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  • Joseph Nebus 6:00 pm on Sunday, 12 March, 2017 Permalink | Reply
    Tags: , Basic Instructions, , Little Iodine, Phoebe and her Unicorn, Piranha Club,   

    Reading the Comics, March 6, 2017: Blackboards Edition 


    I can’t say there’s a compelling theme to the first five mathematically-themed comics of last week. Screens full of mathematics turned up in a couple of them, so I’ll run with that. There were also just enough strips that I’m splitting the week again. It seems fair to me and gives me something to remember Wednesday night that I have to rush to complete.

    Jimmy Hatlo’s Little Iodine for the 1st of January, 1956 was rerun on the 5th of March. The setup demands Little Iodine pester her father for help with the “hard homework” and of course it’s arithmetic that gets to play hard work. It’s a word problem in terms of who has how many apples, as you might figure. Don’t worry about Iodine’s boss getting fired; Little Iodine gets her father fired every week. It’s their schtick.

    Little Iodine wakes her father early after a night at the lodge. 'You got to help me with my [hard] homework.' 'Ooh! My head! Wha'?' 'The first one is, if John has twice as many apples as Tom and Sue put together ... ' 'Huh? kay! Go on, let's get this over with.' They work through to morning. Iodine's teacher sees her asleep in class and demands she bring 'a note from your parents as to why you sleep in school instead of at home!' She goes to her father's office where her father's boss is saying, 'Well, Tremblechin, wake up! The hobo hotel is three blocks south and PS: DON'T COME BACK!'

    Jimmy Hatlo’s Little Iodine for the 1st of January, 1956. I guess class started right back up the 2nd, but it would’ve avoided so much trouble if she’d done her homework sometime during the winter break. That said, I never did.

    Dana Simpson’s Phoebe and her Unicorn for the 5th mentions the “most remarkable of unicorn confections”, a sugar dodecahedron. Dodecahedrons have long captured human imaginations, as one of the Platonic Solids. The Platonic Solids are one of the ways we can make a solid-geometry analogue to a regular polygon. Phoebe’s other mentioned shape of cubes is another of the Platonic Solids, but that one’s common enough to encourage no sense of mystery or wonder. The cube’s the only one of the Platonic Solids that will fill space, though, that you can put into stacks that don’t leave gaps between them. Sugar cubes, Wikipedia tells me, have been made only since the 19th century; the Moravian sugar factory director Jakub Kryštof Rad got a patent for cutting block sugar into uniform pieces in 1843. I can’t dispute the fun of “dodecahedron” as a word to say. Many solid-geometric shapes have names that are merely descriptive, but which are rendered with Greek or Latin syllables so as to sound magical.

    Bud Grace’s Piranha Club for the 6th started a sequence in which the Future Disgraced Former President needs the most brilliant person in the world, Bud Grace. A word balloon full of mathematics is used as symbol for this genius. I feel compelled to point out Bud Grace was a physics major. But while Grace could as easily have used something from the physics department to show his deep thinking abilities, that would all but certainly have been rendered as equation and graphs, the stuff of mathematics again.

    At the White Supremacist House: 'I have the smartest people I could find to help me run this soon-to-be-great-again country, but I'm worried that they're NOT SMART ENOUGH! I want the WORLD'S SMARTEST GENIUS to be my SPECIAL ADVISOR!' Meanwhile, cartoonist Bud Grace thinks of stuff like A = pi*r^2 and a^2 + b^2 = c^2 and tries working out 241 times 365, 'carry the one ... hmmmm ... '

    Bud Grace’s Piranha Club for the 6th of March, 2017. 241 times 635 is 153,035 by the way. I wouldn’t work that out in my head if I needed the number. I might work out an estimate of how big it was, in which case I’d do this: 241 is about 250, which is one-quarter of a thousand. One-quarter of 635 is something like 150, which times a thousand is 150,000. If I needed it exactly I’d get a calculator. Unless I just needed something to occupy my mind without having any particular emotional charge.

    Scott Meyer’s Basic Instructions rerun for the 6th is aptly titled, “How To Unify Newtonian Physics And Quantum Mechanics”. Meyer’s advice is not bad, really, although generic enough it applies to any attempts to reconcile two different models of a phenomenon. Also there’s not particularly a problem reconciling Newtonian physics with quantum mechanics. It’s general relativity and quantum mechanics that are so hard to reconcile.

    Still, Basic Instructions is about how you can do a thing, or learn to do a thing. It’s not about how to allow anything to be done for the first time. And it’s true that, per quantum mechanics, we can’t predict exactly what any one particle will do at any time. We can say what possible things it might do and how relatively probable they are. But big stuff, the stuff for which Newtonian physics is relevant, involve so many particles that the unpredictability becomes too small to notice. We can see this as the Law of Large Numbers. That’s the probability rule that tells us we can’t predict any coin flip, but we know that a million fair tosses of a coin will not turn up 800,000 tails. There’s more to it than that (there’s always more to it), but that’s a starting point.

    Michael Fry’s Committed rerun for the 6th features Albert Einstein as the icon of genius. Natural enough. And it reinforces this with the blackboard full of mathematics. I’m not sure if that blackboard note of “E = md3” is supposed to be a reference to the famous Far Side panel of Einstein hearing the maid talk about everything being squared away. I’ll take it as such.

     
  • Joseph Nebus 6:00 pm on Thursday, 23 February, 2017 Permalink | Reply
    Tags: , , compulsions, , Over The Hedge, , Super-Fun-Pak Comix, Wide Open   

    Reading the Comics, February 15, 2017: SMBC Does Not Cut In Line Edition 


    On reflection, that Saturday Morning Breakfast Cereal I was thinking about was not mathematically-inclined enough to be worth including here. Helping make my mind up on that was that I had enough other comic strips to discuss here that I didn’t need to pad my essay. Yes, on a slow week I let even more marginal stuff in. Here’s the comic I don’t figure to talk about. Enjoy!

    Jack Pullan’s Boomerangs rerun for the 16th is another strip built around the “algebra is useless in real life” notion. I’m too busy noticing Mom in the first panel saying “what are you doing play [sic] video games?” to respond.

    Ruben Bolling’s Super-Fun-Pak Comix excerpt for the 16th is marginal, yeah, but fun. Numeric coincidence and numerology can sneak into compulsions with terrible ease. I can believe easily the need to make the number of steps divisible by some favored number.

    Rich Powell’s Wide Open for the 16th is a caveman science joke, and it does rely on a chalkboard full of algebra for flavor. The symbols come tantalizingly close to meaningful. The amount of kinetic energy, K or KE, of a particle of mass m moving at speed v is indeed K = \frac{1}{2} m v^2 . Both 16 and 32 turn up often in the physics of falling bodies, at least if we’re using feet to measure. a = -\frac{k}{m} x turns up in physics too. It comes from the acceleration of a mass on a spring. But an equation of the same shape turns up whenever you describe things that go through tiny wobbles around the normal value. So the blackboard is gibberish, but it’s a higher grade of gibberish than usual.

    Rick Detorie’s One Big Happy rerun for the 17th is a resisting-the-word-problem joke, made fresher by setting it in little Ruthie’s playing at school.

    T Lewis and Michael Fry’s Over The Hedge for the 18th mentions the three-body problem. As Verne the turtle says, it’s a problem from physics. The way two objects — sun and planet, planet and moon, pair of planets, whatever — orbit each other if they’re the only things in the universe is easy. You can describe it all perfectly and without using more than freshman physics majors know. Introduce a third body, though, and we don’t know anymore. Chaos can happen.

    Emphasis on can. There’s no good way to solve the “general” three-body problem, the one where the star and planets can have any sizes and any starting positions and any starting speeds. We can do well for special cases, though. If you have a sun, a planet, and a satellite — each body negligible compared to the other — we can predict orbits perfectly well. If the bodies have to stay in one plane of motion, instead of moving in three-dimensional space, we can do pretty well. If we know two of the bodies orbit each other tightly and the third is way off in the middle of nowhere we can do pretty well.

    But there’s still so many interesting cases for which we just can’t be sure chaos will not break out. Three interacting bodies just offer so much more chance for things to happen. (To mention something surely coincidental, it does seem to be a lot easier to write good comedy, or drama, with three important characters rather than two. Any pair of characters can gang up on the third, after all. I notice how much more energetic Over The Hedge became when Hammy the Squirrel joined RJ and Verne as the core cast.)

    Dave Whamond’s Reality Check for the 18th is your basic mathematics-illiteracy joke, done well enough.

     
  • Joseph Nebus 6:00 pm on Wednesday, 1 February, 2017 Permalink | Reply
    Tags: cables, , , podcasts   

    Mathematics Stuff To Read Or Listen To 


    I concede January was a month around here that could be characterized as “lazy”. Not that I particularly skimped on the Reading the Comics posts. But they’re relatively easy to do: the comics tell me what to write about, and I could do a couple paragraphs on most anything, apparently.

    While I get a couple things planned out for the coming month, though, here’s some reading for other people.

    The above links to a paper in the Proceedings of the National Academy of Sciences. It’s about something I’ve mentioned when talking about knot before. And it’s about something everyone with computer cables or, like the tweet suggests, holiday lights finds. The things coil up. Spontaneous knotting of an agitated string by Dorian M Raymer and Douglas E Smith examines when these knots are likely to form, and how likely they are. It’s not a paper for the lay audience, but there are a bunch of fine pictures. The paper doesn’t talk about Christmas lights, no matter what the tweet does, but the mathematics carries over to this.

    MathsByAGirl, meanwhile, had a post midmonth listing a couple of mathematics podcasts. I’m familiar with one of them, BBC Radio 4’s A Brief History of Mathematics, which was a set of ten-to-twenty-minute sketches of historically important mathematics figures. I’ll trust MathsByAGirl’s taste on other podcasts. I’d spent most of this month finishing off a couple of audio books (David Hackett Fischer’s Washington’s Crossing which I started listening to while I was in Trenton for a week, because that’s the sort of thing I think is funny, and Robert Louis Stevenson’s Doctor Jekyll and Mister Hyde And Other Stories) and so fell behind on podcasts. But now there’s some more stuff to listen forward to.

    And then I’ll wrap up with this from KeplerLounge. It looks to be the start of some essays about something outside the scope of my Why Stuff Can Orbit series. (Which I figure to resume soon.) We start off talking about orbits as if planets were “point masses”. Which is what the name suggests: a mass that fills up a single point, with no volume, no shape, no features. This makes the mathematics easier. The mathematics is just as easy if the planets are perfect spheres, whether hollow or solid. But real planets are not perfect spheres. They’re a tiny bit blobby. And they’re a little lumpy as well. We can ignore that if we’re doing rough estimates of how orbits work. But if we want to get them right we can’t ignore that anymore. And this essay describes some of how we go about dealing with that.

     
  • Joseph Nebus 6:00 pm on Tuesday, 27 December, 2016 Permalink | Reply
    Tags: , , , Riemann hypothesis,   

    The End 2016 Mathematics A To Z: Xi Function 


    I have today another request from gaurish, who’s also been good enough to give me requests for ‘Y’ and ‘Z’. I apologize for coming to this a day late. But it was Christmas and many things demanded my attention.

    Xi Function.

    We start with complex-valued numbers. People discovered them because they were useful tools to solve polynomials. They turned out to be more than useful fictions, if numbers are anything more than useful fictions. We can add and subtract them easily. Multiply and divide them less easily. We can even raise them to powers, or raise numbers to them.

    If you become a mathematics major then somewhere in Intro to Complex Analysis you’re introduced to an exotic, infinitely large sum. It’s spoken of reverently as the Riemann Zeta Function, and it connects to something named the Riemann Hypothesis. Then you remember that you’ve heard of this, because if you’re willing to become a mathematics major you’ve read mathematics popularizations. And you know the Riemann Hypothesis is an unsolved problem. It proposes something that might be true or might be false. Either way has astounding implications for the way numbers fit together.

    Riemann here is Bernard Riemann, who’s turned up often in these A To Z sequences. We saw him in spheres and in sums, leading to integrals. We’ll see him again. Riemann just covered so much of 19th century mathematics; we can’t talk about calculus without him. Zeta, Xi, and later on, Gamma are the famous Greek letters. Mathematicians fall back on them because the Roman alphabet just hasn’t got enough letters for our needs. I’m writing them out as English words instead because if you aren’t familiar with them they look like an indistinct set of squiggles. Even if you are familiar, sometimes. I got confused in researching this some because I did slip between a lowercase-xi and a lowercase-zeta in my mind. All I can plead is it’s been a hard week.

    Riemann’s Zeta function is famous. It’s easy to approach. You can write it as a sum. An infinite sum, but still, those are easy to understand. Pick a complex-valued number. I’ll call it ‘s’ because that’s the standard. Next take each of the counting numbers: 1, 2, 3, and so on. Raise each of them to the power ‘s’. And take the reciprocal, one divided by those numbers. Add all that together. You’ll get something. Might be real. Might be complex-valued. Might be zero. We know many values of ‘s’ what would give us a zero. The Riemann Hypothesis is about characterizing all the possible values of ‘s’ that give us a zero. We know some of them, so boring we call them trivial: -2, -4, -6, -8, and so on. (This looks crazy. There’s another way of writing the Riemann Zeta function which makes it obvious instead.) The Riemann Hypothesis is about whether all the proper, that is, non-boring values of ‘s’ that give us a zero are 1/2 plus some imaginary number.

    It’s a rare thing mathematicians have only one way of writing. If something’s been known and studied for a long time there are usually variations. We find different ways to write the problem. Or we find different problems which, if solved, would solve the original problem. The Riemann Xi function is an example of this.

    I’m going to spare you the formula for it. That’s in self-defense. I haven’t found an expression of the Xi function that isn’t a mess. The normal ways to write it themselves call on the Zeta function, as well as the Gamma function. The Gamma function looks like factorials, for the counting numbers. It does its own thing for other complex-valued numbers.

    That said, I’m not sure what the advantages are in looking at the Xi function. The one that people talk about is its symmetry. Its value at a particular complex-valued number ‘s’ is the same as its value at the number ‘1 – s’. This may not seem like much. But it gives us this way of rewriting the Riemann Hypothesis. Imagine all the complex-valued numbers with the same imaginary part. That is, all the numbers that we could write as, say, ‘x + 4i’, where ‘x’ is some real number. If the size of the value of Xi, evaluated at ‘x + 4i’, always increases as ‘x’ starts out equal to 1/2 and increases, then the Riemann hypothesis is true. (This has to be true not just for ‘x + 4i’, but for all possible imaginary numbers. So, ‘x + 5i’, and ‘x + 6i’, and even ‘x + 4.1 i’ and so on. But it’s easier to start with a single example.)

    Or another way to write it. Suppose the size of the value of Xi, evaluated at ‘x + 4i’ (or whatever), always gets smaller as ‘x’ starts out at a negative infinitely large number and keeps increasing all the way to 1/2. If that’s true, and true for every imaginary number, including ‘x – i’, then the Riemann hypothesis is true.

    And it turns out if the Riemann hypothesis is true we can prove the two cases above. We’d write the theorem about this in our papers with the start ‘The Following Are Equivalent’. In our notes we’d write ‘TFAE’, which is just as good. Then we’d take which ever of them seemed easiest to prove and find out it isn’t that easy after all. But if we do get through we declare ourselves fortunate, sit back feeling triumphant, and consider going out somewhere to celebrate. But we haven’t got any of these alternatives solved yet. None of the equivalent ways to write it has helped so far.

    We know some some things. For example, we know there are infinitely many roots for the Xi function with a real part that’s 1/2. This is what we’d need for the Riemann hypothesis to be true. But we don’t know that all of them are.

