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Physonomics/Mathemology Crew: The Three Body Problem/0-1 Line & Chaos Theory
c4tch
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Registered User regular
I'm no physicist or mathematician, but I've recently been reading up on the three body problem and have some questions and would love to start a discussion of such.
Ok so, as far as I understand it the three body problem states that given objects interacting with one another based on classical Newtonian mechanics, other than actually calculating the trajectory, velocity, place, etc. of an object in relation to the other objects in the scenario, it is IMPOSSIBLE to predict ahead of time what will happen.
It is my further understanding that this is due to the nature of the chaos theory such that small changes in initial values result in massive differences between potential states later in the system. An example of this would be a mountain upon which a raindrop falls - if it lands slightly to the right of the peak it washes out to the atlantic, if it lands slightly to the left, the pacific ocean. This difference may be a fraction of an inch, but results in a massive difference later on.
I accept that the three body problem is impossible to solve because of the nature of chaos. But I ran across something yesterday that has got me thinking quite a lot, so much so that I wanted to get some better minds on it to maybe explain some of it away to put me a little more at ease.
So even the Greeks thought of this long ago: it's this concept of a line that goes from 0 to 1, and that with fractions like 3/7, or 1/2, or even 445423/100494349, you can describe an EXACT position on the line. But also, you must remember that you have irrational numbers (non-terminating, non-repeating decimals) that also may fall somewhere on that line, as values between 0 and 1. They are just as real as the rationals, but cannot be so aptly described.
MY QUESTION is based on the idea that "hey, we live in the real world, if I were presented with a giant line from values 0 to 1, and were asked to place my finger down on that line anywhere, what are my odds (given that there is a perfect computer that can perfectly calculate the exact position of my finger on this line to many, many, many, many decimal places) of landing on a rational number? What are my odds of placing my finger so that it lands EXACTLY on 1/3 or 4/5 on that line?"
I'd say it's damned near impossible, wouldn't you? Of all the infinite values present on that line, there seems to be far greater potential to "land" on an irrational number than a rational one. But also suppose that I land on a fraction that is something like 4697458713854675/147895478941320045197? Now take that concept, and extend it further. Now extend it further. Further. You can have rational numbers that will appear, even on an atomic-interaction level, to be non-terminating, non-repeating decimals, just like irrational numbers.
The reason I say all of this is that it seems to me that in this universe, everything would by logic be operating on the same scheme as my finger touching that line, whether it be the orbit of the planets or the trajectory of a comet, and as such no wonder we consider things so unpredictable in the three-body problem: everything is based on irrational/seemingly irrational numbers, with interactions down to an atomic and subatomic level all behaving "irrationally" from which on an individual, single calculation can be observed, but not predicted further.
There are surely flaws in my logic, but I just love talking about this stuff. So, if I've made any sense to you at all, I'd love for you to weigh in on this.
Ok so, as far as I understand it the three body problem states that given objects interacting with one another based on classical Newtonian mechanics, other than actually calculating the trajectory, velocity, place, etc. of an object in relation to the other objects in the scenario, it is IMPOSSIBLE to predict ahead of time what will happen.
It is my further understanding that this is due to the nature of the chaos theory such that small changes in initial values result in massive differences between potential states later in the system. An example of this would be a mountain upon which a raindrop falls - if it lands slightly to the right of the peak it washes out to the atlantic, if it lands slightly to the left, the pacific ocean. This difference may be a fraction of an inch, but results in a massive difference later on.
I accept that the three body problem is impossible to solve because of the nature of chaos. But I ran across something yesterday that has got me thinking quite a lot, so much so that I wanted to get some better minds on it to maybe explain some of it away to put me a little more at ease.
So even the Greeks thought of this long ago: it's this concept of a line that goes from 0 to 1, and that with fractions like 3/7, or 1/2, or even 445423/100494349, you can describe an EXACT position on the line. But also, you must remember that you have irrational numbers (non-terminating, non-repeating decimals) that also may fall somewhere on that line, as values between 0 and 1. They are just as real as the rationals, but cannot be so aptly described.
MY QUESTION is based on the idea that "hey, we live in the real world, if I were presented with a giant line from values 0 to 1, and were asked to place my finger down on that line anywhere, what are my odds (given that there is a perfect computer that can perfectly calculate the exact position of my finger on this line to many, many, many, many decimal places) of landing on a rational number? What are my odds of placing my finger so that it lands EXACTLY on 1/3 or 4/5 on that line?"
I'd say it's damned near impossible, wouldn't you? Of all the infinite values present on that line, there seems to be far greater potential to "land" on an irrational number than a rational one. But also suppose that I land on a fraction that is something like 4697458713854675/147895478941320045197? Now take that concept, and extend it further. Now extend it further. Further. You can have rational numbers that will appear, even on an atomic-interaction level, to be non-terminating, non-repeating decimals, just like irrational numbers.
