether the magnetic field is from a permanent magnet moving with respect to the circuit or from the changing current in another circuit. The results of many experiments can be summarized in the Faraday induction law.

I have always admired the 19th century Victorian physicists, such as Faraday, Maxwell and Kelvin. Michael Faraday, in particular, is a hero of mine (it is good to have heroes; they help you stay inspired when the mundane chores of life distract you). I had the pleasure of quoting from Faraday’s Experimental Researches in Electricity in an editorial I wrote in 2005 for the journal Heart Rhythm:  “Michael Faraday and Painless Defibrillation.” I tried to get a picture of Faraday included as part of the editorial, but alas the journal editor removed it. The article described a heart defibrillator having a design that included a type of Faraday cage.

Michael Faraday, arguably the greatest experimental physicist who ever lived, first demonstrated the shielding effect of a hollow conductor in 1836 by building a 12 ft × 12 ft × 12 ft cubic chamber out of metal. We would now call it a “Faraday cage.”

“I went into the cube and lived in it, and using lighted candles, electrometers, and all other tests of electrical states, I could not find the least influence upon them, or indication of anything particular given by them, though all the time the outside of the cube was powerfully charged, and large sparks and brushes were darting off from every part of its outer surface.” [Faraday M. Experimental Researches in Electricity. Paragraph 1174. Reprinted in: Hutchins RM, editor. Great Books of the Western World, Volume 45. Encyclopedia Britannica, Chicago, 1952.]

Faraday cages are used to shield sensitive electronic equipment. The metal skin of an airplane, acting as a Faraday cage, protects passengers from injury by lightning. Researchers perform electrophysiology experiments inside a Faraday cage to prevent external noise from contaminating the data. A rather spectacular example of shielding can be seen in the Boston Museum of Science, where a van de Graaff generator of over one million volts produces a dramatic display of lightning, while the operator stands nearby—safe inside a Faraday cage.

Why this little physics lesson? In this issue of Heart Rhythm, Jayam et al. [Jayam V, Zviman M, Jayanti V, Roguin A, Halperin H, Berger RD. “Internal Defibrillation with Minimal Skeletal Muscle Activation: A New Paradigm Toward Painless Defibrillation,” Heart Rhythm, Volume 2, Pages 1108–1113, 2005] describe a new electrode system for internal defibrillation that eliminates the skeletal muscle activation and pain associated with a shock. The central feature of their design is a Faraday cage: a conducting sock fitted over the epicardial surface of the heart…

In Section 8.7, Russ and I describe what may be the most important biomedical application of Faraday’s work: magnetic stimulation.

Since a changing magnetic field generates an induced electric field, it is possible to stimulate nerve or muscle cells without using electrodes. The advantage is that for a given induced current deep within the brain, the currents in the scalp that are induced by the magnetic field are far less than the currents that would be required for electrical stimulation. Therefore transcranial magnetic stimulation (TMS) is relatively painless. Magnetic stimulation can be used to diagnose central nervous system diseases that slow the conduction velocity in motor nerves without changing the conduction velocity in sensory nerves [Hallett and Cohen (1989)]. It could be used to monitor motor nerves during spinal cord surgery, and to map motor brain function. Because TMS is noninvasive and nearly painless, it can be used to study learning and plasticity (changes in brain organization over time). Recently, researchers have suggested that repetitive TMS might be useful for treating depression and other mood disorders.

I worked on magnetic stimulation for many years while at the National Institutes of Health in the 1990s. It was a pleasure to explore an application of Faraday induction; it is my kind of biological physics.

Faraday’s name can be found in a few other places in our book. It first appears in Chapter 3, when the Faraday constant is defined: F = 96,485 Coulombs per mole. It also appears in an abbreviated form in the unit of capacitance: a farad (F).

I suppose by now the reader realizes that I like Mike. But is he a biological physicist? Doubters might want to look at another physics blog: http://skullsinthestars.com/2010/05/15/shocking-michael-faraday-does-biology-1839. Faraday apparently did studies on the electrodynamics of electric fish. So, yes, I claim him as a biological physicist, and the question mark in the title of this blog post is unnecessary.

Isaac Newton, Biological Physicist?

Arguably the greatest physicist of all time (and probably the greatest scientist of all time) is Isaac Newton (1643–1727). Newton is so famous that the English put him on their one pound note (although I gather nowadays they use a coin instead of paper currency for one pound). Given Newton’s influence, it is fair to ask what his role is in the 4th edition of Intermediate Physics for Medicine and Biology. One way Newton (along with Leibniz) contributes to nearly every page of our book is through the invention of calculus (or, as I prefer, “the calculus”). Russ Hobbie states in the preface of our book that “calculus is used without apology.”

