Classical Mechanics

The Laplace-Runge-Lenz Vector

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Greetings from The Amaz!ng Meeting!

This is the happiest I have yet been, here in the city I am learning to hate, the city with nothing to offer me and nothing to enjoy — except Joshua, Rebecca, PZ, Phil and nine hundred other friends. Yes, Gentle Reader, you know me well enough to guess that my most truly recreational experience here in Las Vegas has been sleeping late, eating a Toblerone in bed, tidying up a blag post from my drafts pile, and missing Michael Shermer’s talk. Joshua just “texted” me, as the kids say, with the following message: “Ok. Shermer needs his Powerpoint privileges revoked. How many text-dump slides is he going to use?”

Call me psychic, Gentle Reader, or at least give me a little credit for remembering what I read. . . .

Anyway, because I’m here to have fun, it’s time to talk physics. With equations.

When we studied the hydrogen atom, we found that an interaction potential which fell off inversely with distance was shape-invariant, implying all sorts of nice symmetry properties of the hydrogen atom’s state space. The classical analogue of this situation would be two objects interacting via an inverse-square force (remember that force is given by the derivative of the potential). Having recently taken a rather madcap tour of the history of classical mechanics, we can probe a little more deeply and investigate one item in more technical detail. Today’s subject will be defining and appreciating the Laplace-Runge-Lenz vector, which as we said earlier was not discovered by Laplace, Runge or Lenz. After finding out that this vector quantity is conserved, we’ll take a quick look at equations which define ellipses and then show that an inverse-square law of gravity can yield elliptical orbits. If any portion of this post is in error, please return the unused portion for a full refund.
Continue reading The Laplace-Runge-Lenz Vector →

Video

Sagan on Eratosthenes

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After getting himself all grumpy about the ways in which statistics are abused, Joshua Hall decided to relax with a little Carl Sagan.

Fun fact: the philosopher Poseidonios of Apameia (c. 135–51 BCE) repeated Eratosthenes’s experiment about a century and a half later. He observed that on the island of Rhodes, the bright star Canopus was just touching the horizon, while at the same time in Alexandria, the star was a few degrees above the horizon. Because the Earth curves between the two places, the star was seen from different vantage points, and thus the angle between Rhodes and Alexandria could be found. Poseidonios was luckier than he knew: both his figure for the distance between Rhodes and Alexandria and his measurement of Canopus’s position were wrong, but the two wrongnesses canceled each other out, giving a reasonable final answer.

(And yes, I employ BCE/CE dating just to irritate people.)

Astronomy, Bad Math, Classical Mechanics, Favorites, Popularization, String Theory

The Necessity of Mathematics

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Today, everything from international finance to teenage sexuality flows on a global computer network which depends upon semiconductor technology which, in turn, could not have been developed without knowledge of the quantum principles of solid-state physics. Today, we are damaging our environment in ways which require all our fortitude and ingenuity just to comprehend, let alone resolve. More and more people are becoming convinced that our civilization requires wisdom in order to survive, the sort of wisdom which can only come from scientific literacy; thus, an increasing number of observers are trying to figure out why science has been taught so poorly and how to fix that state of affairs. Charles Simonyi draws a distinction between those who merely “popularize” a science and those who promote the public understanding of it. We might more generously speak of bad popularizers and good ones, but the distinction between superficiality and depth is a real one, and we would do well to consider what criteria separate the two.

Opinions on how to communicate science are as diverse as the communicators. In this Network age, anyone with a Web browser and a little free time can join the conversation and become part of the problem — or part of the solution, if you take an optimistic view of these newfangled media. Certain themes recur, and tend to drive people into one or another loose camp of like-minded fellows: what do you do when scientific discoveries clash with someone’s religious beliefs? Why do news stories sensationalize or distort scientific findings, and what can we do about it? What can we do when the truth, as best we can discern it, is simply not politic?