    The Xi function isn’t entirely about what it can tell us for the Zeta function. The Xi function has its own exotic and wonderful properties. In a 2009 paper on arxiv.org, for example, Drs Yang-Hui He, Vishnu Jejjala, and Djordje Minic describe how if the zeroes of the Xi function are all exactly where we expect them to be then we learn something about a particular kind of string theory. I admit not knowing just what to say about a genus-one free energy of the topological string past what I have read in this paper. In another paper they write of how the zeroes of the Xi function correspond to the description of the behavior for a quantum-mechanical operator that I just can’t find a way to describe clearly in under three thousand words.

    But mathematicians often speak of the strangeness that mathematical constructs can match reality so well. And here is surely a powerful one. We learned of the Riemann Hypothesis originally by studying how many prime numbers there are compared to the counting numbers. If it’s true, then the physics of the universe may be set up one particular way. Is that not astounding?

     
    • gaurish 5:34 am on Wednesday, 28 December, 2016 Permalink | Reply

      Yes it’s astounding. You have a very nice talent of talking about mathematical quantities without showing formulas :)

      Liked by 1 person

      • Joseph Nebus 5:15 am on Thursday, 5 January, 2017 Permalink | Reply

        You’re most kind, thank you. I’ve probably gone overboard in avoiding formulas lately though.

        Like

  • Joseph Nebus 6:00 pm on Friday, 11 November, 2016 Permalink | Reply
    Tags: , , , , , ,   

    The End 2016 Mathematics A To Z: Ergodic 


    This essay follows up on distributions, mentioned back on Wednesday. This is only one of the ideas which distributions serve. Do you have a word you’d like to request? I figure to close ‘F’ on Saturday afternoon, and ‘G’ is already taken. But give me a request for a free letter soon and I may be able to work it in.

    Ergodic.

    There comes a time a physics major, or a mathematics major paying attention to one of the field’s best non-finance customers, first works on a statistical mechanics problem. Instead of keeping track of the positions and momentums of one or two or four particles she’s given the task of tracking millions of particles. It’s listed as a distribution of all the possible values they can have. But she still knows what it really is. And she looks at how to describe the way this distribution changes in time. If she’s the slightest bit like me, or anyone I knew, she freezes up this. Calculate the development of millions of particles? Impossible! She tries working out what happens to just one, instead, and hopes that gives some useful results.

    And then it does.

    It’s a bit much to call this luck. But it is because the student starts off with some simple problems. Particles of gas in a strong box, typically. They don’t interact chemically. Maybe they bounce off each other, but she’s never asked about that. She’s asked about how they bounce off the walls. She can find the relationship between the volume of the box and the internal gas pressure on the interior and the temperature of the gas. And it comes out right.

    She goes on to some other problems and it suddenly fails. Eventually she re-reads the descriptions of how to do this sort of problem. And she does them again and again and it doesn’t feel useful. With luck there’s a moment, possibly while showering, that the universe suddenly changes. And the next time the problem works out. She’s working on distributions instead of toy little single-particle problems.

    But the problem remains: why did it ever work, even for that toy little problem?

    It’s because some systems of things are ergodic. It’s a property that some physics (or mathematics) problems have. Not all. It’s a bit hard to describe clearly. Part of what motivated me to take this topic is that I want to see if I can explain it clearly.

    Every part of some system has a set of possible values it might have. A particle of gas can be in any spot inside the box holding it. A person could be in any of the buildings of her city. A pool ball could be travelling in any direction on the pool table. Sometimes that will change. Gas particles move. People go to the store. Pool balls bounce off the edges of the table.

    These values will have some kind of distribution. Look at where the gas particle is now. And a second from now. And a second after that. And so on, to the limits of human knowledge. Or to when the box breaks open. Maybe the particle will be more often in some areas than in others. Maybe it won’t. Doesn’t matter. It has some distribution. Over time we can say how often we expect to find the gas particle in each of its possible places.

    The same with whatever our system is. People in buildings. Balls on pool tables. Whatever.

    Now instead of looking at one particle (person, ball, whatever) we have a lot of them. Millions of particle in the box. Tens of thousands of people in the city. A pool table that somehow supports ten thousand balls. Imagine they’re all settled to wherever they happen to be.

    So where are they? The gas particle one is easy to imagine. At least for a mathematics major. If you’re stuck on it I’m sorry. I didn’t know. I’ve thought about boxes full of gas particles for decades now and it’s hard to remember that isn’t normal. Let me know if you’re stuck, and where you are. I’d like to know where the conceptual traps are.

    But back to the gas particles in a box. Some fraction of them are in each possible place in the box. There’s a distribution here of how likely you are to find a particle in each spot.

    How does that distribution, the one you get from lots of particles at once, compare to the first, the one you got from one particle given plenty of time? If they agree the system is ergodic. And that’s why my hypothetical physics major got the right answers from the wrong work. (If you are about to write me to complain I’m leaving out important qualifiers let me say I know. Please pretend those qualifiers are in place. If you don’t see what someone might complain about thank you, but it wouldn’t hurt to think of something I might be leaving out here. Try taking a shower.)

    The person in a building is almost certainly not an ergodic system. There’s buildings any one person will never ever go into, however possible it might be. But nearly all buildings have some people who will go into them. The one-person-with-time distribution won’t be the same as the many-people-at-once distribution. Maybe there’s a way to qualify things so that it becomes ergodic. I doubt it.

    The pool table, now, that’s trickier to say. For a real pool table no, of course not. An actual ball on an actual table rolls to a stop pretty soon, either from the table felt’s friction or because it drops into a pocket. Tens of thousands of balls would form an immobile heap on the table that would be pretty funny to see, now that I think of it. Well, maybe those are the same. But they’re a pretty boring same.

    Anyway when we talk about “pool tables” in this context we don’t mean anything so sordid as something a person could play pool on. We mean something where the table surface hasn’t any friction. That makes the physics easier to model. It also makes the game unplayable, which leaves the mathematical physicist strangely unmoved. In this context anyway. We also mean a pool table that hasn’t got any pockets. This makes the game even more unplayable, but the physics even easier. (It makes it, really, like a gas particle in a box. Only without that difficult third dimension to deal with.)

    And that makes it clear. The one ball on a frictionless, pocketless table bouncing around forever maybe we can imagine. A huge number of balls on that frictionless, pocketless table? Possibly trouble. As long as we’re doing imaginary impossible unplayable pool we could pretend the balls don’t collide with each other. Then the distributions of what ways the balls are moving could be equal. If they do bounce off each other, or if they get so numerous they can’t squeeze past one another, well, that’s different.

    An ergodic system lets you do this neat, useful trick. You can look at a single example for a long time. Or you can look at a lot of examples at one time. And they’ll agree in their typical behavior. If one is easier to study than the other, good! Use the one that you can work with. Mathematicians like to do this sort of swapping between equivalent problems a lot.

    The problem is it’s hard to find ergodic systems. We may have a lot of things that look ergodic, that feel like they should be ergodic. But proved ergodic, with a logic that we can’t shake? That’s harder to do. Often in practice we will include a note up top that we are assuming the system to be ergodic. With that “ergodic hypothesis” in mind we carry on with our work. It gives us a handle on a lot of problems that otherwise would be beyond us.

     
  • Joseph Nebus 6:00 pm on Wednesday, 9 November, 2016 Permalink | Reply
    Tags: , , , , , Josiah Willard Gibbs   

    The End 2016 Mathematics A To Z: Distribution (statistics) 


    As I’ve done before I’m using one of my essays to set up for another essay. It makes a later essay easier. What I want to talk about is worth some paragraphs on its own.

    Distribution (statistics)

    The 19th Century saw the discovery of some unsettling truths about … well, everything, really. If there is an intellectual theme of the 19th Century it’s that everything has an unsettling side. In the 20th Century craziness broke loose. The 19th Century, though, saw great reasons to doubt that we knew what we knew.

    But one of the unsettling truths grew out of mathematical physics. We start out studying physics the way Galileo or Newton might have, with falling balls. Ones that don’t suffer from air resistance. Then we move up to more complicated problems, like balls on a spring. Or two balls bouncing off each other. Maybe one ball, called a “planet”, orbiting another, called a “sun”. Maybe a ball on a lever swinging back and forth. We try a couple simple problems with three balls and find out that’s just too hard. We have to track so much information about the balls, about their positions and momentums, that we can’t solve any problems anymore. Oh, we can do the simplest ones, but we’re helpless against the interesting ones.

    And then we discovered something. By “we” I mean people like James Clerk Maxwell and Josiah Willard Gibbs. And that is that we can know important stuff about how millions and billions and even vaster numbers of things move around. Maxwell could work out how the enormously many chunks of rock and ice that make up Saturn’s rings move. Gibbs could work out how the trillions of trillions of trillions of trillions of particles of gas in a room move. We can’t work out how four particles move. How is it we can work out how a godzillion particles move?

    We do it by letting go. We stop looking for that precision and exactitude and knowledge down to infinitely many decimal points. Even though we think that’s what mathematicians and physicists should have. What we do instead is consider the things we would like to know. Where something is. What its momentum is. What side of a coin is showing after a toss. What card was taken off the top of the deck. What tile was drawn out of the Scrabble bag.

    There are possible results for each of these things we would like to know. Perhaps some of them are quite likely. Perhaps some of them are unlikely. We track how likely each of these outcomes are. This is called the distribution of the values. This can be simple. The distribution for a fairly tossed coin is “heads, 1/2; tails, 1/2”. The distribution for a fairly tossed six-sided die is “1/6 chance of 1; 1/6 chance of 2; 1/6 chance of 3” and so on. It can be more complicated. The distribution for a fairly tossed pair of six-sided die starts out “1/36 chance of 2; 2/36 chance of 3; 3/36 chance of 4” and so on. If we’re measuring something that doesn’t come in nice discrete chunks we have to talk about ranges: the chance that a 30-year-old male weighs between 180 and 185 pounds, or between 185 and 190 pounds. The chance that a particle in the rings of Saturn is moving between 20 and 21 kilometers per second, or between 21 and 22 kilometers per second, and so on.

    We may be unable to describe how a system evolves exactly. But often we’re able to describe how the distribution of its possible values evolves. And the laws by which probability work conspire to work for us here. We can get quite precise predictions for how a whole bunch of things behave even without ever knowing what any thing is doing.

    That’s unsettling to start with. It’s made worse by one of the 19th Century’s late discoveries, that of chaos. That a system can be perfectly deterministic. That you might know what every part of it is doing as precisely as you care to measure. And you’re still unable to predict its long-term behavior. That’s unshakeable too, although statistical techniques will give you an idea of how likely different behaviors are. You can learn the distribution of what is likely, what is unlikely, and how often the outright impossible will happen.

    Distributions follow rules. Of course they do. They’re basically the rules you’d imagine from looking at and thinking about something with a range of values. Something like a chart of how many students got what grades in a class, or how tall the people in a group are, or so on. Each possible outcome turns up some fraction of the time. That fraction’s never less than zero nor greater than 1. Add up all the fractions representing all the times every possible outcome happens and the sum is exactly 1. Something happens, even if we never know just what. But we know how often each outcome will.

    There is something amazing to consider here. We can know and track everything there is to know about a physical problem. But we will be unable to do anything with it, except for the most basic and simple problems. We can choose to relax, to accept that the world is unknown and unknowable in detail. And this makes imaginable all sorts of problems that should be beyond our power. Once we’ve given up on this precision we get precise, exact information about what could happen. We can choose to see it as a moral about the benefits and costs and risks of how tightly we control a situation. It’s a surprising lesson to learn from one’s training in mathematics.

     
  • Joseph Nebus 6:00 pm on Wednesday, 2 November, 2016 Permalink | Reply
    Tags: , , eigenvalues, , , , ,   

    The End 2016 Mathematics A To Z: Algebra 


    So let me start the End 2016 Mathematics A To Z with a word everybody figures they know. As will happen, everybody’s right and everybody’s wrong about that.

    Algebra.

    Everybody knows what algebra is. It’s the point where suddenly mathematics involves spelling. Instead of long division we’re on a never-ending search for ‘x’. Years later we pass along gifs of either someone saying “stop asking us to find your ex” or someone who’s circled the letter ‘x’ and written “there it is”. And make jokes about how we got through life without using algebra. And we know it’s the thing mathematicians are always doing.

    Mathematicians aren’t always doing that. I expect the average mathematician would say she almost never does that. That’s a bit of a fib. We have a lot of work where we do stuff that would be recognizable as high school algebra. It’s just we don’t really care about that. We’re doing that because it’s how we get the problem we are interested in done. the most recent few pieces in my “Why Stuff can Orbit” series include a bunch of high school algebra-style work. But that was just because it was the easiest way to answer some calculus-inspired questions.

    Still, “algebra” is a much-used word. It comes back around the second or third year of a mathematics major’s career. It comes in two forms in undergraduate life. One form is “linear algebra”, which is a great subject. That field’s about how stuff moves. You get to imagine space as this stretchy material. You can stretch it out. You can squash it down. You can stretch it in some directions and squash it in others. You can rotate it. These are simple things to build on. You can spend a whole career building on that. It becomes practical in surprising ways. For example, it’s the field of study behind finding equations that best match some complicated, messy real data.

    The second form is “abstract algebra”, which comes in about the same time. This one is alien and baffling for a long while. It doesn’t help that the books all call it Introduction to Algebra or just Algebra and all your friends think you’re slumming. The mathematics major stumbles through confusing definitions and theorems that ought to sound comforting. (“Fermat’s Little Theorem”? That’s a good thing, right?) But the confusion passes, in time. There’s a beautiful subject here, one of my favorites. I’ve talked about it a lot.

    We start with something that looks like the loosest cartoon of arithmetic. We get a bunch of things we can add together, and an ‘addition’ operation. This lets us do a lot of stuff that looks like addition modulo numbers. Then we go on to stuff that looks like picking up floor tiles and rotating them. Add in something that we call ‘multiplication’ and we get rings. This is a bit more like normal arithmetic. Add in some other stuff and we get ‘fields’ and other structures. We can keep falling back on arithmetic and on rotating tiles to build our intuition about what we’re doing. This trains mathematicians to look for particular patterns in new, abstract constructs.

    Linear algebra is not an abstract-algebra sort of algebra. Sorry about that.

    And there’s another kind of algebra that mathematicians talk about. At least once they get into grad school they do. There’s a huge family of these kinds of algebras. The family trait for them is that they share a particular rule about how you can multiply their elements together. I won’t get into that here. There are many kinds of these algebras. One that I keep trying to study on my own and crash hard against is Lie Algebra. That’s named for the Norwegian mathematician Sophus Lie. Pronounce it “lee”, as in “leaning”. You can understand quantum mechanics much better if you’re comfortable with Lie Algebras and so now you know one of my weaknesses. Another kind is the Clifford Algebra. This lets us create something called a “hypercomplex number”. It isn’t much like a complex number. Sorry. Clifford Algebra does lend to a construct called spinors. These help physicists understand the behavior of bosons and fermions. Every bit of matter seems to be either a boson or a fermion. So you see why this is something people might like to understand.

    Boolean Algebra is the algebra of this type that a normal person is likely to have heard of. It’s about what we can build using two values and a few operations. Those values by tradition we call True and False, or 1 and 0. The operations we call things like ‘and’ and ‘or’ and ‘not’. It doesn’t sound like much. It gives us computational logic. Isn’t that amazing stuff?

