{"id":337,"date":"2006-06-21T13:23:17","date_gmt":"2006-06-21T13:23:17","guid":{"rendered":"http:\/\/scienceblogs.com\/principles\/2006\/06\/21\/assume-a-spherical-cow\/"},"modified":"2006-06-21T13:23:17","modified_gmt":"2006-06-21T13:23:17","slug":"assume-a-spherical-cow","status":"publish","type":"post","link":"http:\/\/chadorzel.com\/principles\/2006\/06\/21\/assume-a-spherical-cow\/","title":{"rendered":"Assume a Spherical Cow"},"content":{"rendered":"

Over at bento-box, there’s a nice response<\/a> to my recent post about simulations<\/a>. He makes the very good point that the Sandia press release in question could sensibly be read as referring to the fact that recent computer technology requires fewer simplifying approximations:<\/p>\n

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Well, it isn’t really until quite recently that computers have gotten fast enough that many of these approximations can be toned-down. Simulations are starting to match up on a more than qualitative level with experiment on more than simple and uninteresting systems. But computers have been around for a long time and there is a long history of simulating chemical systems. There is a long history of “bad” simulations, lots and lots of computer generated results that were either totally wrong or only qualitatively good at best. That is to say, there is a long history of the simulation being wrong, but the theory being “correct”. So to look at simulations as a test of the theory, at least in chemical simulations, one had to wait till fairly recently to find this a reasonable prospect.<\/p>\n<\/blockquote>\n

I’m still highly dubious about the idea of using simulations as a “test” of theory, no matter how few approximations you make. I’ll probably have more to say about this later, but I’ve got to get to my day job, so I’m going to cop out a bit, and re-post some thoughts on the question of simplifying approximations from back in 2003<\/a>, below the fold and after the obligatory DonorsChoose icon:<\/p>\n

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There’s an old joke in physics circles about a dairy farmer who, in a fit of desperation over the fact that his cows won’t give enough milk, consults a theoretical physicist about the problem. The physicist listens to him, asks a few questions, and then says he’ll take the assignment. A few weeks later, he calls up the farmer, and says “I’ve got the answer.” They arrange for him to give a presentation of his solution to the milk shortage.<\/p>\n

When the day for the presentation arrives, he begins his talk by saying, “First, we assume a spherical cow…”<\/p>\n

OK, it’s not a great<\/strong> joke, and it’s the sort of joke that’s only funny to physics majors. Because this is a common feature of physics lectures– you begin considering the problem using simplifying assumptions which are often bizarrely unrealistic. Any object you need to model is first considered as a sphere, if not a point. Strings are assumed to be massless, surfaces frictionless, and mirrors and lenses loss-less.<\/p>\n

(This is sometimes taken to absurd extremes– a former colleague claimed to have had great success modeling the growth of his baby daughter as a sphere accreting cells at a fixed rate…)<\/p>\n

None of these assumptions are remotely realistic, but in their own way, they’re absolutely essential. Students are often baffled and bored by the simplified cases– one of the best students from last term’s introductory mechanics<\/a> class remarked at the end of the term that “we neglected friction and air resistance, and all the interesting stuff.” But if you don’t make those assumptions, you can never discover the underlying laws and general principles that make physics such a successful science.<\/p>\n

The ability to abstract away the “interesting stuff” and get down to the basic principles is one of the qualifications of the Great Names in physics. In a way, this is what Einstein did, by demolishing the idea of simultaneity<\/a> (though another way of looking at it would be to say that he removed a simplifying assumption that people didn’t know they were making). And going back even farther, it was the specific failure to consider the “interesting stuff” my student commented on that let Newton and Galileo get the whole field of physics started.<\/p>\n

Newton’s Laws have been around for long enough that they’ve sort of become part of the atmosphere surrounding us. Phrases like “an object at rest tends to remain at rest” or “for every action there is an equal and opposite reaction” have attained the same sort of unconscious quote status as various Shakespeare references (“there’s a method to his madness” and the like)– say the phrase to a randomly chosen person, and it will at least sound familiar, even if they don’t understand what it means.<\/p>\n

Of course, if you stop to think about it, the full statement of Newton’s first law– “an object at rest tends to remain at rest, an object in motion tends to continue in motion in a straight line at constant speed, unless acted on by an external force”– is a little tricky to really see in action. Half of it is trivial– stationary objects rarely start into spontaneous motion– but the other half is actually not that obvious. Pick an object near you, and start it moving– odds are, it will stop moving in fairly short order. Thrown objects will be hauled down to the ground by gravity, while sliding objects are subject to friction. It takes a day like today, when there’s a quarter-inch layer of ice on everything<\/strong> to really make you believe in Newton’s first law.<\/p>\n

Plenty of people tried to develop general theories of motion before Newton came around, and all of them got tripped up by their inability to see past the inescapable forces of friction and gravity. Aristotle<\/a> is probably the most famous of these failures, and while Galileo did better<\/a>, he didn’t get all the way there. Newton is justly famous for being the first to realize that gravity was also an external force, and get to the laws that bear his name.<\/p>\n

The step of ignoring friction and gravity is the absolutely critical moment in early physics. It’s what lets us get from an incomplete and unsatisfying description of the everyday world to a broad and elegant description of everything. If you believe that the natural state of inanimate objects is to be motionless on the ground, you won’t do too badly at describing the sort of objects Aristotle had to work with in ancient Greece, but you’ll have a terrible time trying to describe the motion of planets, stars, and galaxies, or the behavior of atoms and molecules. To get rules that apply to everything, you need the imagination to start with the ideal case, and work back to the more complicated reality.<\/p>\n

This extends well beyond mechanics, of course. In dealing with electricity and magnetism, you start with the assumption of perfect conductors and insulators, and work back to real materials. In quantum mechanics, you start with impenetrable potential barriers, and work back to finite potential values. In atomic physics, everything starts with two-level atoms, despite the fact that, as my former boss was once (mis)quoted, “There are no two-level atoms, and sodium is not one of them.”<\/p>\n

You need to work the simple, idealized problems first, to get the general rules, and then apply those rules to whatever more realistic situation you’re interested in. Daft as it may seem, you need to solve the spherical cow problem before you can understand the problems of a cow shaped like, well, a cow. And sometimes, genius lies in knowing how to look at a cow and see a sphere.<\/p>\n

(Of course, hand in hand with knowing how to make the “spherical cow” approximation to render a problem soluble goes the knowledge of when not<\/strong> to make those kind of approximations. Excessive zeal in making simplifying assumptions is the bane of more than a few physicists, and whole fields of social science. But that’s a topic for another day…)<\/p>\n","protected":false},"excerpt":{"rendered":"

Over at bento-box, there’s a nice response to my recent post about simulations. He makes the very good point that the Sandia press release in question could sensibly be read as referring to the fact that recent computer technology requires fewer simplifying approximations: Well, it isn’t really until quite recently that computers have gotten fast… Continue reading Assume a Spherical Cow<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"1","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-337","post","type-post","status-publish","format-standard","hentry","category-physics","entry"],"_links":{"self":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/comments?post=337"}],"version-history":[{"count":0,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/337\/revisions"}],"wp:attachment":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/media?parent=337"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/categories?post=337"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/tags?post=337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}