The reason I say all of this is that it seems to me that in this universe, everything would by logic be operating on the same scheme as my finger touching that line, whether it be the orbit of the planets or the trajectory of a comet, and as such no wonder we consider things so unpredictable in the three-body problem: everything is based on irrational/seemingly irrational numbers, with interactions down to an atomic and subatomic level all behaving "irrationally" from which on an individual, single calculation can be observed, but not predicted further.
There are surely flaws in my logic, but I just love talking about this stuff. So, if I've made any sense to you at all, I'd love for you to weigh in on this.
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I wouldn't necessarily call the bounds of the range irrational, either. You have the Planck length to contend with as a lower bound to how precisely your finger can fall. Really, given what your finger is actually made of, once you get down to ridiculously small units the bounds of the range are just ill-defined anyway. You can't really extend the precision indefinitely without being dishonest about what you're measuring.
So no, you're quite wrong.
EDIT: About the number line thing, anyway. Yes, many physical systems are chaotic if you want their precise behavior.
And yes. Yes it does.
the "no true scotch man" fallacy.
well suspend your disbelief for a moment and imagine that my finger is capable of falling, down to the plank length, on a specific point.
and also, if you really want to talk in terms of the plank length, I could just as easily say since we all live in a 3 dimension-observable universe, that it is very much like a 3D grid from which at any plank-box you could tell me the exact energy, movement, etc...
with that in mind, the three-body problem is about interactions, and interactions are based on distance, and the distances between two objects regardless of the plank length will most definitely be irrational/observed as irrational.
Is Zeno's arrow paradox the one with halving distances and the turtle and etc?
You don't need the Planck length to solve that.
Man, not having precise measurement isn't the same as irrationality.
EDIT: Carl Friedrich Gauss made Zeno his bitch. Don't know if he was the first one.
are you talking about the idea that any system if left on its own capable of coming back to its original position is considered stable and therefore finite?
Actually, the tortoise paradox is more relevant to the OP. And yeah, you're right.
the "no true scotch man" fallacy.
Actually no I was applying finite in a way that you can't, so disregard finiteness. I was saying that even in space that's discrete, i.e. totally without irrationality, you can still have behavior that is heavily dependent on initial conditions. Irrationality need not factor in.
but even in those conditions where all is rational, even two lines with a measurement of 1 at perpendicular angles give a third line between them that is irrational.
math is just a language we're using to describe the universe, and it is entirely relativistic. I'm saying that regardless of the language, the "irrationality" is inescapable, even in your discrete space, right?
If my physics are correct, then yes the trajectory would ultimately be unpredictable. But if we could measure that accurately it would probably stay close enough to the real value for a long time anyway.
But I don't know.
And regardless, numbers as a system are theoretically infinitely divisible even if space is not, so we could "imagine" half of the indivisible distance, even though such a half is impossible, and such imagining is probably based on an incoherent conception of physical reality.
Just throwing that out there in case smarter people think it's relevant.
Just wanted to point out this is not true.
Math is an artificial system based on postulates (that is, things that cannot be proven). Not many postulates but they are there.
It just so happens that math is a useful tool in physics but it is not in any way correct to call it a 'language'. It is a consistent logical system based on certain postulates.
BTW: With regards to your confusion as to irrational vs. rational numbers look up the terms "countable" and "uncountable". The space of rational numbers is a countable infinity (ie: there is an isomorphism between that space and the space of integers). Real numbers are uncountable (ie: there is no such isomorphism).
This basically means that while they are both infinite there are still a lot more Reals than Rationals. They are different kinds of infinite.
Which is the tortoise paradox? I've probably heard it, but I suck at names.
edit: Is this the "turtles all the way down" problem of not being able to infinitely subdivide particles into their components? Eg, "What is an atom made of? Okay, what's a proton made of? Okay, what's a quark made of? Okay, what's that made of?" ad nauseum.
Man chasing a tortoise will always be behind because even as he's catching up, the tortoise will have moved ahead a little, and in the time it takes to cover that distance the tortoise will have moved a bit more etc etc.
Ah, yes, that one. Isn't that fundamentally the same as Zeno? Ie, solvable by infinitesimals?
In general though, even though it may be useful heuristic for everyday life, you can't really think of real physical space as a three dimensional Cartesian space of real numbers (or R^3). I'm not an expert on physics, but I know they have evidence that it is a whole lot weirder than that.
Depends on your purpose. You can think of it that way just fine if you're instructing someone where to park their car or how to build a house. Just like you can think of Newtonian physics as hunky-dory for figuring out what'll happen to you if you plow into a brick wall at 60mph in your Beemer.
There's little reason to believe that any scientific formulation we've come up with is any more than a very useful approximation of the how the universe works. What you use depends on how much precision you need in your answer for a given purpose. And the beauty of the Planck constants is that we don't even really need to obsess with Ultimate Precision or Perfect Models. If something can reliably predict behavior down to within the Planck Length, or Planck Time, or Planck Credit Rating, or what have you, you're golden.