Rather than trying to find a new and juicy angle on these oft-repeated questions, this essay will attempt to explore another direction, one which I believe has received insufficient attention. We might grandiosely call this a foray into the philosophy of science popularization. The topic I wish to explore is the role mathematics plays in understanding and doing science, and how we disable ourselves if our “explanations” of science do not include mathematics. The fact that too many people don’t know statistics has already been mourned, but the problem runs deeper than that. To make my point clear, I’d like to focus on a specific example, one drawn from classical physics. Once we’ve explored the idea in question, extensions to other fields of inquiry will be easier to make. To make life as easy as possible, we’re going to step back a few centuries and look at a development which occurred when the modern approach to natural science was in its infancy.

Our thesis will be the following: that if one does not understand or refuses to deal with mathematics, one has fatally impaired one’s ability to follow the physics, because not only are the ideas of the physics expressed in mathematical form, but also the relationships among those ideas are established with mathematical reasoning.

This is a strong assertion, and a rather pessimistic one, so we turn to a concrete example to investigate what it means. Our example comes from the study of planetary motion and begins with Kepler’s Three Laws.

KEPLER’S THREE LAWS

Johannes Kepler (1571–1630) discovered three rules which described the motions of the planets. He distilled them from the years’ worth of data collected by his contemporary, the Danish astronomer Tycho Brahe (1546–1601). The story of their professional relationship is one of clashing personalities, set against a backdrop of aristocracy, ruin and war. From that drama, we boil away the biography and extract some items of geometry:
Continue reading The Necessity of Mathematics →

Video

Hydrogen Atoms in Motion

Still working on diagrams. . . Cairochemist, who has just recently started co-blogging at Skulls in the Stars, presents us a video showing hydrogen atoms migrating across a copper-palladium surface. Yes, understanding how catalysts work might help in arranging reactions for new energy sources, but even if it didn’t, being able to watch atoms move has a certain appeal of its own.

Oh, and Skulls in the Stars also points us to The Giant’s Shoulders, which we’re all hoping will become a new monthly event.

Video

Who’s Next?

2 Comments

Drawing diagrams for my next post. . . I begin to procrastinate. . . and find that Dana is now embedding Tom Lehrer videos. Given my background, this one has particular resonance for me:

The Tom Lehrer songbook — yes, I own the Tom Lehrer songbook — has alternate lyrics which include the couplet, “Japan will have its own device / transistorized at half the price.”

Popularization

Help, Help, I’m Being Oppressed!

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I just noticed a post at Michael White’s Adaptive Complexity entitled, “Bad Science Journalism and the Myth of the Oppressed Underdog” (9 March 2008). It’s good. Go read it.

[I]n our culture we love the underdog, who sticks to his or her guns, in spite of heavy opposition. In this narrative, we have heroes, villains, and a famous, brilliant scientist proven wrong.

I’m sure you could pick out instances in science history where this story is true, but more often it is not. You wouldn’t know this from the pages of our major news media though; in fact you’d probably get the impression that the underdog narrative is the way science works. And many journalists may think that too; after all, most of them read (or misread) Thomas Kuhn when they were in college, and Kuhn brought this kind of narrative to a new high.

This is the narrative which I called the “David vs. Goliath of Scientific Establishment” story. White gives a specific example, concerning the public presentation of evolutionary biology; in physics, the problem can be even more fun, since the actual relationship between David’s revolutionary idea and Goliath’s orthodoxy often requires a hefty dose of mathematical reasoning to understand. When a typical statement by a physicist trying to sort the mess out might read, “Any embedding of your gauge group in either noncompact real form of E8 will always give you a nonchiral fermion spectrum,” well. . . the temptation will always be to plump for the “social” angle and emphasize the personalities of the physicists involved. The “oppressed underdog” story emerges quite naturally.

To be fair, or at least spread the misery around, biology can suffer from Popularizer Mathophobia, too. White’s example comes from sexual selection theory and relates to the role game theory might play in understanding the dynamics of a population. Oooh! Math! Scary!

(My “typical statement” is a paraphrase of Jacques Distler, who pointed out this problem almost six years ago.)