    So if someone says “algebra” she might mean any of these. A normal person in a non-academic context probably means high school algebra. A mathematician speaking without further context probably means abstract algebra. If you hear something about “matrices” it’s more likely that she’s speaking of linear algebra. But abstract algebra can’t be ruled out yet. If you hear a word like “eigenvector” or “eigenvalue” or anything else starting “eigen” (or “characteristic”) she’s more probably speaking of abstract algebra. And if there’s someone’s name before the word “algebra” then she’s probably speaking of the last of these. This is not a perfect guide. But it is the sort of context mathematicians expect other mathematicians notice.

     
    • John Friedrich 2:13 am on Thursday, 3 November, 2016 Permalink | Reply

      The cruelest trick that happened to me was when a grad school professor labeled the Galois Theory class “Algebra”. Until then, the lowest score I’d ever gotten in a math class was a B. After that, I decided to enter the work force and abandon my attempts at a master’s degree.

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      • Joseph Nebus 3:32 pm on Friday, 4 November, 2016 Permalink | Reply

        Well, it’s true enough that it’s part of algebra. But I’d feel uncomfortable plunging right into that without the prerequisites being really clear. I’m not sure I’ve even run into a nice clear pop-culture explanation of Galois Theory past some notes about how there’s two roots to a quadratic equation and see how they mirror each other.

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  • Joseph Nebus 6:00 pm on Friday, 28 October, 2016 Permalink | Reply
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    Why Stuff Can Orbit, Part 7: ALL the Circles 


    Previously:

    And some supplemental reading:


    Last time around I showed how to do a central-force problem for normal gravity. That’s one where a planet, or moon, or satellite, or whatever is drawn towards the center of space. It’s drawn by a potential energy that equals some constant times the inverse of the distance from the origin. That is, V(r) = C r-1. With a little bit of fussing around we could find out what distance from the center lets a circular orbit happen. And even Kepler’s Third Law, connecting how long an orbit takes to how big it must be.

    There are two natural follow-up essays. One is to work out elliptical orbits. We know there are such things; all real planets and moons have them, and nearly all satellites do. The other is to work out circular orbits for another easy-to-understand example, like a mass on a spring. That’s something with a potential energy that looks like V(r) = C r2.

    I want to do the elliptical orbits later on. The mass-on-a-spring I could do now. So could you, if you look follow last week’s essay and just change the numbers a little. But, you know, why bother working out one problem? Why not work out a lot of them? Why not work out every central-force problem, all at once?

    Because we can’t. I mean, I can describe how to do that, but it isn’t going to save us much time. Like, the quadratic formula is great because it’ll give you the roots of a quadratic polynomial in one step. You don’t have to do anything but a little arithmetic. We can’t get a formula that easy if we try to solve for every possible potential energy.

    But we can work out a lot of central-force potential energies all at once. That is, we can solve for a big set of similar problems, a “family” as we call them. The obvious family is potential energies that are powers of the planet’s distance from the center. That is, they’re potential energies that follow the rule

    V(r) = C r^n

    Here ‘C’ is some number. It might depend on the planet’s mass, or the sun’s mass. Doesn’t matter. All that’s important is that it not change over the course of the problem. So, ‘C’ for Constant. And ‘n’ is another constant number. Some numbers turn up a lot in useful problems. If ‘n’ is -1 then this can describe gravitational attraction. If ‘n’ is 2 then this can describe a mass on a spring. This ‘n’ can be any real number. That’s not an ideal choice of letter. ‘n’ usually designates a whole number. By using that letter I’m biasing people to think of numbers like ‘2’ at the expense of perfectly legitimate alternatives such as ‘2.1’. But now that I’ve made that explicit maybe we won’t make a casual mistake.

    So what I want is to find where there are stable circular orbits for an arbitrary radius-to-a-power force. I don’t know what ‘C’ and ‘n’ are, but they’re some numbers. To find where a planet can have a circular orbit I need to suppose the planet has some mass, ‘m’. And that its orbit has some angular momentum, a number called ‘L’. From this we get the effective potential energy. That’s what the potential energy looks like when we remember that angular momentum has to be conserved.

    V_{eff}(r) = C r^n + \frac{L^2}{2m} r^{-2}

    To find where a circular orbit can be we have to take the first derivative of Veff with respect to ‘r’. The circular orbit can happen at a radius for which this first derivative equals zero. So we need to solve this:

    \frac{dV_{eff}}{dr} = n C r^{n-1} - 2\frac{L^2}{2m} r^{-3} = 0

    That derivative we know from the rules of how to take derivatives. And from this point on we have to do arithmetic. We want to get something which looks like ‘r = (some mathematics stuff here)’. Hopefully it’ll be something not too complicated. And hey, in the second term there, the one with L2 in it, we have a 2 in the numerator and a 2 in the denominator. So those cancel out and that’s simpler. That’s hopeful, isn’t it?

    n C r^{n-1} - \frac{L^2}{m}r^{-3} = 0

    OK. Add \frac{L^2}{m}r^{-3} to both sides of the equation; we’re used to doing that. At least in high school algebra we are.

    n C r^{n-1} = \frac{L^2}{m}r^{-3}

    Not looking much better? Try multiplying both left and right sides by ‘r3‘. This gets rid of all the ‘r’ terms on the right-hand side of the equation.

    n C r^{n+2} = \frac{L^2}{m}

    Now we’re getting close to the ideal of ‘r = (some mathematics stuff)’. Divide both sides by the constant number ‘n times C’.

    r^{n+2} = \frac{L^2}{n C m}

    I know how much everybody likes taking (n+2)-nd roots of a quantity. I’m sure you occasionally just pick an object at random — your age, your telephone number, a potato, a wooden block — and find its (n+2)-nd root. I know. I’ll spoil some of the upcoming paragraphs to say that it’s going to be more useful knowing ‘rn + 2‘ than it is knowing ‘r’. But I’d like to have the radius of a circular orbit on the record. Here it is.

    r = \left(\frac{L^2}{n C m}\right)^{\frac{1}{n + 2}}

    Can we check that this is right? Well, we can at least check that things aren’t wrong. We can check against the example we already know. That’s the gravitational potential energy problem. For that one, ‘C’ is the number ‘G M m’. That’s the gravitational constant of the universe times the mass of the sun times the mass of the planet. And for gravitational potential energy, ‘n’ is equal to -1. This implies that, for a gravitational potential energy problem, we get a circular orbit when

    r_{grav} = \left(\frac{L^2}{n G M m^2}\right)^{\frac{1}{1}}

    I’m labelling it ‘rgrav‘ to point out it’s the radius of a circular orbit for gravitational problems. Might or might not need that in the future, but the label won’t hurt anything.

    Go ahead and guess whether that agrees with last week’s work. I’m feeling confident.

    OK, so, we know where a circular orbit might turn up for an arbitrary power function potential energy. Is it stable? We know from the third “Why Stuff Can Orbit” essay that it’s not a sure thing. We can have potential energies that don’t have any circular orbits. So it must be possible there are unstable orbits.

    Whether our circular orbit is stable demands we do the same work we did last time. It will look a little harder to start, because there’s one more variable in it. What had been ‘-1’ last time is now an ‘n’, and stuff like ‘-2’ becomes ‘n-1’. Is that actually harder? Really?

    So here’s the second derivative of the effective potential:

    \frac{d^2V_{eff}}{dr^2} = (n-1)nCr^{n - 2} + 3\frac{L^2}{m}r^{-4}

    My first impulse when I worked this out was to take the ‘r’ for a circular orbit, the thing worked out five paragraphs above, and plug it in to that expression. This is madness. Don’t do it. Or, you know, go ahead and start doing it and see how long it takes before you regret the errors of your ways.

    The non-madness-inducing way to work out if this is a positive number? It involves noticing r^{n-2} is the same number as r^{n+2}\cdot r^{-4} . So we have this bit of distribution-law magic:

    \frac{d^2V_{eff}}{dr^2} = (n-1)nCr^{n + 2}r^{-4} + 3\frac{L^2}{m}r^{-4}

    \frac{d^2V_{eff}}{dr^2} = \left((n-1)nCr^{n + 2} + 3\frac{L^2}{m}\right) \cdot r^{-4}

    I’m sure we all agree that’s better, right? No, honestly, let me tell you why this is better. When will this expression be true?

    \left((n-1)nCr^{n + 2} + 3\frac{L^2}{m}\right) \cdot r^{-4} > 0

    That’s the product of two expressions. One of them is ‘r-4‘. ‘r’ is the radius of the planet’s orbit. That has to be a positive number. It’s how far the planet is from the origin. The number can’t be anything but positive. So we don’t have to worry about that.

    SPOILER: I just palmed a card there. Did you see me palm a card there? Because I totally did. Watch for where that card turns up. It’ll be after this next bit.

    So let’s look at the non-card-palmed part of this. We’re going to have a stable equilibrium when the other factor of that mess up above is positive. We need to know when this is true:

    (n-1)nCr^{n + 2} + 3\frac{L^2}{m}  > 0

    OK. Well. We do know what ‘rn+2‘ is. Worked that out … uhm … twelve(?) paragraphs ago. I’ll say twelve and hope I don’t mess that up in editing. Anyway, what’s important is r^{n+2} = \frac{L^2}{n C m} . So we put that in where ‘rn+2‘ appeared in that above expression.

    (n-1)nC\frac{L^2}{n C m} + 3 \frac{L^2}{m} > 0

    This is going to simplify down some. Look at that first term, with an ‘n C’ in the numerator and again in the denominator. We’re going to be happier soon as we cancel those out.

    (n-1)\frac{L^2}{m} + 3\frac{L^2}{m} > 0

    And now we get to some fine distributive-law action, the kind everyone likes:

    \left( (n-1) + 3 \right)\frac{L^2}{m} > 0

    Well, we know \frac{L^2}{m} has to be positive. The angular momentum ‘L’ might be positive or might be negative but its square is certainly positive. The mass ‘m’ has to be a positive number. So we’ll get a stable equilibrium whenever (n - 1) + 3 is greater than 0. That is, whenever n > -2 . Done.

    No we’re not done. That’s nonsense. We knew that going in. We saw that a couple essays ago. If your potential energy were something like, say, V(r) = -2 r^3 you wouldn’t have any orbits at all, never mind stable orbits. But 3 is certainly greater than -2. So what’s gone wrong here?

    Let’s go back to that palmed card. Remember I mentioned how the radius of our circular orbit was a positive number. This has to be true, if there is a circular orbit. What if there isn’t one? Do we know there is a radius ‘r’ that the planet can orbit the origin? Here’s the formula giving us that circular orbit’s radius once again:

    r = \left(\frac{L^2}{n C m}\right)^{\frac{1}{n + 2}}

    Do we know that’s going to exist? … Well, sure. That’s going to be some meaningful number as long as we avoid obvious problems. Like, we can’t have the power ‘n’ be equal to zero, because dividing by zero is all sorts of bad. Also we can’t have the constant ‘C’ be zero, again because dividing by zero is bad.

    Not a problem, though. If either ‘C’ or ‘n’ were zero, or if both were, then the original potential energy would be a constant number. V(r) would be equal to ‘C’ (if ‘n’ were zero), or ‘0’ (if ‘C’ were zero). It wouldn’t change with the radius ‘r’. This is a case called the ‘free particle’. There’s no force pushing the planet in one direction or another. So if the planet were not moving it would never start. If the planet were already moving, it would keep moving in the same direction in a straight line. No circular orbits.

    Similarly if ‘n’ were equal to ‘-2’ there’d be problems because the power we raise that parenthetical expression to would be equal to one divided by zero, which is bad. Is there anything else that could be trouble there?

    What if the thing inside parentheses is a negative number? I may not know what ‘n’ is. I don’t. We started off by supposing we didn’t know beyond that it was a number. But I do know that the n-th root of a negative number is going to be trouble. It might be negative. It might be complex-valued. But it won’t be a positive number. And we need a radius that’s a positive number. So that’s the palmed card. To have a circular orbit at all, positive or negative, we have to have:

    \frac{L^2}{n C m} > 0

    ‘L’ is a regular old number, maybe positive, maybe negative. So ‘L2‘ is a positive number. And the mass ‘m’ is a positive number. We don’t know what ‘n’ and C’ are. But as long as their product is positive we’re good. The whole equation will be true. So ‘n’ and ‘C’ can both be negative numbers. We saw that with gravity: V(r) = -\frac{GMm}{r} . ‘G’ is the gravitational constant of the universe, a positive number. ‘M’ and ‘m’ are masses, also positive.

    Or ‘n’ and ‘C’ can both be positive numbers. That turns up with spring problems: V(r) = K r^2 , where ‘K’ is the ‘spring constant’. That’s some positive number again.

    That time we found potential energies that didn’t have orbits? They were ones that had a positive ‘C’ and negative ‘n’, or a negative ‘C’ and positive ‘n’. The case we just worked out doesn’t have circular orbits. It’s nice to have that sorted out at least.

    So what does it mean that we can’t have a stable orbit if ‘n’ is less than or equal to -2? Even if ‘C’ is negative? It turns out that if you have a negative ‘C’ and big negative ‘n’, like say -5, the potential energy drops way down to something infinitely large and negative at smaller and smaller radiuses. If you have a positive ‘C’, the potential energy goes way up at smaller and smaller radiuses. For large radiuses the potential drops to zero. But there’s never the little U-shaped hill in the middle, the way you get for gravity-like potentials or spring potentials or normal stuff like that. Yeah, who would have guessed?

    What if we do have a stable orbit? How long does an orbit take? How does that relate to the radius of the orbit? We used this radius expression to work out Kepler’s Third Law for the gravity problem last week. We can do that again here.

    Last week we worked out what the angular momentum ‘L’ had to be in terms of the radius of the orbit and the time it takes to complete one orbit. The radius of the orbit we called ‘r’. The time an orbit takes we call ‘T’. The formula for angular momentum doesn’t depend on what problem we’re doing. It just depends on the mass ‘m’ of what’s spinning around and how it’s spinning. So:

    L = 2\pi m \frac{r^2}{T}

    And from this we know what ‘L2‘ is.

    L^2 = 4\pi^2 m^2 \frac{r^4}{T^2}

    That’s convenient because we have an ‘L2‘ term in the formula for what the radius is. I’m going to stick with the formula we got for ‘rn+2‘ because that is so, so much easier to work with than ‘r’ by itself. So we go back to that starting point and then substitute what we know ‘L2‘ to be in there.

    r^{n + 2} = \frac{L^2}{n C m}

    This we rewrite as:

    r^{n + 2} = \frac{4 \pi^2 m^2}{n C m}\frac{r^4}{T^2}

    Some stuff starts cancelling out again. One ‘m’ in the numerator and one in the denominator. Small thing but it makes our lives a bit better. We can multiply the left side and the right side by T2. That’s more obviously an improvement. We can divide the left side and the right side by ‘rn + 2‘. And yes that is too an improvement. Watch all this:

    r^{n + 2} = \frac{4 \pi^2 m}{n C}\frac{r^4}{T^2}

    T^2 \cdot r^{n + 2} = \frac{4 \pi^2 m}{n C}r^4

    T^2  = \frac{4 \pi^2 m}{n C}r^{2 - n}

    And that last bit is the equivalent of Kepler’s Third Law for our arbitrary power-law style force.

    Are we right? Hard to say offhand. We can check that we aren’t wrong, at least. We can check against the gravitational potential energy. For this ‘n’ is equal to -1. ‘C’ is equal to ‘-G M m’. Make those substitutions; what do we get?

    T^2  = \frac{4 \pi^2 m}{(-1) (-G M m)}r^{2 - (-1)}

    T^2  = \frac{4 \pi^2}{G M}r^{3}

    Well, that is what we expected for this case. So the work looks good, this far. Comforting.

