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When they say motions of three bodies are random and unpredictable, I assume they mean not able to be modeled with a closed form equation? Seems like the motion
by thinkski 5y ago
When they say motions of three bodies are random and unpredictable, I assume they mean not able to be modeled with a closed form equation? Seems like the motions would still be entirely deterministic — could still predict the locations of the bodies computationally, given your computer computes faster than reality (at least for a reality with only 3 bodies), no?
- ordu 5y agoThey would be deterministic. But they are unpredictable. Minuscule fluctuations (coming from influence of some much smaller bodies, that are not in the model, or approximations in calculations) can lead to dramatic differences in the outcome. So theoretically speaking, they are deterministic, but practically they are unpredictable.
- Aeolun 5y agoHuh? If this is the case, wouldn’t a 2 body problem also be practically impossible to calculate? I mean, you can certainly still predict to a certain (probably high) level of accuracy, but ultimately that motion is also influenced by factors outside your model.
- deleted 5y ago[deleted]
- colechristensen 5y agoThe question is what happens with a small perturbation and how does it "grow". For a two body problem, you nudge one of the bodies and it is forever off by a small amount, but your predictions into infinity require only a small adjustment to compensate. For a three body problem you nudge one of the bodies and only for a very short time do your previous predictions stay true, the change amplifies until nothing you thought might happen before the nudge means anything at all, and a common occurrence is one of the bodies being ejected.
- stavros 5y ago> a common occurrence is one of the bodies being ejected. So the problem eventually solves itself?
- motoboi 5y agoWell, not if the ejected body contains you.
- TheOtherHobbes 5y agoBeing ejected from your model is unquestionably a solution of sorts, although it does Raise Questions.
- dnautics 5y agoYes, but but also the problem really is: "given some initial condition space, can we estimate the likelihood of ejection (how many conditions in the condition space eject) in X timeframe"?
- onlyrealcuzzo 5y ago> For a two body problem, you nudge one of the bodies and it is forever off by a small amount, but your predictions into infinity require only a small adjustment to compensate. How does this even work? Where is a place in the universe where there are only two bodies?? Where would the nudge come from if not from a third body?! A ghost?
- mannerheim 5y agoYou're taking it too literally. The 'nudge' means a change in initial conditions, and could result e.g. from measurement uncertainty, not necessarily a literal nudge. It also doesn't necessarily matter that much if there are more than two bodies, if the gravitational influence of other bodies is small enough, then you can model as a two body problem.
- 5y ago
- alephu5 5y agoFor a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in advance with a precise albeit imperfect measurement of initial conditions. With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.
- Grustaf 5y agoNo that's not it. The two body problem has a closed, analytical solution, the three body one doesn't, so you need to simulate it. It's a fundamentally different approach.
- extropy 5y agoBoth of you are correct.
- l332mn 5y agoThat's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.
- Grustaf 5y agoIt is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case. Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.
- johncolanduoni 5y agoThere are iterative methods/systems that stabilize over time. For example, symplectic integrators on tame problems oscillate lightly around the true energy of the system over time. The issue here is the properties of the underlying problem, not the set of solution methods.
- Grustaf 5y agoNo, the fundamental difference is that the two-body problem can be solved analytically, you can write down a formula. For three bodies and up you only have numerical solutions, simulations, and they will break down over time.
- bottled_poe 5y agoI think this is accurate and I don’t understand why you are being downvoted.
- phreeza 5y agoThe problem is that a three-body system is inherently unstable/chaotic. So even if you run a numerical simulation with the same granularity for a two-body and a three-body system, the three-body simulation will degrade much faster than the two-body system. This is unrelated to the fact that there is a closed form solution, there are many stable, non-chaotic dynamical systems that don't have a closed form solution.
- MontyCarloHall 5y agoConversely, there are also extremely simple closed-form recurrence relations that exhibit chaotic behavior, e.g. the logistic map.
- phreeza 5y agoTrue but a recurrence relationship is not the same as a closed form solution. The differential equation for a three body problem is also very simple.
- MontyCarloHall 5y agoOf course. My point was just that you can evaluate a recurrence relation exactly (i.e. with zero numerical error) and still get chaotic behavior. OP’s mistaken point was that the three body problem’s chaos arises solely from numerical error during simulation, which is untrue.
- sasaf5 5y agoBetter stated, the 2-body problem can be solved with a finite number of standard operations, i.e. a closed-form expression. This solution does not exist for the 3-body problem.
- evanb 5y agoThis is not a requirement for a system to lack chaos, nor is it a metric by which we can judge if a system DOES have chaos.
- aaaaaaaaaaab 5y agoNo. A small error in the initial conditions of a 2-body system produces a small error in the result. In case of a 3-body system, a small error will result in drastically different outcomes. The phase space of a 2-body system is nice and smooth, but a 3-body system’s is more like a fractal.
- bsf_ 5y agoIn principle - yes. Except that we can change our frame of reference, and treat the two body problem as a pseudo one body problem (the lab frame becomes the center of mass of one of the bodies). One cannot do this for the three body problem, which gives us at best a pseudo two body problem.
- mjburgess 5y agoRandom doesnt mean non-deterministic anyway. X is random with respect to Y, if knowing Y makes no difference to your predicting that X. QM systems are indeterminate, they are random in the above sense /because/ they are indeterminate. But that isnt what random means.