Wobosphere fun

Please Her With Your Genome

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One of the nice things about science blogs is that they can help transmit the “life stories” of the scientific lifestyle, making a written record of the hitherto unwritten folklore our community generates. In an earlier epoch, each student had to learn the hard way the lesson that whatever you do in lab will not work the first time. Now, the student can receive a tiny foretaste of the pain which awaits and thus, perhaps, live better for it. So, in that spirit, I offer the “life lesson” for this week:

Do not oversleep and miss a meeting because the meeting announcement was sent to the e-mail address you don’t use because it’s continually broken, or else you too may draw the short straw in absentia and find yourself in charge of assembling a volume of conference proceedings. I enjoy working with the written word, perhaps more so than many of my fellow physics boffins, but I can easily think of more interesting things to edit than an overpriced book which nobody will buy and fewer will read. Hooray, academia.

And here’s another thing: the downside of working in an “interdisciplinary” environment (other than the episodic crises of “I’m not doing real science!” conscience and the assertions from specialists in field X that they can solve problems and lead revolutions in field Y without actually having the ground-level knowledge of Y) is that you might be one of the select few who know LaTeX. This is a death sentence: much as being the guy/girl who knows how the Web server works means that all the Web-related tasks fall onto your lap, being the one who knows LaTeX brings all sorts of “little,” er, “action items” your way.

But it’s not all bad. Even digging through the conference e-mail pile for the revised papers people have sent in has a few compensations. There’s the Big Name who doesn’t want to bother with the “Procrustean bed requirements” imposed by the publisher and forgoes inclusion, but even better, there’s the entertaining spam attracted to an e-mail address which was, after all, publicly announced on the conference Web site without the slightest bit of preventive obfuscation.

“We offer the clone-by-phone service of your dreams,” proclaims the e-mail. In just twelve days, their company will “synthesize and clone a 1 kb gene, with all mutations, deletions, fusions, epitope tags and codon optimizations you can think of.” Increase the size of your DNA by four micrograms in only two weeks!

The same firm — which, on casual investigation, looks otherwise legitimate — has been spamming other addresses. The linguist Mark Liberman got a similar message, for example. (“I expect they got my name and email address from some bioinformatics conference I attended, or something like that,” he notes.)

MEMO TO GEN SCRIPT: Mass, unsolicited mailings are not the way to generate good will or impress the world with your netiquette. Thanks for the laugh, though.

Upon reflection, this opens an entire new avenue of exploration. What other scientific services can be hawked by spam, and what would the messages sound like?

Neuroscience, Popularization

Brain Damage and Journalism

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Dan Hurley writes of Katherine P. Rankin’s neurological research on sarcasm,

To her surprise, though, the magnetic resonance scans revealed that the part of the brain lost among those who failed to perceive sarcasm was not in the left hemisphere of the brain, which specializes in language and social interactions, but in a part of the right hemisphere previously identified as important only to detecting contextual background changes in visual tests.

That’s from a New York Times piece, “The Science of Sarcasm (Not That You Care),” 3 June 2008. The abstract for Rankin et al.‘s presentation says, in part,

This study provides lesion data suggesting that the right posterior temporal lobe and dorsomedial frontal cortex are associated with recognizing and interpreting sarcastic irony using paralinguistic vocal and facial cues, consistent with functional imaging research examining neural correlates of voice prosody, facial emotion recognition, and perspective taking.

Trust the alchemy of science journalism to turn a result consistent with prior research into a great surprise. As Vaughan Bell points out, by the early 1980s people had already found out that damage to the brain’s right hemisphere can cause “disorders of affective language,” i.e., problems with recognizing emotion in speech. More recently, Shamay-Tsoory et al. (2005) found that lesions in the ventromedial section of the right prefrontal cortex impaired test subjects’ abilities to handle tasks which required understanding sarcasm.

This is the sort of gaffe which makes neurocurmudgeons file a story under “chaff” instead of “wheat.” I wonder: can fMRI detect a cortical lesion which makes all interesting and worthwhile research be perceived as revolutionary? The problem is not just restricted to neuro-journalism, of course — revolution disease is a general trope, right up there with “Think of the children!” and “David vs. the Scientific Goliath.” One begins to appreciate why Eric Roston, author of The Carbon Age (2008), said of his research practices, “I wanted to avoid secondary literature, media,” even though this meant a masochistic journey through peer-reviewed articles.

As for the research itself. . . . To my knowledge (and that of the Neurocritic), Rankin et al.‘s results have not yet appeared in a journal, only at an American Academy of Neurology conference, so pickings are as yet rather thin. No doubt more details will be available anon.

