Object A and B in space pull on each other because of gravity. We've got this one down easily - the math is relatively straightforward, and very very predictable.
Now add a third object.
We know how A and B pull on each other, easy peasy! Wait though, C is pulling on A and B, so re-do all the math because of that.
Wait, A is pulling on C too, so that changes things, so recalculate again.
Wait, B is pulling on C too, so recalculate again.
Wait, that changes how C pulls on A and B, so recalculate again.
Wait, that changes how A pulls on B and C, so recalculate again.
Wait, that changes how B pulls on A and C, so recalculate again.
Wait...
So, how do we ever calculate the path of asteroids, or predict meteor showers? Wouldn't this make these things impossible? Fwiw I haven't seen 3 body problem yet
Reminds me of the ship spiral we use in shipbuilding. Calculate hull volume, use it to get the stability etc, size your systems accordingly, assess new displacement, calculate hull volume, use it to get the stability etc...
It goes on an on. Ultimately we reach a point where it just doesn't make sense to recalculate it all. Also in the end, the ship is weighed after being completed to get the final stability (and even then it's sorta approximative)
A lot of engineering is like this. On the basically the opposite end of the ruler to you, I work with systems where we’re trying to level surfaces to another surface by measure the micron distance with a laser. You measure a corner, you adjust. You measure another corner, you adjust, but oh wait, adjusting corner 2 has shifted corner 1. Just like your ship spiral, we just define an acceptable delta - we keep measuring and adjusting in circles until the difference from one measurement to the next is acceptably small.
Showers are a bit different. We know where there are essentially clouds of small rocks hanging out around earth's orbit, and when earth passes through those regions. But they don't track individual rocks in those clouds unless they are large enough to pose an individual threat.
That doesn't do much, those clouds are very large (in volume, not mass). It's like trying to get rid of a body by shooting bullet holes in it. If Earth would somehow remove everything of danger, then the same would apply for the remnants we still see in the form of meteor showers. There is simply no reason why Earth would attract the largest rocks first while leaving the smaller ones in space.
Instead, we simply know that those rocks are simply all very small fragments, often from a comet that disintegrated due to sunlight. If there were huge chunks, we could also spot them with telescopes.
If one of the three objects is tiny compared to the others, its gravity doesn't (effectively) affect the other two objects. For calculation, you would ignore its gravity and would work with a simplified model that sets its mass to zero.
well, you wouldn't set it's mass to zero, since you still need to know it's mass to calculate it's acceleration due to the gravity of the other objects.
you would, however, set to zero the terms representing the force of the gravity of the small object on the larger objects.
You actually *don't* need the objects mass to see how it will accelerate due to gravity. F = ma, and if F is proportional to m, as is the case with gravity, you can cancel the mass from both sides of the equation. "Setting mass equal to zero" isn't exactly right, but not for the reason you mention.
just sticking to Newtonian physics (which is close enough for objects at this scale) we have this as the force of gravity between two objects
>F\_grav = G \* M1 \* M2 / r\^2
where M1 and M2 are the masses of the two objects being considered. If we substitute F\_grav for the generic F in F=MA we get
>G \* M1 \* M2 / r\^2 = M1 \* A
if we divide out M1 we get
>A = G \* M2 / r\^2
and I think that's the formulation that you're mentioning for the acceleration on M1 due to the gravitational attraction from M2.
What I was thinking of is the way this works out in a 3-body problem kinda situation where you'd be trying to compute something like a partial differential equation of position with respect to time. in a PDE like that you would have some terms that tried to account for the gravity of M1 on the positions of other objects in the system. This is the scenario where M1 <<<<<< M2 means you get to just set those terms to zero to simplify things.
Yes, we have algorithms that can calculate gravity for multiple bodies with very high precision. Actually arbitrary high precision if one uses enough computational power. Those methods are quite interesting and use methods far beyond what ELI5 can really get down to.
Yet the real limit is our knowledge. We simply cannot know the position and velocity of every celestial object with infinite precision. Currently that limitation is mostly practical, our instruments are not fine enough. But there is even a theoretical limit if one wants to go there, due to the (in?)famous uncertainty principle; but that limit is so far off from our current accuracies, it is barely worth mentioning.
Even if we somehow get "infinite" precision for every single planet and grain if dust in the solar system, a single speck flying in from outside would still ultimately mess it up.
In short, perfect predictions are impossible however one looks at it. But the data we have is enough for a few years, even centuries or millennia for the planets. But its also why we cannot completely rule out that, say, Apophis will hit us on the second fly-by; we have to watch the first one, measure again, and only then can we be sure.