Yes, they're all basically examples of the same conceptual problem.
the "no true scotch man" fallacy.
Anyway, as for the number line question: This is why mathematics and reality are mutually exclusive. There is no such thing in reality as a number line. Not in the reality you can "touch," anyway. Lines, triangles, circles, hell, even numbers, these things have no concrete basis. They aren't real. Real numbers aren't even real. You could never draw me or show me a circle whose circumference divided by its diameter was exactly pi. Precise enough measurement would always show it to be not quite a perfect circle. We've calculated pi to trillions of decimal places, but you could take pi to only 39 decimal places and that number (which isn't pi) accurately calculates measurements of the entire observeable universe relative to the precision of a single hydrogen atom. Circles and pi don't exist. They are all imaginary tools, concepts we ourselves have imagined and defined, ones that apply quite consistently to help us figure things out about reality.
Well, yes. Though calculus is sort of based on infinitesimals, inasmuch as the Fundamental Theorem of Calculus is "Hey, look what happens when you add up a bunch of 'em together!"
Sorry, I'm fixated on that thread. I'll drop it eventually.
I agree about infinitessimals except that we had a thread like years ago where I was touting them and everyone was saying they didn't exist and weren't meaningful in any real mathematics system.
Is it like lim (x->0) x ?
Of course, I also think "2" exists as a real thing, so, you know.
And I maintain that science is just a bunch of anecdotes in much the same way a diamond is just a bunch of carbon atoms. Yes, they need to be the right kind of atoms, organized just so, meticulously arranged. But you look at a single molecule, and guess what? Fucking carbon atom.
You are right that multi-body orbital mechanics problems are chaotic, but that has nothing to do with the insolubility of the n-body problem for n>2, and you are wrong in saying that the impossibility of finding a closed-form solution means that prediction is impossible—it doesn't. It just means we use a numerical approximation (of whatever accuracy is desired; down to fractions of a millimeter if necessary).
Basically an infinitely small nothingth of data that, when added to an infinite number of other infinitely small nothingths of data, combine to form something finite and tangible.
But an infintesimal as explained to me in high school was basically what differentiates lim(x->0)x and -(lim(x->0)x)
Does Magritte know you're smoking his pipe?
the "no true scotch man" fallacy.
There are a lot of problems like that in differential equations. Fuckers be hard.
Edit: Yes, this is indeed the case.
I wasn't aware you could only count objects that were perfectly identical. Now if you'll excuse me, I'm going to go buy a carton of an egg and another egg and another egg and another egg and another egg and another egg and another egg and another egg and another egg and another egg and another egg and another egg.
I think they are something that shows up in different number systems than the real numbers. I haven't actually used them, but I heard of them in that context.
CR covered it well. There could be an analytical solution, but it's complex as hell if so, and I wouldn't bet money on it.
Zen master lays his hand on your egg and says, "This is not an egg."
the "no true scotch man" fallacy.
Ok, so yes a possible rigorous definition would be the lim(x->0) x or something similar.
Edit: Aaaaah, I see now. "infinitesimals" is an idea used to explain limits without getting into the rigorous definition. EG: in highshcool or 1st year college calculus. I was wondering why it didn't ring a bell.
The rigorous definition is indeed kind of hard to really wrap your brain around.
Edit: from an earlier post of mine
(images stolen from wikipedia)
is true (and thus the limit is said to exist) if and only if
(my translation from shorthand)
The function f(x) has a limit L as x approaches c if and only if for every epsilon greater than zero there must exist a delta (greater than zero) such that for every x where 0 < | x - c | < delta it must be the case that | f(x) - L | < epsilon.
*Where f is defined on an open interval containing c (though it may exclude the value c) and c, L, delta, epsilon and x are real numbers.
Note: Remember that the notation |x| means "absolute value of x" and thus | x - c | is really just the distance between x and c. So the above is saying that for even really tiny values of epsilon you can always find some value of the function that is that close to the limit. Pretty much.
Also note that the footnoting about real numbers and open intervals which seems like the mathematical equivalent of a EULA (eg: pointless boilerplate) is actually very important. This definition is itself based (via a surprisingly few other proofs) on the fundamental axioms of the real numbers. Doesn't work unless you can assume those axioms are true.
And it is vital to remember that reality does not share the same principles of the "Real" numbers. Look closely enough and its very hard to find something in nature which is actually continuous or conforms to the notion of "open intervals". The real world is full of (is almost entirely made up of) discontinuities. Dirac came up with a system called "distributions" (to work around the restrictions in the idea of "functions") for dealing with data that has a lot of such discontinuities.
That happened to me once, only instead of a zen master it was a hobo and instead of my egg it was my ass and I think he actually phrased it as "You got a purty mouth."
Before that, I totally did not grok the whole distribution thing at all.