     
  • Joseph Nebus 6:00 pm on Friday, 21 October, 2016 Permalink | Reply
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    Why Stuff Can Orbit, Part 6: Circles and Where To Find Them 


    Previously:

    And some supplemental reading:


    So now we can work out orbits. At least orbits for a central force problem. Those are ones where a particle — it’s easy to think of it as a planet — is pulled towards the center of the universe. How strong that pull is depends on some constants. But it only changes as the distance the planet is from the center changes.

    What we’d like to know is whether there are circular orbits. By “we” I mean “mathematical physicists”. And I’m including you in that “we”. If you’re reading this far you’re at least interested in knowing how mathematical physicists think about stuff like this.

    It’s easiest describing when these circular orbits exist if we start with the potential energy. That’s a function named ‘V’. We write it as ‘V(r)’ to show it’s an energy that changes as ‘r’ changes. By ‘r’ we mean the distance from the center of the universe. We’d use ‘d’ for that except we’re so used to thinking of distance from the center as ‘radius’. So ‘r’ seems more compelling. Sorry.

    Besides the potential energy we need to know the angular momentum of the planet (or whatever it is) moving around the center. The amount of angular momentum is a number we call ‘L’. It might be positive, it might be negative. Also we need the planet’s mass, which we call ‘m’. The angular momentum and mass let us write a function called the effective potential energy, ‘Veff(r)’.

    And we’ll need to take derivatives of ‘Veff(r)’. Fortunately that “How Differential Calculus Works” essay explains all the symbol-manipulation we need to get started. That part is calculus, but the easy part. We can just follow the rules already there. So here’s what we do:

    • The planet (or whatever) can have a circular orbit around the center at any radius which makes the equation \frac{dV_{eff}}{dr} = 0 true.
    • The circular orbit will be stable if the radius of its orbit makes the second derivative of the effective potential, \frac{d^2V_{eff}}{dr^2} , some number greater than zero.

    We’re interested in stable orbits because usually unstable orbits are boring. They might exist but any little perturbation breaks them down. The mathematician, ordinarily, sees this as a useless solution except in how it describes different kinds of orbits. The physicist might point out that sometimes it can take a long time, possibly millions of years, before the perturbation becomes big enough to stand out. Indeed, it’s an open question whether our solar system is stable. While it seems to have gone millions of years without any planet changing its orbit very much we haven’t got the evidence to say it’s impossible that, say, Saturn will be kicked out of the solar system anytime soon. Or worse, that Earth might be. “Soon” here means geologically soon, like, in the next million years.

    (If it takes so long for the instability to matter then the mathematician might allow that as “metastable”. There are a lot of interesting metastable systems. But right now, I don’t care.)

    I realize now I didn’t explain the notation for the second derivative before. It looks funny because that’s just the best we can work out. In that fraction \frac{d^2V_{eff}}{dr^2} the ‘d’ isn’t a number so we can’t cancel it out. And the superscript ‘2’ doesn’t mean squaring, at least not the way we square numbers. There’s a functional analysis essay in there somewhere. Again I’m sorry about this but there’s a lot of things mathematicians want to write out and sometimes we can’t find a way that avoids all confusion. Roll with it.

    So that explains the whole thing clearly and easily and now nobody could be confused and yeah I know. If my Classical Mechanics professor left it at that we’d have open rebellion. Let’s do an example.

    There are two and a half good examples. That is, they’re central force problems with answers we know. One is gravitation: we have a planet orbiting a star that’s at the origin. Another is springs: we have a mass that’s connected by a spring to the origin. And the half is electric: put a positive electric charge at the center and have a negative charge orbit that. The electric case is only half a problem because it’s the same as the gravitation problem except for what the constants involved are. Electric charges attract each other crazy way stronger than gravitational masses do. But that doesn’t change the work we do.

    This is a lie. Electric charges accelerating, and just orbiting counts as accelerating, cause electromagnetic effects to happen. They give off light. That’s important, but it’s also complicated. I’m not going to deal with that.

    I’m going to do the gravitation problem. After all, we know the answer! By Kepler’s something law, something something radius cubed something G M … something … squared … After all, we can look up the answer!

    The potential energy for a planet orbiting a sun looks like this:

    V(r) = - G M m \frac{1}{r}

    Here ‘G’ is a constant, called the Gravitational Constant. It’s how strong gravity in the universe is. It’s not very strong. ‘M’ is the mass of the sun. ‘m’ is the mass of the planet. To make sense ‘M’ should be a lot bigger than ‘m’. ‘r’ is how far the planet is from the sun. And yes, that’s one-over-r, not one-over-r-squared. This is the potential energy of the planet being at a given distance from the sun. One-over-r-squared gives us how strong the force attracting the planet towards the sun is. Different thing. Related thing, but different thing. Just listing all these quantities one after the other means ‘multiply them together’, because mathematicians multiply things together a lot and get bored writing multiplication symbols all the time.

    Now for the effective potential we need to toss in the angular momentum. That’s ‘L’. The effective potential energy will be:

    V_{eff}(r) = - G M m \frac{1}{r} + \frac{L^2}{2 m r^2}

    I’m going to rewrite this in a way that means the same thing, but that makes it easier to take derivatives. At least easier to me. You’re on your own. But here’s what looks easier to me:

    V_{eff}(r) = - G M m r^{-1} + \frac{L^2}{2 m} r^{-2}

    I like this because it makes every term here look like “some constant number times r to a power”. That’s easy to take the derivative of. Check back on that “How Differential Calculus Works” essay. The first derivative of this ‘Veff(r)’, taken with respect to ‘r’, looks like this:

    \frac{dV_{eff}}{dr} = -(-1) G M m r^{-2} -2\frac{L^2}{2m} r^{-3}

    We can tidy that up a little bit: -(-1) is another way of writing 1. The second term has two times something divided by 2. We don’t need to be that complicated. In fact, when I worked out my notes I went directly to this simpler form, because I wasn’t going to be thrown by that. I imagine I’ve got people reading along here who are watching these equations warily, if at all. They’re ready to bolt at the first sign of something terrible-looking. There’s nothing terrible-looking coming up. All we’re doing from this point on is really arithmetic. It’s multiplying or adding or otherwise moving around numbers to make the equation prettier. It happens we only know those numbers by cryptic names like ‘G’ or ‘L’ or ‘M’. You can go ahead and pretend they’re ‘4’ or ‘5’ or ‘7’ if you like. You know how to do the steps coming up.

    So! We allegedly can have a circular orbit when this first derivative is equal to zero. What values of ‘r’ make true this equation?

    G M m r^{-2} - \frac{L^2}{m} r^{-3} = 0

    Not so helpful there. What we want is to have something like ‘r = (mathematics stuff here)’. We have to do some high school algebra moving-stuff-around to get that. So one thing we can do to get closer is add the quantity \frac{L^2}{m} r^{-3} to both sides of this equation. This gets us:

    G M m r^{-2} = \frac{L^2}{m} r^{-3}

    Things are getting better. Now multiply both sides by the same number. Which number? r3. That’s because ‘r-3‘ times ‘r3‘ is going to equal 1, while ‘r-2‘ times ‘r3‘ will equal ‘r1‘, which normal people call ‘r’. I kid; normal people don’t think of such a thing at all, much less call it anything. But if they did, they’d call it ‘r’. We’ve got:

    G M m r = \frac{L^2}{m}

    And now we’re getting there! Divide both sides by whatever number ‘G M’ is, as long as it isn’t zero. And then we have our circular orbit! It’s at the radius

    r = \frac{L^2}{G M m^2}

    Very good. I’d even say pretty. It’s got all those capital letters and one little lowercase. Something squared in the numerator and the denominator. Aesthetically pleasant. Stinks a little that it doesn’t look like anything we remember from Kepler’s Laws once we’ve looked them up. We can fix that, though.

    The key is the angular momentum ‘L’ there. I haven’t said anything about how that number relates to anything. It’s just been some constant of the universe. In a sense that’s fair enough. Angular momentum is conserved, exactly the same way energy is conserved, or the way linear momentum is conserved. Why not just let it be whatever number it happens to be?

    (A note for people who skipped earlier essays: Angular momentum is not a number. It’s really a three-dimensional vector. But in a central force problem with just one planet moving around we aren’t doing any harm by pretending it’s just a number. We set it up so that the angular momentum is pointing directly out of, or directly into, the sheet of paper we pretend the planet’s orbiting in. Since we know the direction before we even start work, all we have to car about is the size. That’s the number I’m talking about.)

    The angular momentum of a thing is its moment of inertia times its angular velocity. I’m glad to have cleared that up for you. The moment of inertia of a thing describes how easy it is to start it spinning, or stop it spinning, or change its spin. It’s a lot like inertia. What it is depends on the mass of the thing spinning, and how that mass is distributed, and what it’s spinning around. It’s the first part of physics that makes the student really have to know volume integrals.

    We don’t have to know volume integrals. A single point mass spinning at a constant speed at a constant distance from the origin is the easy angular momentum to figure out. A mass ‘m’ at a fixed distance ‘r’ from the center of rotation moving at constant speed ‘v’ has an angular momentum of ‘m’ times ‘r’ times ‘v’.

    So great; we’ve turned ‘L’ which we didn’t know into ‘m r v’, where we know ‘m’ and ‘r’ but don’t know ‘v’. We’re making progress, I promise. The planet’s tracing out a circle in some amount of time. It’s a circle with radius ‘r’. So it traces out a circle with perimeter ‘2 π r’. And it takes some amount of time to do that. Call that time ‘T’. So its speed will be the distance travelled divided by the time it takes to travel. That’s \frac{2 \pi r}{T} . Again we’ve changed one unknown number ‘L’ for another unknown number ‘T’. But at least ‘T’ is an easy familiar thing: it’s how long the orbit takes.

    Let me show you how this helps. Start off with what ‘L’ is:

    L = m r v = m r \frac{2\pi r}{T} = 2\pi m \frac{r^2}{T}

    Now let’s put that into the equation I got eight paragraphs ago:

    r = \frac{L^2}{G M m^2}

    Remember that one? Now put what I just said ‘L’ was, in where ‘L’ shows up in that equation.

    r = \frac{\left(2\pi m \frac{r^2}{T}\right)^2}{G M m^2}

    I agree, this looks like a mess and possibly a disaster. It’s not so bad. Do some cleaning up on that numerator.

    r = \frac{4 \pi^2 m^2}{G M m^2} \frac{r^4}{T^2}

    That’s looking a lot better, isn’t it? We even have something we can divide out: the mass of the planet is just about to disappear. This sounds bizarre, but remember Kepler’s laws: the mass of the planet never figures into things. We may be on the right path yet.

    r = \frac{4 \pi^2}{G M} \frac{r^4}{T^2}

    OK. Now I’m going to multiply both sides by ‘T2‘ because that’ll get that out of the denominator. And I’ll divide both sides by ‘r’ so that I only have the radius of the circular orbit on one side of the equation. Here’s what we’ve got now:

    T^2 = \frac{4 \pi^2}{G M} r^3

    And hey! That looks really familiar. A circular orbit’s radius cubed is some multiple of the square of the orbit’s time. Yes. This looks right. At least it looks reasonable. Someone else can check if it’s right. I like the look of it.

    So this is the process you’d use to start understanding orbits for your own arbitrary potential energy. You can find the equivalent of Kepler’s Third Law, the one connecting orbit times and orbit radiuses. And it isn’t really hard. You need to know enough calculus to differentiate one function, and then you need to be willing to do a pile of arithmetic on letters. It’s not actually hard. Next time I hope to talk about more and different … um …

    I’d like to talk about the different … oh, dear. Yes. You’re going to ask about that, aren’t you?

    Ugh. All right. I’ll do it.

    How do we know this is a stable orbit? Well, it just is. If it weren’t the Earth wouldn’t have a Moon after all this. Heck, the Sun wouldn’t have an Earth. At least it wouldn’t have a Jupiter. If the solar system is unstable, Jupiter is probably the most stable part. But that isn’t convincing. I’ll do this right, though, and show what the second derivative tells us. It tells us this is too a stable orbit.

    So. The thing we have to do is find the second derivative of the effective potential. This we do by taking the derivative of the first derivative. Then we have to evaluate this second derivative and see what value it has for the radius of our circular orbit. If that’s a positive number, then the orbit’s stable. If that’s a negative number, then the orbit’s not stable. This isn’t hard to do, but it isn’t going to look pretty.

    First the pretty part, though. Here’s the first derivative of the effective potential:

    \frac{dV_{eff}}{dr} = G M m r^{-2} - \frac{L^2}{m} r^{-3}

    OK. So the derivative of this with respect to ‘r’ isn’t hard to evaluate again. This is again a function with a bunch of terms that are all a constant times r to a power. That’s the easiest sort of thing to differentiate that isn’t just something that never changes.

    \frac{d^2 V_{eff}}{dr^2} = -2 G M m r^{-3} - (-3)\frac{L^2}{m} r^{-4}

    Now the messy part. We need to work out what that line above is when our planet’s in our circular orbit. That circular orbit happens when r = \frac{L^2}{G M m^2} . So we have to substitute that mess in for ‘r’ wherever it appears in that above equation and you’re going to love this. Are you ready? It’s:

    -2 G M m \left(\frac{L^2}{G M m^2}\right)^{-3} + 3\frac{L^2}{m}\left(\frac{L^2}{G M m^2}\right)^{-4}

    This will get a bit easier promptly. That’s because something raised to a negative power is the same as its reciprocal raised to the positive of that power. So that terrible, terrible expression is the same as this terrible, terrible expression:

    -2 G M m \left(\frac{G M m^2}{L^2}\right)^3 + 3 \frac{L^2}{m}\left(\frac{G M m^2}{L^2}\right)^4

    Yes, yes, I know. Only thing to do is start hacking through all this because I promise it’s going to get better. Putting all those third- and fourth-powers into their parentheses turns this mess into:

    -2 G M m \frac{G^3 M^3 m^6}{L^6} + 3 \frac{L^2}{m} \frac{G^4 M^4 m^8}{L^8}

    Yes, my gut reaction when I see multiple things raised to the eighth power is to say I don’t want any part of this either. Hold on another line, though. Things are going to start cancelling out and getting shorter. Group all those things-to-powers together:

    -2 \frac{G^4 M^4 m^7}{L^6} + 3 \frac{G^4 M^4 m^7}{L^6}

    Oh. Well, now this is different. The second derivative of the effective potential, at this point, is the number

    \frac{G^4 M^4 m^7}{L^6}

    And I admit I don’t know what number that is. But here’s what I do know: ‘G’ is a positive number. ‘M’ is a positive number. ‘m’ is a positive number. ‘L’ might be positive or might be negative, but ‘L6‘ is a positive number either way. So this is a bunch of positive numbers multiplied and divided together.

    So this second derivative what ever it is must be a positive number. And so this circular orbit is stable. Give the planet a little nudge and that’s all right. It’ll stay near its orbit. I’m sorry to put you through that but some people raised the, honestly, fair question.

    So this is the process you’d use to start understanding orbits for your own arbitrary potential energy. You can find the equivalent of Kepler’s Third Law, the one connecting orbit times and orbit radiuses. And it isn’t really hard. You need to know enough calculus to differentiate one function, and then you need to be willing to do a pile of arithmetic on letters. It’s not actually hard. Next time I hope to talk about the other kinds of central forces that you might get. We only solved one problem here. We can solve way more than that.

     
    • howardat58 6:18 pm on Friday, 21 October, 2016 Permalink | Reply

      I love the chatty approach.

      Like

      • Joseph Nebus 5:03 am on Saturday, 22 October, 2016 Permalink | Reply

        Thank you. I realized doing Theorem Thursdays over the summer that it was hard to avoid that voice, and then that it was fun writing in it. So eventually I do learn, sometimes.

