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  • Joseph Nebus 4:00 pm on Tuesday, 11 July, 2017 Permalink | Reply
    Tags: , , , , , , , phase space,   

    Why Stuff Can Orbit, Part 11: In Search Of Closure 


    Previously:

    And the supplemental reading:


    I’m not ready to finish the series off yet. But I am getting closer to wrapping up perturbed orbits. So I want to say something about what I’m looking for.

    In some ways I’m done already. I showed how to set up a central force problem, where some mass gets pulled towards the center of the universe. It can be pulled by a force that follows any rule you like. The rule has to follow some rules. The strength of the pull changes with how far the mass is from the center. It can’t depend on what angle the mass makes with respect to some reference meridian. Once we know how much angular momentum the mass has we can find whether it can have a circular orbit. And we can work out whether that orbit is stable. If the orbit is stable, then for a small nudge, the mass wobbles around that equilibrium circle. It spends some time closer to the center of the universe and some time farther away from it.

    I want something a little more, else I can’t carry on this series. I mean, we can make central force problems with more things in them. What we have now is a two-body problem. A three-body problem is more interesting. It’s pretty near impossible to give exact, generally true answers about. We can save things by only looking at very specific cases. Fortunately one is a sun, planet, and moon, where each object is much more massive than the next one. We see a lot of things like that. Four bodies is even more impossible. Things start to clear up if we look at, like, a million bodies, because our idea of what “clear” is changes. I don’t want to do that right now.

    Instead I’m going to look for closed orbits. Closed orbits are what normal people would call “orbits”. We’re used to thinking of orbits as, like, satellites going around and around the Earth. We know those go in circles, or ellipses, over and over again. They don’t, but the difference between a closed orbit and what they do is small enough we don’t need to care.

    Here, “orbit” means something very close to but not exactly what normal people mean by orbits. Maybe I should have said something about that before. But the difference hasn’t counted for much before.

    Start off by thinking of what we need to completely describe what a particular mass is doing. You need to know the central force law that the mass obeys. You need to know, for some reference time, where it is. You also need to know, for that same reference time, what its momentum is. Once you have that, you can predict where it should go for all time to come. You can also work out where it must have been before that reference time. (This we call “retrodicting”. Or “predicting the past”. With this kind of physics problem time has an unnerving symmetry. The tools which forecast what the mass will do in the future are exactly the same as those which tell us what the mass has done in the past.)

    Now imagine knowing all the sets of positions and momentums that the mass has had. Don’t look just at the reference time. Look at all the time before the reference time, and look at all the time after the reference time. Imagine highlighting all the sets of positions and momentums the mass ever took on or ever takes on. We highlight them against the universe of all the positions and momentums that the mass could have had if this were a different problem.

    What we get is this ribbon-y thread that passes through the universe of every possible setup. This universe of every possible setup we call a “phase space”. It’s easy to explain the “space” part of that name. The phase space obeys the rules we’d expect from a vector space. It also acts in a lot of ways like the regular old space that we live in. The “phase” part I’m less sure how to justify. I suspect we get it because this way of looking at physics problems comes from statistical mechanics. And in that field we’re looking, often, at the different ways a system can behave. This mathematics looks a lot like that of different phases of matter. The changes between solids and liquids and gases are some of what we developed this kind of mathematics to understand, in fact. But this is speculation on my part. I’m not sure why “phase” has attached to this name. I can think of other, harder-to-popularize reasons why the name would make sense too. Maybe it’s the convergence of several reasons. I’d love to hear if someone has a good etymology. If one exists; remember that we still haven’t got the story straight about why ‘m’ stands for the slope of a line.

    Anyway, this ribbon of all the arrangements of position and momentum that the mass does ever at any point have we call a “trajectory”. We call it a trajectory because it looks like a trajectory. Sometimes mathematics terms aren’t so complicated. We also call it an “orbit” since very often the problems we like involve trajectories that loop around some interesting area. It looks like a planet orbiting a sun.

    A “closed orbit” is an orbit that gets back to where it started. This means you can take some reference time, and wait. Eventually the mass comes back to the same position and the same momentum that you saw at that reference time. This might seem unavoidable. Wouldn’t it have to get back there? And it turns out, no, it doesn’t. A trajectory might wander all over phase space. This doesn’t take much imagination. But even if it doesn’t, if it stays within a bounded region, it could still wander forever without repeating itself. If you’re not sure about that, please consider an old sequence I wrote inspired by the Aardman Animation film Arthur Christmas. Also please consider seeing the Aardman Animation film Arthur Christmas. It is one of the best things this decade has offered us. The short version is, though, that there is a lot of room even in the smallest bit of space. A trajectory is, in a way, a one-dimensional thing that might get all coiled up. But phase space has got plenty of room for that.