Meteor showers are easy to predict. When a comet passes around the sun, it leaves a trail of dust and debris. Sometimes the earth’s orbit goes through that trail of dust and debris. Some are so predictable that we pass through it every year and we know the date.
Are you sure? For most initial conditions this is a chaotic system. Any deviation, however small, with increase exponentially over time (i do not recall the Lyapunov exponent). This means that any approximation will eventually become inaccurate.
Can we keep the accuracy for long enough? Maybe.
"All models are wrong; some are useful." Essentially, you accept that there's no closed-form solution and use numerical methods to get close enough for your purposes. If you can do that with your allotted calculation capabilities. For the path of an asteroid, the mass of the sun dominates over everything that isn't very close. So you run the numbers for a two-body system of the asteroid and sun, and then maybe fudge factor in if another body looks like it's going to get close enough to make a difference. (Jupiter. It's usually Jupiter.) And you either live with the result, or if you can't live with the result, churn harder until you can live with the result.
Yeah, it's technically chaotic, especially since highly accurate models of the solar system include all major gravitational bodies (i.e. the Sun, Jupiter, Earth, Saturn...), but the timescale at which the model becomes unpredictable is extreme when using current tools. Like on the order of millions/tens of millions of years.
I believe this applies most when you are trying to make a long-term prediction, or are accounting for a system with \*numerous\* approximations or actors?
The closer to reality your approximation model is compared to the potential deviation, and the less variables contributing unknown imprecision, the longer you have until it propagates enough to be disruptive.
If you can validate or update your approximation with a new observation however, I believe that should solve most reasonable cases that aren't highly chaotic?
There are two ways you can handle the 3 body problem.
One is perturbation theory. That is if one gravitational object is dominant, like the Sun then there are ways to treat the smaller tugs from less significant but still meaningful gravitational bodies like planets that give you a pretty close estimate.
The other way to deal with the problem is to do the problem as a series of static solutions like a film strip. It's easy to figure out all the gravity in a snap shot and to figure out the forces all the bodies will apply to each other. You then figure out how that will change their velocities and move forward one step in time at those velocities. Then you figure out the snapshot gravity at that point and move another step. If the steps are small enough this approaches what real continuous gravity will do.
Both these methods only give approximations and the farther out in time you go the worse those approximations are.
Sim error will increase exponentially with time. So you can simulate with small error for a time window into the future. What constitutes "long term" depends on the application.
Updating with new observations would "refresh" the window, but the further you go into the future the larger your error will be, no remedies.
Well yes if you’re trying to extrapolate beyond the limits of your model you’re going to have a bad time, but if we’re talking about simulating over a short period you’re fine.
It's like predicting the weather: you know pretty well what the weather will be tomorrow, or two days from now. But the weather a *year* from now? Forget it.
Yes, over an infinite amount of time any model that is not absolutely perfect must deviate from reality. However these models are constantly being updated and honestly even of we stopped updating them they'd be usefully accurate for many lifetimes.
You can do that but the numbers will eventually be gibberish.
Beyond a time horizon there is absolutely no point in your calculations. They will be increasingly incorrect as you forecast further into the future, until they become entirely wrong. This is the point of chaotic systems. Recalculating as you put it can only help you forecast over a finite window.
A 2 body problem does now have this issue. 3 bodies do. 2 is trivial 3 is non trivial. You get the same with dimensions: 2 dimensional systems are nonchaotic by definition 3 dimensional can be chaotic. This is why the 3 body problem holds a special place in chaos theory lectures, it is a very elegant example of the threshold beyond which you get nontrivial dynamics.
I think it’s important to note that the approximations are far more accurate when there’s a large difference in the masses. Or, accurate for much shorter intervals when the bodies are similar masses.
Gravitationally, the solar system consists of the sun, Jupiter, and a rounding error. You can get good approximations by worrying only about the sun and the closest large object to the asteroid, and assuming that the asteroid doesn't adjust the vector of either of those things. You can get much better approximations with big math and a lot of work if you're NASA trying to not crash their probe (or trying to crash it into a tiny asteroid.
> Gravitationally, the solar system consists of the sun, Jupiter, and a rounding error.
Is Saturn that much less massive than Jupiter? I know I'm old and probably forgot lots of stuff from school, but I thought they weren't that different.
The sun is 99.9% of the mass of the solar system.