        Like

  • Joseph Nebus 6:00 pm on Friday, 14 October, 2016 Permalink | Reply
    Tags: , , , , , harmonic motion, perturbations,   

    How Mathematical Physics Works: Another Course In 2200 Words 


    OK, I need some more background stuff before returning to the Why Stuff Can Orbit series. Last week I explained how to take derivatives, which is one of the three legs of a Calculus I course. Now I need to say something about why we take derivatives. This essay won’t really qualify you to do mathematical physics, but it’ll at least let you bluff your way through a meeting with one.

    We care about derivatives because we’re doing physics a smart way. This involves thinking not about forces but instead potential energy. We have a function, called V or sometimes U, that changes based on where something is. If we need to know the forces on something we can take the derivative, with respect to position, of the potential energy.

    The way I’ve set up these central force problems makes it easy to shift between physical intuition and calculus. Draw a scribbly little curve, something going up and down as you like, as long as it doesn’t loop back on itself. Also, don’t take the pen from paper. Also, no corners. That’s just cheating. Smooth curves. That’s your potential energy function. Take any point on this scribbly curve. If you go to the right a little from that point, is the curve going up? Then your function has a positive derivative at that point. Is the curve going down? Then your function has a negative derivative. Find some other point where the curve is going in the other direction. If it was going up to start, find a point where it’s going down. Somewhere in-between there must be a point where the curve isn’t going up or going down. The Intermediate Value Theorem says you’re welcome.

    These points where the potential energy isn’t increasing or decreasing are the interesting ones. At least if you’re a mathematical physicist. They’re equilibriums. If whatever might be moving happens to be exactly there, then it’s not going to move. It’ll stay right there. Mathematically: the force is some fixed number times the derivative of the potential energy there. The potential energy’s derivative is zero there. So the force is zero and without a force nothing’s going to change. Physical intuition: imagine you laid out a track with exactly the shape of your curve. Put a marble at this point where the track isn’t rising and isn’t falling. Does the marble move? No, but if you’re not so sure about that read on past the next paragraph.

    Mathematical physicists learn to look for these equilibriums. We’re taught to not bother with what will happen if we release this particle at this spot with this velocity. That is, you know, not looking at any particular problem someone might want to know. We look instead at equilibriums because they help us describe all the possible behaviors of a system. Mathematicians are sometimes characterized as lazy in spirit. This is fair. Mathematicians will start out with a problem looking to see if it’s just like some other problem someone already solved. But the flip side is if one is going to go to the trouble of solving a new problem, she’s going to really solve it. We’ll work out not just what happens from some one particular starting condition. We’ll try to describe all the different kinds of thing that could happen, and how to tell which of them does happen for your measly little problem.

    If you actually do have a curvy track and put a marble down on its equilibrium it might yet move. Suppose the track is rising a while and then falls back again; putting the marble at top and it’s likely to roll one way or the other. If it doesn’t it’s probably because of friction; the track sticks a little. If it were a really smooth track and the marble perfectly round then it’d fall. Give me this. But even with a perfectly smooth track and perfectly frictionless marble it’ll still roll one way or another. Unless you put it exactly at the spot that’s the top of the hill, not a bit to the left or the right. Good luck.

    What’s happening here is the difference between a stable and an unstable equilibrium. This is again something we all have a physical intuition for. Imagine you have something that isn’t moving. Give it a little shove. Does it stay about like it was? Then it’s stable. Does it break? Then it’s unstable. The marble at the top of the track is at an unstable equilibrium; a little nudge and it’ll roll away. If you had a marble at the bottom of a track, inside a valley, then it’s a stable equilibrium. A little nudge will make the marble rock back and forth but it’ll stay nearby.

    Yes, if you give it a crazy big whack the marble will go flying off, never to be seen again. We’re talking about small nudges. No, smaller than that. This maybe sounds like question-begging to you. But what makes for an unstable equilibrium is that no nudge is too small. The nudge — perturbation, in the trade — will just keep growing. In a stable equilibrium there’s nudges small enough that they won’t keep growing. They might not shrink, but they won’t grow either.

    So how to tell which is which? Well, look at your potential energy and imagine it as a track with a marble again. Where are the unstable equilibriums? They’re the ones at tops of hills. Near them the curve looks like a cup pointing down, to use the metaphor every Calculus I class takes. Where are the stable equilibriums? They’re the ones at bottoms of valleys. Near them the curve looks like a cup pointing up. Again, see Calculus I.

    We may be able to tell the difference between these kinds of equilibriums without drawing the potential energy. We can use the second derivative. To find the second derivative of a function you take the derivative of a function and then — you may want to think this one over — take the derivative of that. That is, you take the derivative of the original function a second time. Sometimes higher mathematics gives us terms that aren’t too hard.

    So if you have a spot where you know there’s an equilibrium, look at what the second derivative at that spot is. If it’s positive, you have a stable equilibrium. If it’s negative, you have an unstable equilibrium. This is called “Second Derivative Test”, as it was named by a committee that figured it was close enough to 5 pm and why cause trouble?

    If the second derivative is zero there, um, we can’t say anything right now. The equilibrium may also be an inflection point. That’s where the growth of something pauses a moment before resuming. Or where the decline of something pauses a moment before resuming. In either case that’s still an unstable equilibrium. But it doesn’t have to be. It could still be a stable equilibrium. It might just have a very smoothly flat base. No telling just from that one piece of information and this is why we have to go on to other work.

    But this gets at how we’d like to look at a system. We look for its equilibriums. We figure out which equilibriums are stable and which ones are unstable. With a little more work we can say, if the system starts out like this it’ll stay near that equilibrium. If it starts out like that it’ll stay near this whole other equilibrium. If it starts out this other way, it’ll go flying off to the end of the universe. We can solve every possible problem at once and never have to bother with a particular case. This feels good.

    It also gives us a little something more. You maybe have heard of a tangent line. That’s a line that’s, er, tangent to a curve. Again with the not-too-hard terms. What this means is there’s a point, called the “point of tangency”, again named by a committee that wanted to get out early. And the line just touches the original curve at that point, and it’s going in exactly the same direction as the original curve at that point. Typically this means the line just grazes the curve, at least around there. If you’ve ever rolled a pencil until it just touched the edge of your coffee cup or soda can, you’ve set up a tangent line to the curve of your beverage container. You just didn’t think of it as that because you’re not daft. Fair enough.

    Mathematicians will use tangents because a tangent line has values that are so easy to calculate. The function describing a tangent line is a polynomial and we llllllllove polynomials, correctly. The tangent line is always easy to understand, however hard the original function was. Its value, at the equilibrium, is exactly what the original function’s was. Its first derivative, at the equilibrium, is exactly what the original function’s was at that point. Its second derivative is zero, which might or might not be true of the original function. We don’t care.

    We don’t use tangent lines when we look at equilibriums. This is because in this case they’re boring. If it’s an equilibrium then its tangent line is a horizontal line. No matter what the original function was. It’s trivial: you know the answer before you’ve heard the question.

    Ah, but, there is something mathematical physicists do like. The tangent line is boring. Fine. But how about, using the second derivative, building a tangent … well, “parabola” is the proper term. This is a curve that’s a quadratic, that looks like an open bowl. It exactly matches the original function at the equilibrium. Its derivative exactly matches the original function’s derivative at the equilibrium. Its second derivative also exactly matches the original function’s second derivative, though. Third derivative we don’t care about. It’s so not important here I can’t even finish this sentence in a

    What this second-derivative-based approximation gives us is a parabola. It will look very much like the original function if we’re close to the equilibrium. And this gives us something great. The great thing is this is the same potential energy shape of a weight on a spring, or anything else that oscillates back and forth. It’s the potential energy for “simple harmonic motion”.

    And that’s great. We start studying simple harmonic motion, oh, somewhere in high school physics class because it’s so much fun to play with slinkies and springs and accidentally dropping weights on our lab partners. We never stop. The mathematics behind it is simple. It turns up everywhere. If you understand the mathematics of a mass on a spring you have a tool that relevant to pretty much every problem you ever have. This approximation is part of that. Close to a stable equilibrium, whatever system you’re looking at has the same behavior as a weight on a spring.

    It may strike you that a mass on a spring is itself a central force. And now I’m saying that within the central force problem I started out doing, stuff that orbits, there’s another central force problem. This is true. You’ll see that in a few Why Stuff Can Orbit essays.

    So far, by the way, I’ve talked entirely about a potential energy with a single variable. This is for a good reason: two or more variables is harder. Well of course it is. But the basic dynamics are still open. There’s equilibriums. They can be stable or unstable. They might have inflection points. There is a new kind of behavior. Mathematicians call it a “saddle point”. This is where in one direction the potential energy makes it look like a stable equilibrium while in another direction the potential energy makes it look unstable. Examples of it kind of look like the shape of a saddle, if you haven’t looked at an actual saddle recently. (If you really want to know, get your computer to plot the function z = x2 – y2 and look at the origin, where x = 0 and y = 0.) Well, there’s points on an actual saddle that would be saddle points to a mathematician. It’s unstable, because there’s that direction where it’s definitely unstable.

    So everything about multivariable functions is longer, and a couple bits of it are harder. There’s more chances for weird stuff to happen. I think I can get through most of Why Stuff Can Orbit without having to know that. But do some reading up on that before you take a job as a mathematical physicist.

     
  • Joseph Nebus 6:00 pm on Friday, 7 October, 2016 Permalink | Reply
    Tags: , , ,   

    How Differential Calculus Works 


    I’m at a point in my Why Stuff Can Orbit essays where I need to talk about derivatives. If I don’t then I’m going to be stuck with qualitative talk that’s too hard to focus on anything. But it’s a big subject. It’s about two-fifths of a Freshman Calculus course and one of the two main legs of calculus. So I want to pull out a quick essay explaining the important stuff.

    Derivatives, also called differentials, are about how things change. By “things” I mean “functions”. And particularly I mean functions which have as a domain that’s in the real numbers and a range that’s in the real numbers. That’s the starting point. We can define derivatives for functions with domains or ranges that are vectors of real numbers, or that are complex-valued numbers, or are even more exotic kinds of numbers. But those ideas are all built on the derivatives for real-valued functions. So if you get really good at them, everything else is easy.

    Derivatives are the part of calculus where we study how functions change. They’re blessed with some easy-to-visualize interpretations. Suppose you have a function that describes where something is at a given time. Then its derivative with respect to time is the thing’s velocity. You can build a pretty good intuitive feeling for derivatives that way. I won’t stop you from it.

    A function’s derivative is itself a function. The derivative doesn’t have to be constant, although it’s possible. Sometimes the derivative is positive. This means the original function increases as whatever its variable increases. Sometimes the derivative is negative. This means the original function decreases as its variable increases. If the derivative is a big number this means it’s increasing or decreasing fast. If the derivative is a small number it’s increasing or decreasing slowly. If the derivative is zero then the function isn’t increasing or decreasing, at least not at this particular value of the variable. This might be a local maximum, where the function’s got a larger value than it has anywhere else nearby. It might be a local minimum, where the function’s got a smaller value than it has anywhere else nearby. Or it might just be a little pause in the growth or shrinkage of the function. No telling, at least not just from knowing the derivative is zero.

    Derivatives tell you something about where the function is going. We can, and in practice often do, make approximations to a function that are built on derivatives. Many interesting functions are real pains to calculate. Derivatives let us make polynomials that are as good an approximation as we need and that a computer can do for us instead.

    Derivatives can change rapidly. That’s all right. They’re functions in their own right. Sometimes they change more rapidly and more extremely than the original function did. Sometimes they don’t. Depends what the original function was like.

    Not every function has a derivative. Functions that have derivatives don’t necessarily have them at every point. A function has to be continuous to have a derivative. A function that jumps all over the place doesn’t really have a direction, not that differential calculus will let us suss out. But being continuous isn’t enough. The function also has to be … well, we call it “differentiable”. The word at least tells us what the property is good for, even if it doesn’t say how to tell if a function is differentiable. The function has to not have any corners, points where the direction suddenly and unpredictably changes.

    Otherwise differentiable functions can have some corners, some non-differentiable points. For instance the height of a bouncing ball, over time, looks like a bunch of upside-down U-like shapes that suddenly rebound off the floor and go up again. Those points interrupting the upside-down U’s aren’t differentiable, even though the rest of the curve is.

    It’s possible to make functions that are nothing but corners and that aren’t differentiable anywhere. These are done mostly to win quarrels with 19th century mathematicians about the nature of the real numbers. We use them in Real Analysis to blow students’ minds, and occasionally after that to give a fanciful idea a hard test. In doing real-world physics we usually don’t have to think about them.

    So why do we do them? Because they tell us how functions change. They can tell us where functions momentarily don’t change, which are usually important. And equations with differentials in them, known in the trade as “differential equations”. They’re also known to mathematics majors as “diffy Q’s”, a name which delights everyone. Diffy Q’s let us describe physical systems where there’s any kind of feedback. If something interacts with its surroundings, that interaction’s probably described by differential equations.

    So how do we do them? We start with a function that we’ll call ‘f’ because we’re not wasting more of our lives figuring out names for functions and we’re feeling smugly confident Turkish authorities aren’t interested in us.

    f’s domain is real numbers, for example, the one we’ll call ‘x’. Its range is also real numbers. f(x) is some number in that range. It’s the one that the function f matches with the domain’s value of x. We read f(x) aloud as “the value of f evaluated at the number x”, if we’re pretending or trying to scare Freshman Calculus classes. Really we say “f of x” or maybe “f at x”.

    There’s a couple ways to write down the derivative of f. First, for example, we say “the derivative of f with respect to x”. By that we mean how does the value of f(x) change if there’s a small change in x. That difference matters if we have a function that depends on more than one variable, if we had an “f(x, y)”. Having several variables for f changes stuff. Mostly it makes the work longer but not harder. We don’t need that here.

    But there’s a couple ways to write this derivative. The best one if it’s a function of one variable is to put a little ‘ mark: f'(x). We pronounce that “f-prime of x”. That’s good enough if there’s only the one variable to worry about. We have other symbols. One we use a lot in doing stuff with differentials looks for all the world like a fraction. We spend a lot of time in differential calculus explaining why it isn’t a fraction, and then in integral calculus we do some shady stuff that treats it like it is. But that’s \frac{df}{dx} . This also appears as \frac{d}{dx} f . If you encounter this do not cross out the d’s from the numerator and denominator. In this context the d’s are not numbers. They’re actually operators, which is what we call a function whose domain is functions. They’re notational quirks is all. Accept them and move on.

    How do we calculate it? In Freshman Calculus we introduce it with this expression involving limits and fractions and delta-symbols (the Greek letter that looks like a triangle). We do that to scare off students who aren’t taking this stuff seriously. Also we use that formal proper definition to prove some simple rules work. Then we use those simple rules. Here’s some of them.