- elcomet 5y agoWhat you're describing is not randomness, it's independence. It's hard to define randomness. I think non-determinism is better than your definition.
- mjburgess 5y agoIt isn't hard to define randomness. It's an epistemic condition on the knowability of Y given X. Non-determinism is an incoherent definition of randomness; classical physical processes are entirely deterministic. The point of a coinflip being random is that it is random with respect to the information both observers of the coinflip have. It isnt random with respect to /any/ piece of information. There are almost no processes which are non-deterministic in this sense. Not enough to bother calling them random; and in physics we do not: the word is indeterminate. Randomness has nothing to do with quantum mechanics; it wasn't invented in the 1920s. It's an epistemic condition. The RANDOM variable X, st. X ~ N(mean, std) provides a random number x -- x isnt random with repect to the outcome which produced x; nor is it random with respect to an index of a vector in which it is contained.
- elcomet 5y agoThis does not match the intuitive notion of randomness though. I would say for a given variable to be random, it must not be predictible, given any other variables that humans can know. I don't think it makes sense to say that X is random "with respect to Y", that's just the definition of independence. And a constant variable is independent from all other variables, but it's definitely not random.
- 5y ago
- ninkendo 5y agoThe motion isn’t deterministic if free will exists. Launching a rocket into space decreases earth’s rotation speed ever so slightly, which will have a small impact on the moon’s trajectory due to tidal interactions, and so on.
- athrowaway3z 5y agoPlease define 'free will' before using it in a sentence about determinism.
- thaumasiotes 5y agoIf that was a requirement, discussions of determinism would sound awfully one-sided. ;p
- mxxc 5y agothat case is beyond scope of the three body problem
- martincmartin 5y agoThere are those who believe determinism is not only compatible with free will, but required for free will. https://en.wikipedia.org/wiki/Compatibilism https://en.wikipedia.org/wiki/Compatibilism
- simonh 5y agoIt's got nothing to do with perturbations from additional small bodies, the problem is exactly mathematically calculating the outcome for exactly three bodies within reasonable time frames even if you assume perfect ideal knowledge about the system. It turns out this is excruciatingly hard. The n-body problem where you add further bodies, even very small ones, is even harder.
- kergonath 5y agoI think the issue was more with “random” than with “unpredictable”.
- deleted 5y ago[deleted]
- bo0tzz 5y agoWhile we can simulate three (or more) bodies' gravitational interaction, the chaoticness means that any error in initial state, no matter how small, will be hugely amplified. This makes long-term predictions untractable
- akomtu 5y agoThe initial conditions are usually known up to a small epsilon in practice. I guess this initial error grows exponentially with time, hence "unpredictability".
- DiogenesKynikos 5y agoThis exponential increase in error is described by the Lyapunov exponent of the system: https://en.wikipedia.org/wiki/Lyapunov_exponent https://en.wikipedia.org/wiki/Lyapunov_exponent
- bad_username 5y agoThere are cases in classical mechanics that fail to be deterministic. https://physics.stackexchange.com/questions/403574/what-situations-in-classical-physics-are-non-deterministic https://physics.stackexchange.com/questions/403574/what-situ...
- reedf1 5y agoChaos is the extreme dependence on initial conditions. The three body problem is significantly more chaotic than a two body problem. Welcome to chaos theory!
- mxxc 5y agothere is no closed form, like you say, and the additional complexity is around ergodicity, i.e. solutions that start close to each other might end up very far from each other after a certain point. this is also an issue with computer simulations as the error might accumulate and push solutions away. in practice, given the amount of cosmological computations people do on a daily basis, including those for satellites and rockets, this might not necessarily be that big of an issue, but i don't work with that stuff on a daily basis.
- nine_k 5y agoOn top of theoretically computable, but highly divergent / unstable functions mentioned nearby, there are fully deterministic (non-random, pure) but non-computable functions. The simplest example is a function that answers whether a given Turing machine would stop.
- remram 5y agoYou can only do that with some error, e.g. your simulation needs some time step that controls the tradeoff between computational resources used and accuracy of the results. For any fixed time step, after enough time your simulation will completely diverge from reality.
- HWR_14 5y agoAs I understand it, "random and unpredictable" means impossible to measure the initial state in sufficient precision and/or computationally impossible to calculate in a reasonable amount of time. The actual underlying math is deterministic.
- argvargc 5y agoIt seems the problem is often mis-stated - there is no calculation problem, the problem is in adequately defining/sampling the initial data. It's maybe similar to predicting the weather - we can have all the perfect equations in the world for fluid dynamics and heat flow etc, but until we have system-invisible temperature and humidity sensors for every square millimetre of atmosphere and earth volume, we won't be able to predict the weather very accurately or very far ahead.
- m3kw9 5y agoIf you freeze each point in time , isn’t the next minute step deterministic? Every time you freeze, you would have all the motion vectors to calculate the next moment, if you expand that using a lot of computation power, ca you solve it that way?
- ineptech 5y agoEdward Lorenz summarized chaotic behavior as: "When the present determines the future, but the approximate present does not approximately determine the future." So yes, you could predict the locations computationally to an arbitrary point in the future if you knew their starting locations and velocities with perfect precision; but in practice of course you cannot know anything's position with perfect precision, so your simulation would become inaccurate relatively quickly. edit to add: and I believe that what this paper discusses is not a solution to the above, but rather a way of getting around it by modeling some types of three-body behavior as if it were truly random, rather than chaotically deterministic.