    And sometimes we will get a closed orbit. The mass can wander around the center of the universe and come back to wherever we first noticed it with the same momentum it first had. A that point it’s locked into doing that same thing again, forever. If it could ever break out of the closed orbit it would have had to the first time around, after all.

    Closed orbits, I admit, don’t exist in the real world. Well, the real world is complicated. It has more than a single mass and a single force at work. Energy and momentum are conserved. But we effectively lose both to friction. We call the shortage “entropy”. Never mind. No person has ever seen a circle, and no person ever will. They are still useful things to study. So it is with closed orbits.

    An equilibrium orbit, the circular orbit of a mass that’s at exactly the right radius for its angular momentum, is closed. A perturbed orbit, wobbling around the equilibrium, might be closed. It might not. I mean next time to discuss what has to be true to close an orbit.

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  • Joseph Nebus 3:00 pm on Monday, 11 April, 2016 Permalink | Reply
    Tags: , , , , mapping, , , phase space,   

    A Leap Day 2016 Mathematics A To Z: Surjective Map 


    Gaurish today gives me one more request for the Leap Day Mathematics A To Z. And it lets me step away from abstract algebra again, into the world of analysis and what makes functions work. It also hovers around some of my past talk about functions.

    Surjective Map.

    This request echoes one of the first terms from my Summer 2015 Mathematics A To Z. Then I’d spent some time on a bijection, or a bijective map. A surjective map is a less complicated concept. But if you understood bijective maps, you picked up surjective maps along the way.

    By “map”, in this context, mathematicians don’t mean those diagrams that tell you where things are and how you might get there. Of course we don’t. By a “map” we mean that we have some rule that matches things in one set to things in another. If this sounds to you like what I’ve claimed a function is then you have a good ear. A mapping and a function are pretty much different names for one another. If there’s a difference in connotation I suppose it’s that a “mapping” makes a weaker suggestion that we’re necessarily talking about numbers.

    (In some areas of mathematics, a mapping means a function with some extra properties, often some kind of continuity. Don’t worry about that. Someone will tell you when you’re doing mathematics deep enough to need this care. Mind, that person will tell you by way of a snarky follow-up comment picking on some minor point. It’s nothing personal. They just want you to appreciate that they’re very smart.)

    So a function, or a mapping, has three parts. One is a set called the domain. One is a set called the range. And then there’s a rule matching things in the domain to things in the range. With functions we’re so used to the domain and range being the real numbers that we often forget to mention those parts. We go on thinking “the function” is just “the rule”. But the function is all three of these pieces.

    A function has to match everything in the domain to something in the range. That’s by definition. There’s no unused scraps in the domain. If it looks like there is, that’s because were being sloppy in defining the domain. Or let’s be charitable. We assumed the reader understands the domain is only the set of things that make sense. And things make sense by being matched to something in the range.

    Ah, but now, the range. The range could have unused bits in it. There’s nothing that inherently limits the range to “things matched by the rule to some thing in the domain”.

    By now, then, you’ve probably spotted there have to be two kinds of functions. There’s one in which the whole range is used, and there’s ones in which it’s not. Good eye. This is exactly so.

    If a function only uses part of the range, if it leaves out anything, even if it’s just a single value out of infinitely many, then the function is called an “into” mapping. If you like, it takes the domain and stuffs it into the range without filling the range.

    Ah, but if a function uses every scrap of the range, with nothing left out, then we have an “onto” mapping. The whole of the domain gets sent onto the whole of the range. And this is also known as a “surjective” mapping. We get the term “surjective” from Nicolas Bourbaki. Bourbaki is/was the renowned 20th century mathematics art-collective group which did so much to place rigor and intuition-free bases into mathematics.

    The term pairs up with the “injective” mapping. In this, the elements in the range match up with one and only one thing in the domain. So if you know the function’s rule, then if you know a thing in the range, you also know the one and only thing in the domain matched to that. If you don’t feel very French, you might call this sort of function one-to-one. That might be a better name for saying why this kind of function is interesting.

    Not every function is injective. But then not every function is surjective either. But if a function is both injective and surjective — if it’s both one-to-one and onto — then we have a bijection. It’s a mapping that can represent the way a system changes and that we know how to undo. That’s pretty comforting stuff.

    If we use a mapping to describe how a process changes a system, then knowing it’s a surjective map tells us something about the process. It tells us the process makes the system settle into a subset of all the possible states. That doesn’t mean the thing is stable — that little jolts get worn down. And it doesn’t mean that the thing is settling to a fixed state. But it is a piece of information suggesting that’s possible. This may not seem like a strong conclusion. But considering how little we know about the function it’s impressive to be able to say that much.