Wobosphere fun

Shifting Standards of Inclusion

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SILVER SPRING, MD — Tommy Thayer, 30, is quoted by the Washington Post as saying the following:

It’s a damn shame there hasn’t already been a black president as far as I’m concerned. We are so used to all the presidents being all whites and all men. That’s like telling everyone we are a racist nation. I think people are robbing themselves if they don’t get to know other cultures.

Thayer identifies himself as “all white, 100 percent white, Irish, Italian if you want specifics.” Irish and Italians as members of a cultural or ethnic majority: discuss. Have we as a nation outgrown Grandmother’s fears that Kennedy was taking his orders from the Pope? Would latter-day incarnations of Sacco and Vanzetti be regarded as white troublemakers?

Bad Math

Statistics: Tool of an Elitist Bastard

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I was prepared to enjoy Susan Jacoby‘s Op-Ed in the New York Times, “Best Is the New Worst” (30 May 2008). Yes, the title “Best Is The New Worst” is a snowclone, the sort of trite phrasal template with plug-and-play slots which is the refuge of lazy writing. In fact, “X is the new Y” is a classic of the genre. (Is this headline choice an example of such, or a clever meta-reference intended to mock declining standards? Discuss.) However, this bit was more troubling, and made me look askance:

Another peculiar new use of “elitist” (often coupled with “Luddite”) is its application to any caveats about the Internet as a source of knowledge. After listening to one of my lectures, a college student told me that it was elitist to express alarm that one in four Americans, according to the National Constitution Center, cannot name any First Amendment rights or that 62 percent cannot name the three branches of government. “You don’t need to have that in your head,” the student said, “because you can just look it up on the Web.”

Ahem. Laziness in a student may well be a cause for alarm, but uncritically accepting the spin which innumerate and agenda-driven reportage has placed upon a survey is, I dare to suggest, even worse. I’m going to go out on a limb here and say that these numbers, intimidating though they may be, are in fact completely worthless. They don’t measure what they’re claimed to measure. The reason goes back to the First Amendment itself:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

This breaks down to six different rights. The first half of the religion bit — the Establishment Clause — covers a different territory than the second half, the Free Exercise Clause. It’s possible to have the right to follow your personal religion even in a country which has an established church (hint, hint: the United Kingdom has Muslims). Speech and the press get one “freedom” each, but the people have a “right” to assemble and another “right” to petition.

The National Constitution Center website features a 1997 survey which speaks of “four rights” in the First Amendment, and lists them as “speech,” “religion,” “press” and “assembly.” Now, I’d argue that this obscures the constitutional status of religion, reflecting or enforcing an oversimplified view of the matter, but more importantly, the right to petition vanished down a memory hole.

If it were universally acknowledged that, say, petitioning and assembly were two faces of the same freedom, then I wouldn’t mind so much. However, a 2006 survey by the McCormick Tribune Freedom Museum asked respondents to name their First Amendment rights, and their “correct” answers were the freedoms of religion, speech, press, petition and assembly. When two different love-the-Constitution groups can’t even be consistent on how many freedoms exist, can the results be considered reliable? What, exactly, are we citizens supposed to know, and will we lose credit for knowing the wrong “right” answer? Suppose that the NCC called you one September morning in 1997, and you said that the First Amendment covered speech, religion, press and peaceable assembly. You would be a sterling citizen in 1997, but nine years later, the Freedom Museum would stamp you defective.

Both surveys share another flaw. As was pointed out back in 2006,
several “freedoms” which are likely to be uppermost in an American’s mind are found in other, later amendments: bearing arms, avoiding self-incrimination, avoiding unreasonable search and seizure, not being enslaved (took a while for that one) and so forth. What happens if you remember a whole list of these rights, but can’t recall which amendment they go under?
Continue reading Statistics: Tool of an Elitist Bastard →

Bad Math

Mathification

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Megan Garber writes the following in the Columbia Journalism Review‘s daily blog:

We currently find ourselves in, to put it mildly, a lull in the 2008 campaign’s primary season. The delegate tallies are in limbo. Parsing them seems to require a postgraduate degree in calculus.