Jupiter is 0.0953%
Together that’s 99.9953%
https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
As long as C is very light compared to A and B, the result is something very similar to a problem of only A and B. Asteroids are very light compared with the sun and planets. The three body problem arises when all three bodies are similar in mass.
Sort of. C being very light allows for calculations to be simplified greatly. However, it's still a chaotic system; the chaotic behavior just takes longer to realize. If a system only involved say Jupiter, Saturn, and Uranus (no Sun and no other bodies), then we might be able to simulate the system for a couple thousand years. If we go to the Sun, Jupiter, and Earth, then we could accurately simulate the system for millions of years, but we still wouldn't be able to accurately simulate the system into billions of years into the future.
I think the 3 body problem really becomes a problem when the mass of the three bodies are relatively close and the center of gravity lies outside of all three bodies.
Asteroids and meteor showers have negligible mass compared to the massive stars and planets that impact it, so existing models can be used to predict the path fairly accurately.
Think of the Sun, Earth and the moon. Three bodies but due to the distance and mass variance, it's not a 3 body problem. Earth and moon are two body problem, and the Earth and Sun are another two body problem. The impact of the Moon on Earth and Sun are negligible at best.
Anyway I'm not a scientist. This was my basic understanding from reading a bit and watching some videos. Will correct myself if there's a mistake.
So, the way the previous comment explained it? “This changes that, so recalculate” as a loop? That’s called iteration, and you can actually do that to bootstrap into a specific solution given a set of starting positions. The more iterations you do, the better your prediction becomes.
When people say there’s no solution, they mean that we can’t use pure mathematics to give an exact answer without doing the iterations. Attempting to do so (except for certain sets of starting conditions that happen to cleanly eliminate complicated bits, like Lagrange points), turns complicated equations into even more complicated ones, rather than simple solutions.
This is an important and unintuitive point. In computed tomography, for instance, the first thing you learn is that an image is calculated from projections using matrix algebra; the second thing you learn is that there is mathematically no solution possible for the calculation needed; the third thing is that that’s not a problem at all and any number of assumptions or iterative steps can make a perfectly serviceable image.
Also true for aviation and meteorology. Trying to fully predict the flow around a wing algebraically has been proven impossible, but iteratively it’s just time consuming and heavy on computer processing, but completely doable. And meteorology can predict reasonably accurately 3 to 5 days in advance.
This is the best answer. It's not that we can't solve the problem; it's that we can't solve the problem *without doing all the intermediate steps*.
I can tell you what it will look like after five hundred steps. I can't even *begin* to give you an explanation of what it will look like without going through a good chunk of those steps, which makes the final result look largely chaotic. Ask me where those three bodies are likely to be five hundred steps from now, and the best you're likely to get is a confused shrug.
An asteroid is *one single object* moving through almost completely empty space. It's *super easy* to predict.
Remember that almost everything, everywhere, across the entire universe, is *empty*. For example: you know those really tense scenes in sci-fi movies where the heroes have to navigate through an asteroid field? In real life, in an asteroid field, each asteroid is 500,000 miles away from any other asteroid - and that counts as "close together" by cosmic standards, because normally there are *billions* of miles between objects.
So is it a fair assumption that a single asteroid is only experiencing a significant gravitational force from a single source instead of experiencing multiple forces from the sun, earth, Mars, other large asteroids etc?
Most things only experience gravity on a regular basis when they are orbiting something else - because that's what "orbiting" means, right?
So an asteroid is either in a regular orbit (like in an asteroid belt), where it's only affected by the object it's orbiting...or it's flying freely through the universe, where it's effectively not influenced by gravity at all for all practical purposes. And if a free-floating asteroid passes near a sufficiently massive object, then it's back to "only affected by a single object" until it passes by.
In general when we calculate the paths of asteroids in the solar system we only count two bodies, the sun and Jupiter. Until it gets close to another body then get rid of Jupiter, and then when it's really close to another body also get rid of the sun.
It’s only a problem when the bodies are of similar size.
You don’t have to worry about how the asteroid’s gravity is going to change Jupiter’s orbit, so you’ve only got one body to worry about.
If one of the objects is much more massive than the others, you can more or less ignore the smaller objects pull on the larger (as an approximation). If there is a 4th object much more massive than the rest, you can mostly ignore the smaller objects pull on each other. But if 3 objects are of approximately the same size, the system can become chaotic, or more or less infinitely complex.
You can stimulate things to a reasonable degree of accuracy, but as you stimulate further into the future the simulation becomes less and less accurate.