     
    • davekingsbury 8:21 pm on Tuesday, 12 April, 2016 Permalink | Reply

      Surprised it’s not been translated as ‘onjective’ in English …

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      • Joseph Nebus 3:40 am on Friday, 15 April, 2016 Permalink | Reply

        I’m a bit surprised myself. I suspect the injective/surjective/bijective thing looks too much like French-influenced English to be more exotically translated.

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  • Joseph Nebus 3:05 pm on Friday, 12 June, 2015 Permalink | Reply
    Tags: , , , , , phase space, projection, , traffic   

    A Summer 2015 Mathematics A To Z: into 


    Into.

    The definition of “into” will call back to my A to Z piece on “bijections”. It particularly call on what mathematicians mean by a function. When a mathematician talks about a functions she means a combination three components. The first is a set called the domain. The second is a set called the range. The last is a rule that matches up things in the domain to things in the range.

    We said the function was “onto” if absolutely everything which was in the range got used. That is, if everything in the range has at least one thing in the domain that the rule matches to it. The function that has domain of -3 to 3, and range of -27 to 27, and the rule that matches a number x in the domain to the number x3 in the range is “onto”.

    (More …)

     
    • elkement 6:26 am on Sunday, 14 June, 2015 Permalink | Reply

      Here it finally shows that my math education was originally in German. I haven’t known these properties as ‘onto’ or ‘into’, but only surjective and injective – which are the same words in German. I think we don’t have such nice common language terms for that.

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      • Joseph Nebus 2:27 am on Tuesday, 16 June, 2015 Permalink | Reply

        The sense I got as an undergraduate was there was a divide between (English-speaking) mathematicians who preferred onto and into and those who preferred surjective and injective. I can see the benefits of either choice. Onto and into have that nice common Anglo-Saxon touch, and common words tend to write better. But there’s no good matching common word for bijective. And the injective-surjective-bijective triplet seem to reinforce each word’s meaning and probably help all three concepts stand together better than any one alone does.

        Liked by 1 person

  • Joseph Nebus 3:09 pm on Wednesday, 27 May, 2015 Permalink | Reply
    Tags: , bijections, , , , , phase space, ,   

    A Summer 2015 Mathematics A To Z: bijection 


    Bijection.

    To explain this second term in my mathematical A to Z challenge I have to describe yet another term. That’s function. A non-mathematician’s idea a function is something like “a line with a bunch of x’s in it, and maybe also a cosine or something”. That’s fair enough, although it’s a bit like defining chemistry as “mixing together colored, bubbling liquids until something explodes”.

    By a function a mathematician means a rule describing how to pair up things found in one set, called the domain, with the things found in another set, called the range. The domain and the range can be collections of anything. They can be counting numbers, real numbers, letters, shoes, even collections of numbers or sets of shoes. They can be the same kinds of thing. They can be different kinds of thing.

    (More …)

     
    • Ken Dowell 7:16 pm on Wednesday, 27 May, 2015 Permalink | Reply

      Have to admit I have never heard the term. Thought it was a double ejection.

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      • Joseph Nebus 10:49 pm on Thursday, 28 May, 2015 Permalink | Reply

        It’s not a common word. I don’t think anyone besides an upper-level mathematics undergraduate or a grad student would need to use it.

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    • sheldonk2014 10:23 pm on Wednesday, 27 May, 2015 Permalink | Reply

      When numbers are used in the comics do they have a reason they use those specific ones

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      • Joseph Nebus 10:55 pm on Thursday, 28 May, 2015 Permalink | Reply

        I think the only fair answer is “sometimes”. Occasionally a particular number will be part of the joke. Or an in-joke. For example, the number 47 turns up a lot in Star Trek because of a joke on the writing staff.

        But sometimes the cartoonist feels that, if there’s got to be some mathematical content, then it should be correct. Bill Amend of FoxTrot — a physics major, I feel the need to admit — does this a lot. (He’s also talked about the challenges of writing mathematics comics, and how that relates to teaching.) Some do that because it’s fun. Some because they feel it makes the joke stronger if the mathematics talk is authentic. Some, surely, because they learned this stuff so why not put it to use?

        And there are other cartoonists who just pick numbers because they look pretty, or they sound funny, or for other aesthetic reasons like that. And that’s fine also: comic strips are a form of art and aesthetic grounds have to count.

        But overall there just isn’t a universal rule.