    1. The derivative of something that doesn’t change is 0.
    2. The derivative of xn, where n is any constant real number — positive, negative, whole, a fraction, rational, irrational, whatever — is n xn-1.
    3. If f and g are both functions and have derivatives, then the derivative of the function f plus g is the derivative of f plus the derivative of g. This probably has some proper name but it’s used so much it kind of turns invisible. I dunno. I’d call it something about linearity because “linearity” is what happens when adding stuff works like adding numbers does.
    4. If f and g are both functions and have derivatives, then the derivative of the function f times g is the derivative of f times the original g plus the original f times the derivative of g. This is called the Product Rule.
    5. If f and g are both function and have derivatives we might do something called composing them. That is, for a given x we find the value of g(x), and then we put that value in to f. That we write as f(g(x)). For example, “the sine of the square of x”. Well, the derivative of that with respect to x is the derivative of f evaluated at g(x) times the derivative of g. That is, f'(g(x)) times g'(x). This is called the Chain Rule and believe it or not this turns up all the time.
    6. If f is a function and C is some constant number, then the derivative of C times f(x) is just C times the derivative of f(x), that is, C times f'(x). You don’t really need this as a separate rule. If you know the Product Rule and that first one about the derivatives of things that don’t change then this follows. But who wants to go through that many steps for a measly little result like this?
    7. Calculus textbooks say there’s this Quotient Rule for when you divide f by g but you know what? The one time you’re going to need it it’s easier to work out by the Product Rule and the Chain Rule and your life is too short for the Quotient Rule. Srsly.
    8. There’s some functions with special rules for the derivative. You can work them out from the Real And Proper Definition. Or you can work them out from infinitely-long polynomial approximations that somehow make sense in context. But what you really do is just remember the ones anyone actually uses. The derivative of ex is ex and we love it for that. That’s why e is that 2.71828(etc) number and not something else. The derivative of the natural log of x is 1/x. That’s what makes it natural. The derivative of the sine of x is the cosine of x. That’s if x is measured in radians, which it always is in calculus, because it makes that work. The derivative of the cosine of x is minus the sine of x. Again, radians. The derivative of the tangent of x is um the square of the secant of x? I dunno, look that sucker up. The derivative of the arctangent of x is \frac{1}{1 + x^2} and you know what? The calculus book lists this in a table in the inside covers. Just use that. Arctangent. Sheesh. We just made up the “secant” and the “cosecant” to haze Freshman Calculus students anyway. We don’t get a lot of fun. Let us have this. Or we’ll make you hear of the inverse hyperbolic cosecant.

    So. What’s this all mean for central force problems? Well, here’s what the effective potential energy V(r) usually looks like:

    V_{eff}(r) = C r^n + \frac{L^2}{2 m r^2}

    So, first thing. The variable here is ‘r’. The variable name doesn’t matter. It’s just whatever name is convenient and reminds us somehow why we’re talking about this number. We adapt our rules about derivatives with respect to ‘x’ to this new name by striking out ‘x’ wherever it appeared and writing in ‘r’ instead.

    Second thing. C is a constant. So is n. So is L and so is m. 2 is also a constant, which you never think about until a mathematician brings it up. That second term is a little complicated in form. But what it means is “a constant, which happens to be the number \frac{L^2}{2m} , multiplied by r-2”.

    So the derivative of Veff, with respect to r, is the derivative of the first term plus the derivative of the second. The derivatives of each of those terms is going to be some constant time r to another number. And that’s going to be:

    V'_{eff}(r) = C n r^{n - 1} - 2 \frac{L^2}{2 m r^3}

    And there you can divide the 2 in the numerator and the 2 in the denominator. So we could make this look a little simpler yet, but don’t worry about that.

    OK. So that’s what you need to know about differential calculus to understand orbital mechanics. Not everything. Just what we need for the next bit.

     
    • davekingsbury 1:06 pm on Saturday, 8 October, 2016 Permalink | Reply

      Good article. Just finished Morton Cohen’s biography of Lewis Carroll, who was a great populariser of mathematics, logic, etc. Started a shared poem in tribute to him, here is a cheeky plug, hope you don’t mind!

      https://davekingsbury.wordpress.com/2016/10/08/web-of-life/

      Like

      • Joseph Nebus 12:22 am on Tuesday, 11 October, 2016 Permalink | Reply

        Thanks for sharing and I’m quite happy to have your plug here. I know about Carroll’s mathematics-popularization side; his logic puzzles are particularly choice ones even today. (Granting that deductive logic really lends itself to being funny.)

        Oddly I haven’t read a proper biography of Carroll, or most of the other mathematicians I’m interested in. Which is strange since I’m so very interested in the history and the cultural development of mathematics.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Wednesday, 28 September, 2016 Permalink | Reply
    Tags: , , , , , , ,   

    Why Stuff Can Orbit, Part 5: Why Physics Doesn’t Work And What To Do About It 


    Less way previously:


    My title’s hyperbole, to the extent it isn’t clickbait. Of course physics works. By “work” I mean “model the physical world in useful ways”. If it didn’t work then we would call it “pure” mathematics instead. Mathematicians would study it for its beauty. Physicists would be left to fend for themselves. “Useful” I’ll say means “gives us something interesting to know”. “Interesting” I’ll say if you want to ask what that means then I think you’re stalling.

    But what I mean is that Newtonian physics, the physics learned in high school, doesn’t work. Well, it works, in that if you set up a problem right and calculate right you get answers that are right. It’s just not efficient, for a lot of interesting problems. Don’t ask me about interesting again. I’ll just say the central-force problems from this series are interesting.

    Newtonian, high school type, physics works fine. It shines when you have only a few things to keep track of. In this central force problem we have one object, a planet-or-something, that moves. And only one force, one that attracts the planet to or repels the planet from the center, the Origin. This is where we’d put the sun, in a planet-and-sun system. So that seems all right as far as things go.

    It’s less good, though, if there’s constraints. If it’s not possible for the particle to move in any old direction, say. That doesn’t turn up here; we can imagine a planet heading in any direction relative to the sun. But it’s also less good if there’s a symmetry in what we’re studying. And in this case there is. The strength of the central force only changes based on how far the planet is from the origin. The direction only changes based on what direction the planet is relative to the origin. It’s a bit daft to bother with x’s and y’s and maybe even z’s when all we care about is the distance from the origin. That’s a number we’ve called ‘r’.

    So this brings us to Lagrangian mechanics. This was developed in the 18th century by Joseph-Louis Lagrange. He’s another of those 18th century mathematicians-and-physicists with his name all over everything. Lagrangian mechanics are really, really good when there’s a couple variables that describe both what we’d like to observe about the system and its energy. That’s exactly what we have with central forces. Give me a central force, one that’s pointing directly toward or away from the origin, and that grows or shrinks as the radius changes. I can give you a potential energy function, V(r), that matches that force. Give me an angular momentum L for the planet to have, and I can give you an effective potential energy function, Veff(r). And that effective potential energy lets us describe how the coordinates change in time.

    The method looks roundabout. It depends on two things. One is the coordinate you’re interested in, in this case, r. The other is how fast that coordinate changes in time. This we have a couple of ways of denoting. When working stuff out on paper that’s often done by putting a little dot above the letter. If you’re typing, dots-above-the-symbol are hard. So we mark it as a prime instead: r’. This works well until the web browser or the word processor assumes we want smart quotes and we already had the r’ in quote marks. At that point all hope of meaning is lost and we return to communicating by beating rocks with sticks. We live in an imperfect world.

    What we get out of this is a setup that tells us how fast r’, how fast the coordinate we’re interested in changes in time, itself changes in time. If the coordinate we’re interested in is the ordinary old position of something, then this describes the rate of change of the velocity. In ordinary English we call that the acceleration. What makes this worthwhile is that the coordinate doesn’t have to be the position. It also doesn’t have to be all the information we need to describe the position. For the central force problem r here is just how far the planet is from the center. That tells us something about its position, but not everything. We don’t care about anything except how far the planet is from the center, not yet. So it’s fine we have a setup that doesn’t tell us about the stuff we don’t care about.

    How fast r’ changes in time will be proportional to how fast the effective potential energy, Veff(r), changes with its coordinate. I so want to write “changes with position”, since these coordinates are usually the position. But they can be proxies for the position, or things only loosely related to the position. For an example that isn’t a central force, think about a spinning top. It spins, it wobbles, it might even dance across the table because don’t they all do that? The coordinates that most sensibly describe how it moves are about its rotation, though. What axes is it rotating around? How do those change in time? Those don’t have anything particular to do with where the top is. That’s all right. The mathematics works just fine.

    A circular orbit is one where the radius doesn’t change in time. (I’ll look at non-circular orbits later on.) That is, the radius is not increasing and is not decreasing. If it isn’t getting bigger and it isn’t getting smaller, then it’s got to be staying the same. Not all higher mathematics is tricky. The radius of the orbit is the thing I’ve been calling r all this time. So this means that r’, how fast r is changing with time, has to be zero. Now a slightly tricky part.

    How fast is r’, the rate at which r changes, changing? Well, r’ never changes. It’s always the same value. Anytime something is always the same value the rate of its change is zero. This sounds tricky. The tricky part is that it isn’t tricky. It’s coincidental that r’ is zero and the rate of change of r’ is zero, though. If r’ were any fixed, never-changing number, then the rate of change of r’ would be zero. It happens that we’re interested in times when r’ is zero.

    So we’ll find circular orbits where the change in the effective potential energy, as r changes, is zero. There’s an easy-to-understand intuitive idea of where to find these points. Look at a plot of Veff and imagine this is a smooth track or the cross-section of a bowl or the landscaping of a hill. Imagine dropping a ball or a marble or a bearing or something small enough to roll in it. Where does it roll to a stop? That’s where the change is zero.

    It’s too much bother to make a bowl or landscape a hill or whatnot for every problem we’re interested in. We might do it anyway. Mathematicians used to, to study problems that were too complicated to do by useful estimates. These were “analog computers”. They were big in the days before digital computers made it no big deal to simulate even complicated systems. We still need “analog computers” or models sometimes. That’s usually for problems that involve chaotic stuff like turbulent fluids. We call this stuff “wind tunnels” and the like. It’s all a matter of solving equations by building stuff.

    We’re not working with problems that complicated. There isn’t the sort of chaos lurking in this problem that drives us to real-world stuff. We can find these equilibriums by working just with symbols instead.

     
  • Joseph Nebus 6:00 pm on Saturday, 24 September, 2016 Permalink | Reply
    Tags: handedness, ,   

    Some Thermomathematics Reading 


    I have been writing, albeit more slowly, this month. I’m also reading, also more slowly than usual. Here’s some things that caught my attention.

    One is from Elke Stangl, of the Elkemental blog. “Re-Visiting Carnot’s Theorem” is about one of the centerpieces of thermodynamics. It’s about how much work you can possibly get out of an engine, and how much must be lost no matter how good your engineering is. Thermodynamics is the secret spine of modern physics. It was born of supremely practical problems, many of them related to railroads or factories. And it teaches how much solid information can be drawn about a system if we know nothing about the components of the system. Stangl also brings ASCII art back from its Usenet and Twitter homes. There’s just stuff that is best done as a text picture.

    Meanwhile on the CarnotCycle blog Peter Mandel writes on “Le Châtelier’s principle”. This is related to the question of how temperatures affect chemical reactions: how fast they will be, how completely they’ll use the reagents. How a system that’s reached equilibrium will react to something that unsettles the equilibrium. We call that a perturbation. Mandel reviews the history of the principle, which hasn’t always been well-regarded, and explores why it might have gone under-appreciated for decades.

    And lastly MathsByAGirl has published a couple of essays on spirals. Who doesn’t like them? Three-dimensional spirals, that is, helixes, have some obvious things to talk about. A big one is that there’s such a thing as handedness. The mirror image of a coil is not the same thing as the coil flipped around. This handedness has analogues and implications through chemistry and biology. Two-dimensional spirals, by contrast, don’t have handedness like that. But we’ve groups types of spirals into many different sorts, each with their own beauty. They’re worth looking at.

     
  • Joseph Nebus 6:00 pm on Thursday, 8 September, 2016 Permalink | Reply
    Tags: , , , , , , , , ,   

    Why Stuff Can Orbit, Part 4: On The L 


    Less way previously:


    We were chatting about central forces. In these a small object — a satellite, a planet, a weight on a spring — is attracted to the center of the universe, called the origin. We’ve been studying this by looking at potential energy, a function that in this case depends only on how far the object is from the origin. But to find circular orbits, we can’t just look at the potential energy. We have to modify this potential energy to account for angular momentum. This essay I mean to discuss that angular momentum some.

    Let me talk first about the potential energy. Mathematical physicists usually write this as a function named U or V. I’m using V. That’s what my professor used teaching this, back when I was an undergraduate several hundred thousand years ago. A central force, by definition, changes only with how far you are from the center. I’ve put the center at the origin, because I am not a madman. This lets me write the potential energy as V = V(r).

    V(r) could, in principle, be anything. In practice, though, I am going to want it to be r raised to a power. That is, V(r) is equal to C rn. The ‘C’ here is a constant. It’s a scaling constant. The bigger a number it is the stronger the central force. The closer the number is to zero the weaker the force is. In standard units, gravity has a constant incredibly close to zero. This makes orbits very big things, which generally works out well for planets. In the mathematics of masses on springs, the constant is closer to middling little numbers like 1.

    The ‘n’ here is a deceiver. It’s a constant number, yes, and it can be anything we want. But the use of ‘n’ as a symbol has connotations. Usually when a mathematician or a physicist writes ‘n’ it’s because she needs a whole number. Usually a positive whole number. Sometimes it’s negative. But we have a legitimate central force if ‘n’ is any real number: 2, -1, one-half, the square root of π, any of that is good. If you just write ‘n’ without explanation, the reader will probably think “integers”, possibly “counting numbers”. So it’s worth making explicit when this isn’t so. It’s bad form to surprise the reader with what kind of number you’re even talking about.

    (Some number of essays on we’ll find out that the only values ‘n’ can have that are worth anything are -1, 2, and 7. And 7 isn’t all that good. But we aren’t supposed to know that yet.)

    C rn isn’t the only kind of central force that could exist. Any function rule would do. But it’s enough. If we wanted a more complicated rule we could just add two, or three, or more potential energies together. This would give us V(r) = C_1 r^{n_1} + C_2 r^{n_2} , with C1 and C2 two possibly different numbers, and n1 and n2 two definitely different numbers. (If n1 and n2 were the same number then we should just add C1 and C2 together and stop using a more complicated expression than we need.) Remember that Newton’s Law of Motion about the sum of multiple forces being something vector something something direction? When we look at forces as potential energy functions, that law turns into just adding potential energies together. They’re well-behaved that way.

    And if we can add these r-to-a-power potential energies together then we’ve got everything we need. Why? Polynomials. We can approximate most any potential energy that would actually happen with a big enough polynomial. Or at least a polynomial-like function. These r-to-a-power forces are a basis set for all the potential energies we’re likely to care about. Understand how to work with one and you understand how to work with them all.

    Well, one exception. The logarithmic potential, V(r) = C log(r), is really interesting. And it has real-world applicability. It describes how strongly two vortices, two whirlpools, attract each other. You can write the logarithm as a polynomial. But logarithms are pretty well-behaved functions. You might be better off just doing that as a special case.

    Still, at least to start with, we’ll stick with V(r) = C rn and you know what I mean by all those letters now. So I’m free to talk about angular momentum.

    You’ve probably heard of momentum. It’s got something to do with movement, only sports teams and political campaigns are always gaining or losing it somehow. When we talk of that we’re talking of linear momentum. It describes how much mass is moving how fast in what direction. So it’s a vector, in three-dimensional space. Or two-dimensional space if you’re making the calculations easier. To find what the vector is, we make a list of every object that’s moving. We take its velocity — how fast it’s moving and in what direction — and multiply that by its mass. Mass is a single number, a scalar, and we’re always allowed to multiply a vector by a scalar. This gets us another vector. Once we’ve done that for everything that’s moving, we add all those product vectors together. We can always add vectors together. And this gives us a grand total vector, the linear momentum of the system.

    And that’s conserved. If one part of the system starts moving slower it’s because other parts are moving faster, and vice-versa. In the real world momentum seems to evaporate. That’s because some of the stuff moving faster turns out to be air objects bumped into, or particles of the floor that get dragged along by friction, or other stuff we don’t care about. That momentum can seem to evaporate is what makes its use in talking about ports teams or political campaigns make sense. It also annoys people who want you to know they understand science words better than you. So please consider this my authorization to use “gaining” and “losing” momentum in this sense. Ignore complainers. They’re the people who complain the word “decimate” gets used to mean “destroy way more than ten percent of something”, even though that’s the least bad mutation of an English word’s meaning in three centuries.