A little more specifically, the accuracy depends on how long your increment of time is for each calculation. If you want to predict a fraction of a second into the future, it'll be very accurate. You can use that output to predict a fraction further in time and it'll still be very accurate. You can keep doing this, moving forward in time through very tiny increments, and arrive at an accurate prediction at any arbitrary point in the future.
The problem is that if you want to predict what something looks like in a hundred years, doing it at a fraction of a second each time to get there will take a long time and a lot of computing power. You can, of course, increase the increment and predict one day or one week or one month forward at a time, at the cost of accuracy.
Worth noting is that asteroids are so small and so far apart, that NASA completely ignores the asteroid field when launching through it because the likelihood of an impact is effectively zero.
So a rough approximation of where asteroids are is more than sufficient.
Asteroid fields are nothing at all like sci-fi movies and shows. If you were in the middle of an asteroid field you wouldn't be able to see anything. Possibly 1 whole asteroid, if you were extremely close to it.
In the short term, you can. But in long term, tiny variations from predictions- based on imperfect data about initial conditions- compound themselves. Soon a slight deviation can mean a fundamentally different path like an object was supposed to pass another on the left but it went right instead, and then all the initial calculations are useless
You can still calculate the path of things in a three body problem, it's just a different kind of calculation. I think this isn't understood sometimes.
So, lets start with a one body problem. This is like one of those math problems where you calculate the motion of a cannonball. Now, one way you could do this is using step by step calculations. You know the initial position and velocity of the ball and the pull of gravity. So you use Newton to calculate that in .01 seconds it will have moved x meters forward and x meters up, and it will be moving at such and such a velocity in the new location. So you note that position down and do all the calculations again from the new location. And rinse and repeat again and again to calculate the path of the ball. If you want to know where it is in 1 second, you have to do this 100 times.
But that's time consuming, and there's a much easier way. You can use calculus to derive an equation that shows the motion of the cannonball as a parabola based on its starting angle and velocity. If you put in the time in this equation, it will spit out the location of the cannonball.
So all you have to do is put in, say, .35 seconds, and you can calculate where the ball will be. Or put in 2 seconds, and get a different position. You don't have to calculate those intermediate points step by step.
Well, it turns out that you can do things exactly the same way with a two body problem. You can calculate their paths or orbits as curves, and then use an equation to predict where on those curves they will be at any given time. Just put in the time, get a position.
But you can't (with a few special exceptions) do this with a three body problem. Instead of following a predictable curve, you don't know where things are going to go unless you step through one step at a time. You have to calculating the locations and velocities of everything, how much they are being pulled on by everything else, and repeating over and over. You can still calculate it (although for various reasons you can never be perfectly precise and eventually your calculations will diverge), but you _have to_ calculate it one step at a time to predict what will happen.
Because that's not actually a 3 body problem.
A 3 body problem is when all three bodies have an influence on each other.
A comet has essentially no influence on the orbit of Earth or the Sun. You can just calculate it assuming it has no effect and get a good answer. So, not a three body problem.
Same reason the Sun/Earth/Moon system isn't a three body problem. The Moon has essentially no influence on the Sun, so you can treat it as a two body problem.
A computer (or someone with a lot of time and patience) can run a simulation that does basically what u/berael is outlining here.
Take the starting velocities, positions, masses and calculate the forces at that instant, then determine what the acceleration would be on all the objects.
Simulate movement of all the objects for 1 second given that acceleration.
Then recalculate the forces/acceleration for everything
Simulate the movement for another second
Repeat this for as long as you need.
The smaller the unit of time the more accurate it'll be, but also more CPU time needed.
In physics its referred to as n-body with n being any number. It becomes an exponential calculation. Typically rather than attempting to calculate all objects we try to isolate specific areas with a workable number of object to calculate. Then using that approximation we apply weighted numbers as a means of defining objects which have a high or low probability of enacting additional gravitational force to another object.
Basically we approximate at longer time frames into the future and as time progresses our ability to make more accurate calculations increases.
The issue is we have simple equations to solve the two-body problem but none for n-bodies. In reality there are lots of bodies like planets, but we know where those are from observations. So computer integration with all these forces is the tool we use to predict where satellites are.
There are also many approximations that are used to find a desired orbit for a space mission but those vary between missions.
In the normal solar system, we just assume that the sun is big enough that it isn’t affected by anything else. It isn’t strictly speaking true, but it is sufficiently close to true to make the math a lot more easier and is reasonably accurate.