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    • baffledbaboon 4:00 pm on Thursday, 28 May, 2015 Permalink | Reply

      At first glance you’d think the word “Bijection” would mean “to be ejected twice”

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      • Joseph Nebus 10:58 pm on Thursday, 28 May, 2015 Permalink | Reply

        It’s not far off. There’s a related term, “injection”, which would have made sense to put first if I were not doing this in the a-to-z order. (If I weren’t sticking with alphabetical order I’d have defined “function”, then “injection”, then “bijection”.) In an injective function everything in the domain is matched to something in the range, but it’s possible something in the range is missed.

        In a bijective function, everything in the domain matches exactly one thing in the range. So you can read it as two rules, one that matches everything in the domain to something in the range, and another rule that matches everything in the range to something in the domain. So in a way you can see it as a double injection.

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    • Michelle H 11:08 pm on Sunday, 31 May, 2015 Permalink | Reply

      Makes me think that ‘bijection’ could be a metaphor for a set of ‘perfect’ social grafts (i.e. old-fashion marriage), especially in engineered situations like those that favour monogamy, don’t allow divorce, and only recognize heterosexuality. Hmm, a Jane Austen novel? (Sorry… I recognized your sense of humour and couldn’t refrain.)

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      • Joseph Nebus 8:56 pm on Monday, 1 June, 2015 Permalink | Reply

        No need to apologize. Never fear that.

        I am interested in the metaphors that could be made out of bijection. Functions are all about pairing up things, one from the domain and one from the range. This does suggest parallels to social groupings, although I’m not sure how far the metaphor could be pushed before it stops being enlightening.

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        • Michelle H 9:42 pm on Monday, 1 June, 2015 Permalink | Reply

          Sometimes metaphor is more entertaining than enlightening.

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          • Joseph Nebus 8:02 am on Tuesday, 2 June, 2015 Permalink | Reply

            This is so, although I really love when a metaphor can serve both roles well.

            Liked by 1 person

            • Michelle H 1:23 pm on Tuesday, 2 June, 2015 Permalink | Reply

              In the history of each metaphor, its birth probably brought shock and likely both amusement and disgust. Slowly, it aged to cliché and perhaps ceased to be recognized even as cliché (table’s leg). ‘Bijection’ is a really difficult term for metaphor; I have been thinking on it since reading your post. Putty and country dances might be the best, although one assumes too much experience with Jane Austen.

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              • Joseph Nebus 10:36 pm on Friday, 5 June, 2015 Permalink | Reply

                Bijection is indeed a difficult term to find a metaphor for. But I do appreciate your trying and enjoy your describing of the history of a metaphor. Thank you.

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                • Michelle H 10:08 pm on Monday, 8 June, 2015 Permalink | Reply

                  This might work: transposing music, when a tune is moved from one key to another, everything in the domain has a match in the range (that is to say, every note in the original key must be rewritten into the new one).

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                  • Joseph Nebus 9:24 pm on Tuesday, 9 June, 2015 Permalink | Reply

                    Oh, I think you have it, if not in transpositions then at least arranging a musical piece for a different instrument. A bijective function, among other things, lets you go from the range back to the domain. And some transpositions and arrangements let you reconstruct the original music perfectly.

                    But there are some functions, and some arrangements, that don’t allow that. For example, a song arranged for carousel organs will typically have a narrower range of notes available, and maybe less flexibility in time signatures. A skilled arranger can convert the song so that it sounds right on the new instrument, but you couldn’t reconstruct the original song perfectly from the carousel-organ version. The function matching the original song to the transposed/arranged version is not a bijective one.

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                    • Michelle H 9:58 pm on Tuesday, 9 June, 2015 Permalink | Reply

                      Cool. I was at the piano when the thought settled, but wondered if there would be difficulty transposing among instruments. Indeed, the range is not equal to all instruments. However, this discussion has helped illuminate the movement in bijection, as a kind of reciprocity. There must be a connection here to ratios and proportions, then?

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                      • Joseph Nebus 7:34 pm on Thursday, 11 June, 2015 Permalink

                        There might be a connection to ratios and proportions. I admit at this point I know so little music theory that I’d be wary of making a stupid mistake. I’ll make stupid mistakes, sure, but I do try to limit them to subjects I should know better about.

                        Liked by 1 person

                      • Michelle H 9:59 pm on Thursday, 11 June, 2015 Permalink

                        I thank you for your many exchanges on the topic. My pet idea lately has been excess as discussed in cultural theory, and bijection feels like a strong example of efficiency and quite different, so it attracted me. Thanks again and have a great upcoming weekend.

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                      • Joseph Nebus 4:16 am on Saturday, 13 June, 2015 Permalink

                        You’re quite welcome. I’m glad the topic’s so inspired you.

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