When I search the book for Newton’s name, I find quite a few references to Newton’s laws of motion, and in particular the second law, F = ma. Newton presented his three laws in his masterpiece, the Principia (1687). (Few people have read the Principia, including me, but a good place to learn about it is the book Newton’s Principia for the Common Reader by Subrahmanyan Chandrasekhar) Of course, the unit of force is the newton, so his name pops up often in that context. The only place where we talk about Newton the man is very briefly in the context of light.

A controversy over the nature of light existed for centuries. In the seventeenth century, Sir Isaac Newton explained many properties of light with a particle model. In the early nineteenth century, Thomas Young performed some interference experiments that could be explained only by assuming that light is a wave. By the end of the nineteenth century, nearly all known properties of light, including many of its interactions with matter, could be explained by assuming that light consists of an electromagnetic wave.

Newton’s name also arises when talking about Newtonian fluids (Chapter 1): a fluid in which the shear stress is proportional to the velocity gradient. Not all fluids are Newtonian, with blood being one example. Newton appears again when discussing Newton’s law of cooling (Chapter 3, Problem 45).

Some of Newton’s greatest discoveries are not addressed in our book. For instance, Newton’s universal law of gravity is never mentioned. Except for a few intrepid astronauts, animals live at the surface of the earth where gravity is simply a constant downward force and Newton’s inverse square law is not relevant. I suppose tides influence animals and plants that live near the ocean shore, and the behavior of tides is a classic application of Newtonian gravity, but we never discuss tides in our book. (By the way, harkening back to my vacation in France last summer, the tides at Mont Saint Michel are fascinating to watch. I really must plan a trip to the Bay of Fundy next.) Newton, in his book Optiks, made important contributions to our understanding of color, but Russ and I introduce that subject without referring to him. We don’t discuss telescopes in our book, and thus miss a chance to honor Newton for his invention of the reflecting telescope.

Never at Rest:
A Biography of Isaac Newton,
by Richard Westfall.

A wonderful biography of Newton is Never at Rest, by Richard Westfall. I must admit, Newton is a strange man. His argument with Leibniz about the invention of calculus is perhaps the classic example of an ugly priority dispute. He does not seem to be particularly kind or generous, despite his undeniable genius.

Was Newton a biological physicist? Well, that may be a stretch, but Colin Pennycuick has written a book titled Newton Rules Biology, so we cannot deny his influence. I would say that Newton’s contributions are so widespread and fundamental that they play an important role in all subfields of physics.

Ultraviolet Light Causes Skin Cancer

The New England Journal of Medicine is arguably the premier medical journal in the world. Russ Hobbie is a regular reader, and he sometimes calls my attention to articles that are closely related to topics in the 4th edition of Intermediate Physics for Medicine and Biology. The September 2, 2010 issue of the NEJM contains the article “Indoor Tanning—Science, Behavior, and Policy” (Volume 363, Pages 901–903), by David Fisher and William James. The article begins

The concern arises from increases in the incidence of melanoma and its related mortality. In the United States, the incidence of melanoma is increasing more rapidly than that of any other cancer. From 1992 through 2004, there was a particularly alarming trend in new melanoma diagnoses among girls and women between the ages of 15 and 39. Data from the National Cancer Institute’s Surveillance, Epidemiology, and End Results Registry show an estimated annual increase of 2.7% in this group. Researchers suspect that the increase results at least partially from the expanded use of tanning beds.

Russ and I discuss ultraviolet light in Section 14.9 of Intermediate Physics for Medicine and Biology. In particular, Section 14.9.4 is titled Ultraviolet Light Causes Skin Cancer.

Chronic exposure to ultraviolet radiation causes premature aging of the skin. The skin becomes leathery and wrinkled and loses elasticity. The characteristics of photoaged skin are quite different from skin with normal aging [Kligman (1989)]. UVA radiation was once thought to be harmless. We now understand that UVA radiation contributes substantially to premature skin aging because it penetrates into the dermis. There has been at least one report of skin cancer associated with purely UVA radiation from a cosmetic tanning bed [Lever and Lawrence (1995)]. This can be understood in the context of studies showing that both UVA and UVB suppress the body’s immune system, and that this immunosuppression plays a major role in cancer caused by ultraviolet light [Kripke (2003); Moyal and Fourtanier (2002)]. There are three types of skin cancer. Basal-cell carcinoma (BCC) is most common, followed by squamous cell carcinoma (SCC). These are together called nonmelanoma or nonmelanocytic skin cancer (NMSC). Basal-cell carcinomas can be quite invasive (Fig. 16.44) but rarely metastasize or spread to distant organs. Squamous-cell carcinomas are more prone to metastasis. Melanomas are much more aggressive and frequently metastasize.