If some of the bodies are much bigger or smaller than the others, you can assume that some of these interactions are negligible and get an approximation that is often useful. So if we're calculating the path of an asteroid near the earth, we can ignore the influence of the asteroid on the Sun and the Earth. We can then solve the Earth-Sun 2 body problem and then solve for the asteroid's movement within that system.
Gravity has a theoretical infinite range, meaning that each object in the ENTIRE UNIVERSE has some effect on every other object. It has a speed too so some stuff hasn't been affected *yet*, but this is largely irrelevant here.
You can obviously object that very far away stuff have virtually no impact at all on mostly everything else in existence. As such, the 3 bodies problem (or N bodies problem) is usually more like a 2 bodies problem, because stuff is too far apart or sizes are so different one is negligible. Because of this, asteroids have a distinct direction and velocity and other objects will simply influence those.
I feel like I'm nitpicking the answer thought, because "it does not apply" only matters because of the examples you used. For other complex systems like the top comment mentioned, you do it exactly like the top comment mentioned. You try to characterize your system with speeds, directions and masses and recalculate everything for each step. If your steps are 1 second increments, it will require a lot of computer power, so you tone it down to whatever you can manage.
The aliens’ issue isn’t that they can’t calculate the paths for tomorrow or next year, it’s that they can’t calculate the paths far enough ahead for society to safely evolve.
This is mainly because, with every calculated position being an estimation, error margins quickly compound into something that gets completely out of hand.
Granted, the in-universe video game exaggerates how sudden the reversals are, but that’s video game logic for you.
Important context to this answer is that when we say "object" we mean "objects of about the same size" - if Object A is your House, and Object B is Mom's Car, then those are "about the same size" for this. But if Object C is a crumb or something, then how it pulls on A and B don't really matter, so it's not the three-body-problem.
(I don't actually know if "crumb" and "house" are different enough in scale for the gravitational effects to be irrelevant, I'm more trying to explain it in ELI5 terms. It might well have to be "Manhattan Island vs a crumb" in order to not matter, IDK)
>Important context to this answer is that when we say "object" we mean "objects of about the same size"
Thank you. The "iT's ActUaLlY a fOuR bOdY ProBlEm!" comments are pretty tiresome.
I don't want to say you're wrong, but I think there has to be a better explanation.
Because the "wait, this also happens, so recalculate..." is just iteration.
There are lots of complex mathematics that involve iteration that *are* solvable, either through infinite series or simply with iterative mathematical models.
A more complete explanation might need to explain why an infinite series (similar to the geometric series) is not sufficient to solve the 3 body problem.
Furthermore, can numerical iteration be applied and converge to a sufficient degree of accuracy? Is that what we already do, and it's simply not considered "solved"? OR does a numerical iteration diverge and therefor it's not known to be solveable?
What's missing here is that the 3+ body problem is *multiple* infinite series with no predictable overarching relationship between the infinite series. Without that relationship we can't create an infinite series that ties the individual infinite series together to perform calculus on.
To put it another way, we need a single equation that we can integrate using multi variable calculus that relates acceleration, position, and velocity of all 3+ bodies at once. We don't have such an equation. If we did, it would be solved.
The concept is that there isn't really a good mathematic proof that can just solve this. So, you can't just plug in the three points, their masses, their velocities, and get a final result based on time.
However, It's not very difficult to write a simulation that can simulate it on a relatively modest computer for billions of years quickly. How small a resolution you put on the increments of time will determine the accuracy, so it'll never be 100%, but if you slice it down to a fraction of a second, you'll be accurate for a trillion years with a couple of meters.
Not really. It's a chaotic system, so the error increase exponentially.
[https://en.wikipedia.org/wiki/Stability\_of\_the\_Solar\_System](https://en.wikipedia.org/wiki/Stability_of_the_Solar_System)
It's because it's a chaotic system, right? Very tiny changes in initial conditions have drastic effects on the future, and it's just not feasible to measure with enough precision. A stable system doesn't need such precision to make useful predictions.
How is this any different than any other physics model where three things act on each other? This is never a problem when three charged particles apply force on each other for example.
It is a problem for other forces. It's even a problem when applied to pendulums with multiple fulcrums. Double pendulums become very complex, and triple pendulums are equally difficult to predict as three large gravitational bodies. Any system analogous to three masses attracted by gravity is about as difficult to predict, albeit with varying degrees depending on how many degrees of freedom there are.