    Angular momentum is also a vector. It’s also conserved. We can calculate what that vector is by the same sort of process, that of calculating something on each object that’s spinning and adding it all up. In real applications it can seem to evaporate. But that’s also because the angular momentum is going into particles of air. Or it rubs off grease on the axle. Or it does other stuff we wish we didn’t have to deal with.

    The calculation is a little harder to deal with. There’s three parts to a spinning thing. There’s the thing, and there’s how far it is from the axis it’s spinning around, and there’s how fast it’s spinning. So you need to know how fast it’s travelling in the direction perpendicular to the shortest line between the thing and the axis it’s spinning around. Its angular momentum is going to be as big as the mass times the distance from the axis times the perpendicular speed. It’s going to be pointing in whichever axis direction makes its movement counterclockwise. (Because that’s how physicists started working this out and it would be too much bother to change now.)

    You might ask: wait, what about stuff like a wheel that’s spinning around its center? Or a ball being spun? That can’t be an angular momentum of zero? How do we work that out? The answer is: calculus. Also, we don’t need that. This central force problem I’ve framed so that we barely even need algebra for it.

    See, we only have a single object that’s moving. That’s the planet or satellite or weight or whatever it is. It’s got some mass, the value of which we call ‘m’ because why make it any harder on ourselves. And it’s spinning around the origin. We’ve been using ‘r’ to mean the number describing how far it is from the origin. That’s the distance to the axis it’s spinning around. Its velocity — well, we don’t have any symbols to describe what that is yet. But you can imagine working that out. Or you trust that I have some clever mathematical-physics tool ready to introduce to work it out. I have, kind of. I’m going to ignore it altogether. For now.

    The symbol we use for the total angular momentum in a system is \vec{L} . The little arrow above the symbol is one way to denote “this is a vector”. It’s a good scheme, what with arrows making people think of vectors and it being easy to write on a whiteboard. In books, sometimes, we make do just by putting the letter in boldface, L, which is easier for old-fashioned word processors to do. If we’re sure that the reader isn’t going to forget that L is this vector then we might stop highlighting the fact altogether. That’s even less work to do.

    It’s going to be less work yet. Central force problems like this mean the object can move only in a two-dimensional plane. (If it didn’t, it wouldn’t conserve angular momentum: the direction of \vec{L} would have to change. Sounds like magic, but trust me.) The angular momentum’s direction has to be perpendicular to that plane. If the object is spinning around on a sheet of paper, the angular momentum is pointing straight outward from the sheet of paper. It’s pointing toward you if the object is moving counterclockwise. It’s pointing away from you if the object is moving clockwise. What direction it’s pointing is locked in.

    All we need to know is how big this angular momentum vector is, and whether it’s positive or negative. So we just care about this number. We can call it ‘L’, no arrow, no boldface, no nothing. It’s just a number, the same as is the mass ‘m’ or distance from the origin ‘r’ or any of our other variables.

    If ‘L’ is zero, this means there’s no total angular momentum. This means the object can be moving directly out from the origin, or directly in. This is the only way that something can crash into the center. So if setting L to be zero doesn’t allow that then we know we did something wrong, somewhere. If ‘L’ isn’t zero, then the object can’t crash into the center. If it did we’d be losing angular momentum. The object’s mass times its distance from the center times its perpendicular speed would have to be some non-zero number, even when the distance was zero. We know better than to look for that.

    You maybe wonder why we use ‘L’ of all letters for the angular momentum. I do. I don’t know. I haven’t found any sources that say why this letter. Linear momentum, which we represent with \vec{p} , I know. Or, well, I know the story every physicist says about it. p is the designated letter for linear momentum because we used to use the word “impetus”, as in “impulse”, to mean what we mean by momentum these days. And “p” is the first letter in “impetus” that isn’t needed for some more urgent purpose. (“m” is too good a fit for mass. “i” has to work both as an index and as that number which, squared, gives us -1. And for that matter, “e” we need for that exponentials stuff, and “t” is too good a fit for time.) That said, while everybody, everybody, repeats this, I don’t know the source. Perhaps it is true. I can imagine, say, Euler or Lagrange in their writing settling on “p” for momentum and everybody copying them. I just haven’t seen a primary citation showing this is so.

    (I don’t mean to sound too unnecessarily suspicious. But just because everyone agrees on the impetus-thus-p story doesn’t mean it’s so. I mean, every Star Trek fan or space historian will tell you that the first space shuttle would have been named Constitution until the Trekkies wrote in and got it renamed Enterprise. But the actual primary documentation that the shuttle would have been named Constitution is weak to nonexistent. I’ve come to the conclusion NASA had no plan in mind to name space shuttles until the Trekkies wrote in and got one named. I’ve done less poking around the impetus-thus-p story, in that I’ve really done none, but I do want it on record that I would like more proof.)

    Anyway, “p” for momentum is well-established. So I would guess that when mathematical physicists needed a symbol for angular momentum they looked for letters close to “p”. When you get into more advanced corners of physics “q” gets called on to be position a lot. (Momentum and position, it turns out, are nearly-identical-twins mathematically. So making their symbols p and q offers aesthetic charm. Also great danger if you make one little slip with the pen.) “r” is called on for “radius” a lot. Looking on, “t” is going to be time.

    On the other side of the alphabet, well, “o” is just inviting danger. “n” we need to count stuff. “m” is mass or we’re crazy. “l” might have just been the nearest we could get to “p” without intruding on a more urgently-needed symbol. (“s” we use a lot for parameters like length of an arc that work kind of like time but aren’t time.) And then shift to the capital letter, I expect, because a lowercase l looks like a “1”, to everybody’s certain doom.

    The modified potential energy, then, is going to include the angular momentum L. At least, the amount of angular momentum. It’s also going to include the mass of the object moving, and the radius r that says how far the object is from the center. It will be:

    V_{eff}(r) = V(r) + \frac{L^2}{2 m r^2}

    V(r) was the original potential, whatever that was. The modifying term, with this square of the angular momentum and all that, I kind of hope you’ll just accept on my word. The L2 means that whether the angular momentum is positive or negative, the potential will grow very large as the radius gets small. If it didn’t, there might not be orbits at all. And if the angular momentum is zero, then the effective potential is the same original potential that let stuff crash into the center.

    For the sort of r-to-a-power potentials I’ve been looking at, I get an effective potential of:

    V_{eff}(r) = C r^n + \frac{L^2}{2 m r^2}

    where n might be an integer. I’m going to pretend a while longer that it might not be, though. C is certainly some number, maybe positive, maybe negative.

    If you pick some values for C, n, L, and m you can sketch this out. If you just want a feel for how this Veff looks it doesn’t much matter what values you pick. Changing values just changes the scale, that is, where a circular orbit might happen. It doesn’t change whether it happens. Picking some arbitrary numbers is a good way to get a feel for how this sort of problem works. It’s good practice.

    Sketching will convince you there are energy minimums, where we can get circular orbits. It won’t say where to find them without some trial-and-error or building a model of this energy and seeing where a ball bearing dropped into it rolls to a stop. We can do this more efficiently.

     
  • Joseph Nebus 6:00 pm on Sunday, 28 August, 2016 Permalink | Reply
    Tags: , ,   

    Reading the Comics, August 27, 2016: Calm Before The Term Edition 


    Here in the United States schools are just lurching back into the mode where they have students come in and do stuff all day. Perhaps this is why it was a routine week. Comic Strip Master Command wants to save up a bunch of story problems for us. But here’s what the last seven days sent into my attention.

    Jeff Harris’s Shortcuts educational feature for the 21st is about algebra. It’s got a fair enough blend of historical trivia and definitions and examples and jokes. I don’t remember running across the “number cruncher” joke before.

    Mark Anderson’s Andertoons for the 23rd is your typical student-in-lecture joke. But I do sympathize with students not understanding when a symbol gets used for different meanings. It throws everyone. But sometimes the things important to note clearly in one section are different from the needs in another section. No amount of warning will clear things up for everybody, but we try anyway.

    Tom Thaves’s Frank and Ernest for the 23rd tells a joke about collapsing wave functions, which is why you never see this comic in a newspaper but always see it on a physics teacher’s door. This is properly physics, specifically quantum mechanics. But it has mathematical import. The most practical model of quantum mechanics describes what state a system is in by something called a wave function. And we can turn this wave function into a probability distribution, which describes how likely the system is to be in each of its possible states. “Collapsing” the wave function is a somewhat mysterious and controversial practice. It comes about because if we know nothing about a system then it may have one of many possible values. If we observe, say, the position of something though, then we have one possible value. The wave functions before and after the observation are different. We call it collapsing, reflecting how a universe of possibilities collapsed into a mere fact. But it’s hard to find an explanation for what that is that’s philosophically and physically satisfying. This problem leads us to Schrödinger’s Cat, and to other challenges to our sense of how the world could make sense. So, if you want to make your mark here’s a good problem for you. It’s not going to be easy.

    John Allison’s Bad Machinery for the 24th tosses off a panel full of mathematics symbols as proof of hard thinking. In other routine references John Deering’s Strange Brew for the 26th is just some talk about how hard fractions are.

    While it’s outside the proper bounds of mathematics talk, Tom Toles’s Randolph Itch, 2 am for the 23rd is a delight. My favorite strip of this bunch. Should go on the syllabus.

     
  • Joseph Nebus 6:00 pm on Sunday, 14 August, 2016 Permalink | Reply
    Tags: , polling, , , ,   

    Reading the Comics, August 12, 2016: Skipping Saturday Edition 


    I have no idea how many or how few comic strips on Saturday included some mathematical content. I was away most of the day. We made a quick trip to the Michigan’s Adventure amusement park and then to play pinball in a kind-of competitive league. The park turned out to have every person in the world there. If I didn’t wave to you from the queue on Shivering Timbers I apologize but it hasn’t got the greatest lines of sight. The pinball stuff took longer than I expected too and, long story short, we got back home about 4:15 am. So I’m behind on my comics and here’s what I did get to.

    Tak Bui’s PC and Pixel for the 8th depicts the classic horror of the cleaning people wiping away an enormous amount of hard work. It’s a primal fear among mathematicians at least. Boards with a space blocked off with the “DO NOT ERASE” warning are common. At this point, though, at least, the work is probably savable. You can almost always reconstruct work, and a few smeared lines like this are not bad at all.

    The work appears to be quantum mechanics work. The tell is in the upper right corner. There’s a line defining E (energy) as equal to something including \imath \hbar \frac{\partial}{\partial t}\phi(r, t) . This appears in the time-dependent Schrödinger Equation. It describes how probability waveforms look when the potential energies involved may change in time. These equations are interesting and impossible to solve exactly. We have to resort to approximations, including numerical approximations, all the time. So that’s why the computer lab would be working on this.

    Mark Anderson’s Andertoons! Where would I be without them? Besides short on content. The strip for the 10th depicts a pollster saying to “put the margin of error at 50%”, guaranteeing the results are right. If you follow elections polls you do see the results come with a margin of error, usually of about three percent. But every sampling technique carries with it a margin of error. The point of a sample is to learn something about the whole without testing everything in it, after all. And probability describes how likely it is the quantity measured by a sample will be far from the quantity the whole would have. The logic behind this is independent of the thing being sampled. It depends on what the whole is like. It depends on how the sampling is done. It doesn’t matter whether you’re sampling voter preferences or whether there are the right number of peanuts in a bag of squirrel food.

    So a sample’s measurement will almost never be exactly the same as the whole population’s. That’s just requesting too much of luck. The margin of error represents how far it is likely we’re off. If we’ve sampled the voting population fairly — the hardest part — then it’s quite reasonable the actual vote tally would be, say, one percent different from our poll. It’s implausible that the actual votes would be ninety percent different. The margin of error is roughly the biggest plausible difference we would expect to see.

    Except. Sometimes we do, even with the best sampling methods possible, get a freak case. Rarely noticed beside the margin of error is the confidence level. This is what the probability is that the actual population value is within the sampling error of the sample’s value. We don’t pay much attention to this because we don’t do statistical-sampling on a daily basis. The most normal people do is read election polling results. And most election polls settle for a confidence level of about 95 percent. That is, 95 percent of the time the actual voting preference will be within the three or so percentage points of the survey. The 95 percent confidence level is popular maybe because it feels like a nice round number. It’ll be off only about one time out of twenty. It also makes a nice balance between a margin of error that doesn’t seem too large and that doesn’t need too many people to be surveyed. As often with statistics the common standard is an imperfectly-logical blend of good work and ease of use.

    For the 11th Mark Anderson gives me less to talk about, but a cute bit of wordplay. I’ll take it.

    Anthony Blades’s Bewley for the 12th is a rerun. It’s at least the third time this strip has turned up since I started writing these Reading The Comics posts. For the record it ran also the 27th of April, 2015 and on the 24th of May, 2013. It also suggests mathematicians have a particular tell. Try this out next time you do word problem poker and let me know how it works for you.

    Julie Larson’s The Dinette Set for the 12th I would have sworn I’d seen here before. I don’t find it in my archives, though. We are meant to just giggle at Larson’s characters who bring their penny-wise pound-foolishness to everything. But there is a decent practical mathematics problem here. (This is why I thought it had run here before.) How far is it worth going out of one’s way for cheaper gas? How much cheaper? It’s simple algebra and I’d bet many simple Javascript calculator tools. The comic strip originally ran the 4th of October, 2005. Possibly it’s been rerun since.

    Bill Amend’s FoxTrot Classics for the 12th is a bunch of gags about a mathematics fighting game. I think Amend might be on to something here. I assume mathematics-education contest games have evolved from what I went to elementary school on. That was a Commodore PET with a game where every time you got a multiplication problem right your rocket got closer to the ASCII Moon. But the game would probably quickly turn into people figuring how to multiply the other person’s function by zero. I know a game exploit when I see it.

    The most obscure reference is in the third panel one. Jason speaks of “a z = 0 transform”. This would seem to be some kind of z-transform, a thing from digital signals processing. You can represent the amplification, or noise-removal, or averaging, or other processing of a string of digits as a polynomial. Of course you can. Everything is polynomials. (OK, sometimes you must use something that looks like a polynomial but includes stuff like the variable z raised to a negative power. Don’t let that throw you. You treat it like a polynomial still.) So I get what Jason is going for here; he’s processing Peter’s function down to zero.

    That said, let me warn you that I don’t do digital signal processing. I just taught a course in it. (It’s a great way to learn a subject.) But I don’t think a “z = 0 transform” is anything. Maybe Amend encountered it as an instructor’s or friend’s idiosyncratic usage. (Amend was a physics student in college, and shows his comfort with mathematics-major talk often. He by the way isn’t even the only syndicated cartoonist with a physics degree. Bud Grace of The Piranha Club was also a physics major.) I suppose he figured “z = 0 transform” would read clearly to the non-mathematician and be interpretable to the mathematician. He’s right about that.

     
  • Joseph Nebus 6:00 pm on Sunday, 24 July, 2016 Permalink | Reply
    Tags: , , ,   

    Reading the Comics, July 23, 2016: Familiar Friends Week Edition 


    This past week was refreshing. The mathematics comics appeared at a regular, none-too-excessive pace. And some old familiar friends reappeared. Some were comic strips that haven’t been around in a while. Some were jokes that haven’t been. Enjoy.

    Bill Whitehead’s Free Range for the 17th is the first use of the meth/math lab pun to appear in the comics since September 2014 by my reckoning. And only the second in my Reading the Comics series. I’m surprised too. For all this goes around Twitter and other social media I’d imagine it to make the comics more.