The Skin Cancer Foundation advocates vigorously for the reduction of indoor tanning, and the American Association for Cancer Research has also spoken out against tanning beds. The problem seems to be growing.

Fisher and James conclude their article

An estimated six of every seven melanomas are now being cured, thanks to early detection, but the U.S. Preventive Services Task Force does not recommend skin-cancer screening, since the evidence for its benefit has not been validated in large, prospective, randomized trials. Meanwhile, a number of promising new drugs for metastatic melanoma are progressing slowly through clinical trials to satisfy the FDA’s stringent safety and efficacy criteria—requirements that, remarkably, have not been applied to indoor tanning devices. Relatively few human cancers are tightly linked to a known environmental carcinogen. Given the mechanistic and epidemiologic data, we believe that regulation of this industry may offer one of the most profound cancer-prevention opportunities of our time.

Adrien-Marie Legendre

On page 181 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce Legendre polynomials. The Legendre polynomial P₂(cos(θ)) arises naturally when calculating the extracellular potential in a volume conductor at a position far from an active nerve axon. We include the footnote “You can learn more about Legendre polynomials in texts on differential equations or, for example, in Harris and Stocker (1998).” On page 184, we list the first four Legendre polynomials (and have another footnote referring to Harris and Stocker). Any physics student should memorize at least the first three of these polynomials:

P₀(x) = 1

P₁(x) = x

P₂(x) = (3 x² – 1)/2 .

Legendre polynomials have many interesting properties. They are a solution of Legendre’s differential equation

(1–x²) d²P_n/dx² – 2 x dP_n/dx + n(n+1) P_n = 0 .

You can calculate any Legendre polynomial using Rodrigues formula

P_n(x) = 1/(2ⁿ n!) dⁿ((x²–1)ⁿ)/dxⁿ .

They form an orthogonal set of functions for x over the range from −1 to 1, which is rather too technical of a property to explain in this blog entry, but is very important.

The astute reader might note that Legendre’s differential equation is second order, so there should be two solutions. That is right, but the other solution—called a Legendre function of the second kind, Q_n—is rarely used, and tends to be poorly behaved at x = 1 and x = –1. For instance

Q₀(x) = ½ ln((1+x)/(1–x)) .

A definitive source for information about Legendre polynomials is the Handbook of Mathematical Functions, by Milton Abramowitz and Irene Stegun.

When do Legendre’s polynomials appear in physics? You often find them when working in spherical coordinates, especially when (to use an analogy with the earth) a function depends on latitude but not longitude (axisymmetry). For instance, the general axisymmetric solution to Laplace’s equation in spherical coordinates is a series of powers of the radius r multiplied by Legendre polynomials with x = cos(θ), where θ is measured from the z-axis (or, to use the earth analogy again, from the north pole). Take an introductory class in electricity and magnetism (from, say, the book by Griffiths), and you will use Legendre polynomials all the time.

Why do I bring up Legendre polynomials today? Regular readers of this blog may recall my recent obsession with all things French. Adrien-Marie Legendre (1752–1833) was a French mathematician. Details of his life are given in A Short Account of the History of Mathematics, by Rouse Ball.

Adrian Marie Legendre was born at Toulouse on September 18, 1752, and died at Paris on January 10, 1833. The leading events of his life are very simple and may be summed up briefly. He was educated at the Mazarin College in Paris, appointed professor at the military school in Paris in 1777, was a member of the Anglo-French commission of 1787 to connect Greenwich and Paris geodetically; served on several of the public commissions from 1792 to 1810; was made a professor at the Normal school in 1795; and subsequently held a few minor government appointments. The influence of Laplace was steadily exerted against his obtaining office or public recognition, and Legendre, who was a timid student, accepted the obscurity to which the hostility of his colleague condemned him.

Legendre’s analysis is of a high order of excellence, and is second only to that produced by Lagrange and Laplace, though it is not so original. His chief works are his Géométrie, his Théorie des nombres, his Exercices de calcul intégral, and his Fonctions elliptiques. These include the results of his various papers on these subjects. Besides these he wrote a treatise which gave the rule for the method of least squares, and two groups of memoirs, one on the theory of attractions, and the other on geodetical operations.