I call mathification! The analogue of linguification, this term refers to statements which, as Isabel Lugo puts it, “clearly intend to get across a true point about the real world by making a false point about mathematics.”
Continue reading Mathification →

Bibliophilia

A Gedanken Canon

8 Comments

Originally posted at Pharyngula. Salvaged from the comment threads, edited and re-posted due to popular acclaim.

Let’s try a thought experiment. Suppose that in the near future, our global civilization collapses: maybe we try to cancel out global warming with nuclear winter. Centuries later, cities are being built again, and archaeologists come across books. One series in particular must have been exceedingly important, as copies are found in different languages — French, German, English, Latin — all around the world. Each copy is only a fragment, but by comparing the overlapping portions a complete canon is tentatively identified. Some portions, attested only in surviving electronic records and printed material of lesser quality, seem to be later additions by a community of hands, but then again, portions of the canon itself are judged on literary grounds to be as low-quality as the least inspired of the apocrypha. The first copy found is in Latin, which scholars know to be the oldest of the languages in which translations exist, but further investigation turns up copies in Germanic languages which appear to be older.

Much of the material in these chronicles is of a fantastic nature, with epic battles between good and evil waged in a realm beyond the sight of most citizens, which the chronicles describe as blissfully ignorant. Believers in magic point out that prophecies made in one book are fulfilled in another, a claim which scholars dismiss as valueless. Upon a closer look, however, some places in these fantastic tales can be matched with known cities of the era, attested by archaeology — places like London. What’s more, incidental mentions of technology coincide with what is known about the era just before the Great Collapse.

Should we then have faith in Harrius Potter?

Popularization, Transparency

Medical Journalism is Ill

Gary Schwitzer asks,

Is the news media doing a good job of reporting on new treatments, tests, products, and procedures? Ray Moynihan and colleagues analyzed how often news stories quantified the costs, benefits, and harms of the interventions being discussed, and how often they reported potential conflicts of interest in story sources [1]. Of the 207 newspaper and television stories that they studied, 83 did not report the benefits of medications quantitatively, and of the 124 stories that did quantify the benefits of medications, only 18 presented both relative and absolute benefits. Of all the stories, 53% had no information about potential harms of the treatment, and 70% made no mention of treatment costs. Of 170 stories that cited an expert or a scientific study, 85 (50%) cited at least one with a financial tie to the manufacturer of the drug, a tie that was disclosed in only 33 of the 85 stories.

Moynihan et al. (2000) inspired some Australians to do a similar survey in 2004, which found after six months that Australian print and online news coverage of medical advances was “poor.” Now, Schwitzer has done a more extensive survey of United States media. The punchline is as follows:

In our evaluation of 500 US health news stories over 22 months, between 62%–77% of stories failed to adequately address costs, harms, benefits, the quality of the evidence, and the existence of other options when covering health care products and procedures. This high rate of inadequate reporting raises important questions about the quality of the information US consumers receive from the news media on these health news topics.

Details are available at PLoS Medicine. Now, we just need somebody to pay for a similar survey of non-medical science reporting.

(Tip o’ the fedora to Steve Novella.)

Bibliophilia, Popularization

Gah! But Yay!

I grew up (to the extent that I have grown up) reading the works of Larry Gonick, expositor of science and history in cartoon format. As Cosma Shalizi wrote, “When I think about it, I realize a truly substantial proportion of my basic knowledge of the world derives from reading Larry Gonick’s Cartoon Guides and Cartoon History of the Universe.” So it was with great interest that I read in the Mercury News of 10 April 2008 that Gonick “hopes to work on a cartoon book about calculus” once he’s finished The Cartoon History of the Modern World Part II, which is really the fifth installment in the Cartoon History of the Universe series. However, that Mercury News article commits a serious gaffe:

Gonick hasn’t actually yet put in cartoon form a subject he knows at least as well as history: mathematics.

Ahem. The Cartoon Guide to Statistics (1994), coauthored with Woollcott Smith of Temple University. And that’s not even mentioning his “Science Classics” feature in Discover magazine, a regular two-page comic which often covered mathematical topics. We could go on to list the mathy subjects addressed in his other science books and even in his histories, but really.

"no matter how gifted, you alone cannot change the world"