    Scott Hilburn’s The Argyle Sweater gets back in my review here for the first time since April, to my amazement. Used to be you couldn’t go two weeks without Hilburn looking for my attention. And here’s the first Roman Numerals joke since … I don’t quite feel up to checking just now. I’m going to go ahead and suppose it’s the first one since the last time Samson’s Dark Side Of The Horse was here.

    It’s anachronistic to speak of Ancient Roman students getting ‘C’ grades. Of course it is; it could hardly be otherwise. It’s a joke; how much is that to be worried about? But if I haven’t been mislead the use of letters, A through E-or-F, in student evaluations is an American innovation of the late 19th century. It developed over the 20th century and took over at least American education, in conjunction with the 100-through-0 points evaluation scale. And in parallel to the Grade Point Average, typically with 4.0 as its highest score.

    Samson’s Dark Side Of The Horse makes a comfortable visit back here on the 20th. It’s another counting-sheep and number-representation gag. I love the third panel’s artwork.

    Mark Anderson’s Andertoons for the 22nd is a joke about motivating mathematics study. I believe I’ve mentioned this before, but there was a lovely bit on The Mary Tyler Moore Show along these lines once. Fantastically stupid newsman Ted Baxter was struggling to do some arithmetic until Murray Slaughter gave him the advice: “put a dollar sign in front of it”. Then he had the answer instantly.

    Nat Fakes’s Break of Day for the 22nd brings back mathematics as signifier of the hardest homework a kid can have, or the toughest thing someone can have to think about. Fine enough stuff, although it isn’t really that stunning to think a parent might not understand what the kid’s homework is about. Often the point of an assignment is not to learn how to do something, but to encourage thinking about ways one could do something. That’s a hard assignment to create, and a harder one to do, and a very hard one to help with. As adults we get used to looking at problems as calculations to identify and do as swiftly as possible. That there is value in wandering around the slow routes needs remembering.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd riffs on … I’m not sure exactly. The idea that the sort of meaningless nonsense that makes for good late-night dorm conversations when you’re 20 comes back around to being the cutting edge of theoretical physics, I suppose. It’s funny enough. A complaint often brought against the most cutting edge of theoretical physics is that it’s so abstract that there aren’t any conceivable tests that would say whether a calculation is right or not. In that condition mathematics and theoretical physics merge back into a thread of philosophy and its question of how can we know what it is for something to be true. Once we have a way of discerning whether an idea might happen to be true we’re ejected again from philosophy and into a science. And then the scientist makes a smug, snarky comment about the impossibility of testing philosophical conclusions.

    Since the late 19th century much cutting-edge physics has involved counter-intuitive results. Often they have premises that strain intuition, as see relativity, or that seem to violate it altogether, as see quantum mechanics. But they turn out so very right so very often it’s hard not to feel excited and encouraged by this. Who wouldn’t look for a surprising and counter-intuitive explanation for the world as thrilling and maybe even right idea? I don’t blame anyone for looking to a wild idea like “what if the universe is made of math”. I don’t know what that would mean exactly unless we suppose we do live in a universe of Platonic Forms, in which case perfection runs counter-intuitively to me. I do understand being excited by the question. But the answers probably won’t be that much fun.

     
  • Joseph Nebus 6:00 pm on Saturday, 23 July, 2016 Permalink | Reply
    Tags: , derivations,   

    Anatomizing An Error 


    Though it’s the summer months I’m happy to say the Carnot Cycle thermodynamics blog is still posting. He had been writing about Jacobus Henricus van ‘t Hoff, first recipient of the Nobel Prize in Chemistry. In the 1880s van ‘t Hoff was studying the osmosis. In April’s essay Carnot Cycle described the problem, and how van ‘t Hoff passed up a correct formula describing osmotic pressure in favor of an attractive but wrong alternative.

    In this month’s essay Carnot Cycle continues the topic. It particularly goes over just how van ‘t Hoff got to his mistaken idea. It’s not that he started out wrong. He began from a good start and derived a mistaken formula. The derivation involved a string of assumptions and simplifications and approximations, of the kind that must be made to go from starting principles to a specific problem. He was guided by an idea of what the answer ought to look like, though, and that led him astray. The blog describes what he did and why it would look reasonable in the circumstance. It’s worth reading to see what actual mathematics, the kind that doesn’t have known answers, is like.

     
    • davekingsbury 11:20 pm on Sunday, 24 July, 2016 Permalink | Reply

      So mathematics is essentially exploration of the unknown?

      Like

      • Joseph Nebus 4:36 am on Tuesday, 26 July, 2016 Permalink | Reply

        I’m not sure exactly how I would describe mathematics. Part of it does feel like an exploration of the unknown: we set out basic rules and find implications that aren’t obvious. A lot of work does feel like experimentation and discovery, just as one might do in a science. But it does seem bizarre to imagine that the logical consequences of our chosen premises are unknown; it seems like saying that a chess move might need discovery. I’m not sure how to represent it all. Possibly there’s no representing it all as one thing; there are several strands of thought that run through mathematics, I believe.

        Liked by 1 person

        • davekingsbury 7:24 am on Tuesday, 26 July, 2016 Permalink | Reply

          Thank you for this honest and lucid response. Strikes me it’s a language which avoids the pitfalls of imprecision and emotionality.

          Like

          • Joseph Nebus 5:54 am on Wednesday, 27 July, 2016 Permalink | Reply

            I think avoiding imprecision and emotionality are considered ideals, yes. And a fully mature, cleaned-up mathematical field has got its important work set up and defined in ways that are precise and avoid emotional appeals. But when working out a problem, especially a new and exciting one, there are many provisional definitions and ambiguities discovered late in the paper and all that. Mathematicians are humans and their lives are all over their work, necessarily. We try to look good when strangers peek in, which is again a most human thing to do.

            Like

    • davekingsbury 8:15 am on Wednesday, 27 July, 2016 Permalink | Reply

      Thanks for humanising the world of mathematics for me. You have the skills of a natural teacher.

      Like

  • Joseph Nebus 6:00 pm on Saturday, 9 July, 2016 Permalink | Reply
    Tags: celestial mechanics, ,   

    Why Stuff Can Orbit: Why It’s Waiting 


    I can’t imagine people are going to be surprised to hear this. But I have to put the “Why Stuff Can Orbit” series. It’s about central forces and what circumstances make it possible for something to have a stable orbit. I mean to get back to it. It’s just that the Theorem Thursday posts take up a lot of thinking on my part. They end up running quite long and detailed. I figure to get back to it once I’ve exhausted the Theorem Thursday topics I have in mind, which should be shortly into August.

    It happens I’d run across a WordPress blog that contained the whole of the stable-central-orbits argument, in terse but legitimate terms. I wanted to link to that now but the site’s been deleted for reasons I won’t presume to guess. I have guesses. Sorry.

    But for some other interesting reading, here’s a bit about Immanuel Kant:

    I have long understood, and passed on, that Immanuel Kant had the insight that the laws of physics tell us things about the geometry of space and vice-versa. I haven’t had the chance yet to read Francisco Caruso and Roberto Moreira Xavier’s On Kant’s First Insight into the Problem of Space Dimensionality and its Physical Foundations. But the abstract promises “a conclusion that does not match the usually accepted interpretation of Kant’s reasoning”. I would imagine this to be an interesting introduction to the question, then, and to what might be controversial about Kant and the number of dimensions space should have. Also we need to use the word “tridimensionality” more.

     
  • Joseph Nebus 3:00 pm on Friday, 27 May, 2016 Permalink | Reply
    Tags: , , , , ,   

    Why Stuff Can Orbit, Part 2: Why Stuff Can’t Orbit 


    Previously:

    As I threatened last week, I want to talk some about central forces. They’re forces in which one particle attracts another with a force that depends only on how far apart the two are. Last week’s essay described some of the assumptions behind the model.

    Mostly, we can study two particles interacting as if it were one particle hovering around the origin. The origin is some central reference point. If we’re looking for a circular orbit then we only have to worry about one variable. This would be ‘r’, the radius of the orbit: how far the planet is from the sun it orbits.

    Now, central forces can follow any rule you like. Not in reality, of course. In reality there’s two central forces you ever see. One is gravity (electric attraction or repulsion follows the same rule) and the other is springs. But we can imagine there being others. At the end of this string of essays I hope to show why there’s special things about these gravity and spring-type forces. And by imagining there’s others we can learn something about why we only actually see these.

    So now I’m going to stop talking about forces. I’ll talk about potential energy instead. There’s several reasons for this, but they all come back to this one: energy is easier to deal with. Energy is a scalar, a single number. A force is a vector, which for this kind of physics-based problem is an array of numbers. We have less to deal with if we stick to energy. If we need forces later on we can get them from the energy. We’ll need calculus to do that, but it won’t be the hard parts of calculus.

    The potential energy will be some function. As a central force it’ll depend only on the distance, r, that a particle is from the origin. It’s convenient to have a name for this. So I will use a common name: V(r). V is a common symbol to use for potential energy. U is another. The (r) emphasizes that this is some function which depends on r. V(r) doesn’t commit us to any particular function, not at this point.

    You might ask: why is the potential energy represented with V, or with U? And I don’t really know. Sometimes we’ll use PE to mean potential energy, which is as clear a shorthand name as we could hope for. But a name that’s two letters like that tends to be viewed with suspicion when we have to do calculus work on it. The label looks like the product of P and E, and derivatives of products get tricky. So it’s a less popular label if you know you’re going take the derivative of the potential energy anytime soon. EP can also get used, and the subscript means it doesn’t look like the product of any two things. Still, at least in my experience, U and V are most often used.

    As I say, I don’t know just why it should be them. It might just be that the letters were available when someone wrote a really good physics textbook. If we want to assume there must be some reason behind this letter choice I have seen a plausible guess. Potential energy is used to produce work. Work is W. So potential energy should be a letter close to W. That suggests U and V, both letters that are part of the letter W. (Listen to the name of ‘W’, and remember that until fairly late in the game U and V weren’t clearly distinguished as letters.) But I do not know of manuscript evidence suggesting that’s what anyone every thought. It is at best a maybe useful mnemonic.

    Here’s an advantage that using potential energy will give us: we can postpone using calculus a little. Not for quantitative results. Not for ones that describe exactly where something should orbit. But it’s good for qualitative results. We can answer questions like “is there a circular orbit” and “are there maybe several plausible orbits” just by looking at a picture.

    That picture is a plot of the values of V(r) against r. And that can be anything. I mean it. Take your preferred drawing medium and draw any wiggly curve you like. It can’t loop back or cross itself or something like that, but it can be as smooth or as squiggly as you like. That’s your central-force potential energy V(r).

    Are there any circular orbits for this potential? Calculus gives us the answer, but we don’t need that. For a potential like our V(r), that depend on one variable, we can just look. (We could also do this for a potential that depends on two variables.) Take your V(r). Imagine it’s the sides of a perfectly smooth bowl or track or something. Now imagine dropping a marble or a ball bearing or something nice and smooth on it. Does the marble come to a rest anywhere? That’s your equilibrium. That’s where a circular orbit can happen.

    A generic wiggly shape with a bunch of peaks and troughs.

    Figure 1. A generic yet complicated V(r). Spoiler: I didn’t draw this myself because I figured using Octave was easier than using ArtRage on my iPad.

    We’re using some real-world intuition to skip doing analysis. That’s all right in this case. Newtonian mechanics say that a particle’s momentum changes in the direction of a force felt. If a particle doesn’t change its mass, then that means it accelerates where the force, uh, forces it. And this sort of imaginary bowl or track matches up the potential energy we want to study with a constrained gravitational potential energy.

    My generic V(r) was a ridiculous function. This sort of thing doesn’t happen in the real world. But they might have. Wiggly functions like that were explored in the 19th century by physicists trying to explain chemistry. They hoped complicated potentials would explain why gases expanded when they warmed and contracted when they cooled. The project failed. Atoms follow quantum-mechanics laws that match only vaguely match Newtonian mechanics like this. But just because functions like these don’t happen doesn’t mean we can’t learn something from them.

    We can’t study every possible V(r). Not at once. Not without more advanced mathematics than I want to use right now. What I’d like to do instead is look at one family of V(r) functions. There will be infinitely many different functions here, but they’ll all resemble each other in important ways. If you’ll allow me to introduce two new numbers we can describe them all with a single equation. The new numbers I’ll name C and n. They’re both constants, at least for this problem. They’re some numbers and maybe at some point I’ll care which ones they are, but it doesn’t matter. If you want to pretend that C is another way to write “eight”, go ahead. n … well, you can pretend that’s just another way to write some promising number like “two” for now. I’ll say when I want to be more specific about it.

    The potential energy I want to look at has a form we call a power law, because it’s all about raising a variable to a power. And we only have the one variable, r. So the potential energy looks like this:

    V(r) = C r^n

    There are some values of n that it will turn out are meaningful. If n is equal to 2, then this is the potential energy for two particles connected by a spring. You might complain there are very few things in the world connected to other things by springs. True enough, but a lot of things act as if they were springs. This includes most anything that’s near but pushed away from a stable equilibrium. It’s a potential worth studying.

    If n is equal to -1, then this is the potential energy for two particles attracting each other by gravity or by electric charges. And here there’s an important little point. If the force is attractive, like gravity or like two particles having opposite electric charges, then we need C to be a negative number. If the force is repulsive, like two particles having the same electric charge, then we need C to be a positive number.

    Although n equalling two, and n equalling negative one, are special cases they aren’t the only ones we can imagine. n may be any number, positive or negative. It could be zero, too, but in that case the potential is a flat line and there’s nothing happening there. That’s known as a “free particle”. It’s just something that moves around with no impetus to speed up or slow down or change direction or anything.

    So let me sketch the potentials for positive n, first for a positive C and second for a negative C. Don’t worry about the numbers on either the x- or the y-axes here; they don’t matter. The shape is all we care about right now.

    The curve starts at zero and rises ever upwards as the radius r increases.

    Figure 2. V(r) = C rn for a positive C and a positive n.


    The curve starts at zero and drops ever downwards as the radius r increases.

    Figure 3. V(r) = C rn for a negative C and a positive n.

    Now let me sketch the potentials for a negative n, first for a positive C and second for a negative C.

    The curve starts way high up and keeps dropping, but levelling out, as the radius r increases

    Figure 4. V(r) = C rn for a positive C and a negative n.


    The curve starts way down low and rises, but levelling out, as the radius r increases.

    Figure 5. V(r) = C rn for a negative C and a negative n.

    And now we can look for equilibriums, for circular orbits. If we have a positive n and a positive C, then — well, do the marble-in-a-bowl test. Start from anywhere; the marble rolls down to the origin where it smashes and stops. The only circular orbit is at a radius r of zero.

    With a positive n and a negative C, start from anywhere except a radius r of exactly zero and the marble rolls off to the right, without ever stopping. The only circular orbit is at a radius r of zero.

    With a negative n and a positive C, the marble slides down a hill that gets more shallow but that never levels out. It rolls off getting ever farther from the origin. There’s no circular orbits.

    With a negative n and a negative C, start from anywhere and the marble rolls off to the left. The marble will plummet down that ever-steeper hill. The only circular orbit is at a radius r of zero.

    So for all these cases, with a potential V(r) = C rn, the only possible “orbits” have both particles zero distance apart. Otherwise the orbiting particle smashes right down into the center or races away never to be seen again. Clearly something has gone wrong with this little project.

    If you’ve spotted what’s gone wrong please don’t say what it is right away. I’d like people to ponder it a little before coming back to this next week. That will come, I expect, shortly after the first Theorem Thursday post. If you have any requests for that project, please get them in, the sooner the better.

     
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