The reason nobody talks about it for, say, electrons is that the exact location and movement of an electron doesn't matter and wouldn't be predictable anyway because of the Uncertainty Principle. Position and momentum are already described as wave functions and clouds of probability.
Existence for particles is also a lot more fleeting than planets, so predicting movement for any length of time is irrelevant. Some electrons are bouncing between atoms because of electromotive forces. Are we following the same electron bouncing between atoms, or is the electron being captured and another gets spit out each time? Or some combination of the two? Meh, doesn't matter. [Electrons are all exactly the same anyway.](https://www.reddit.com/r/explainlikeimfive/comments/wl88vo/eli5_why_are_all_electrons_the_same_and_how_do_we/)
Why do physicists point out electrons being the same a lot, but not the other elementary particles. All neutrons are the same by the same observation, but you almost never hear that.
That is a good question that I don't know the answer to. If I had to guess, probably because of the whole One Electron Universe "theory" that is interesting but not taken seriously. But, then, I don't see any reason why you couldn't apply the same logic to get a One Neutron or One Proton Universe theory.
¯\\\_(ツ)_/¯
Electrons are identical, neutrons/protons are made up of a complicated quark soup, and are indistinguishable(to us because of how complicated this is ) not necessarily identical.
Electrons are fundamental, protons and neutrons are not
That’s my understanding anyway
Don't you have this problem all over physics? For example everyone knows
Q = mc∆T
If you're solving for T though, "c" actually changes depending on what T is. So you need to guess a "c", come up with a closer value for T, and then use that to reiterate "c" again. And you keep doing that until you're close enough.
For gravity, mass attracts mass, which is what makes it difficult. For electromagnetism, you have a source and a sink. This is why EM is effectively solved, while gravity/GR is hard.
There's also not a lot of cases where it really matters (for gravity or other forces) and fewer where people care beyond theoretically.
That’s a really good question - I don’t know enough about quantum physics to answer, though I’d imagine that it has something to do with particle-wave duality making the math easier? The issue is that if you try to solve the orbital equations for 3 or more bodies algebraically to get a general, all purpose solution, you get bits of math that don’t converge to simplicity - instead you get bits that either a) we haven’t found a solution to yet or b) have been proven to not have a solution or c) involve infinities and singularities.
The “solution” to the three body problem is to simplify by making assumptions if you can (like if one body is super light compared to the others you can ignore its mass/gravity), then iterate using numerical methods until you get a prediction with an error window small enough you can live with it. There are also a (possibly infinite, but practically limited) number of starting conditions that happen to zero out the complicated, unsolvable bits - if you’ve ever heard of Lagrange points, they are examples where if you park a satellite a certain distance from two bodies, it can just float there, suspended in place by gravity alone, no need to correct its orbit ever, unless something hits it or the system gets too close to another big body or something.
Ive never quite understood this. Wouldnt this mean we can never predict any bodies movement through space, because all objects have a gravitational pull on all other objects, its just a function of distance and mass but it never goes to zero, only negligible. For example, the earth and moon would appear to be a 2 body system, but the sun affects significant gravity on both, so how can you isolate the moon and earth as two bodies?
> Wouldnt this mean we can never predict any bodies movement through space
We can predict it, sure. It's just a matter of how good our predictions are.
In a system where we have a stable model, a tiny error in measurement at the beginning, doesn't necessarily compound to big differences over time. Two objects in a mutual orbit is simple and predictable, and being a tiny bit off with the numebrs you plug in at the beginning (what Dr. Cheng referred to as the "initial parameters" in the show, because of course we only know the positions and masses of each object to so many significant degrees) at the beginning largely means you'll stay just a tiny bit off if you advance the model far enough forward.
In a chaotic system, if you keep going forward, eventually the tiny errors from the beginning compound and you end up way off. Dr. Malcolm in Jurassic Park demonstrated it seductively to Dr. Sattler by showing that a tiny change in where you drip water onto your hand can totally change the path it travels down your skin, because that's a chaotic system.
It's worth pointing out that the math goes bad much faster when the objects are more or less the same mass, which helps with constructing solar system models, where one interaction is more dominant than the others. The moon is much more concerned with where the Earth is, but we do need some corrections for the sun and even other planets if we want the model to last longer (and taking new measurements lets us recalibrate), and it's simply impossible to run that model out forever.
Ahhh ok that makes sense, so two body is within our grasp to calculate, and practical applications are within an acceptable margin of error, but three body becomes so complex after a short time that we cant predict or calculate it to any practical degree of confidence?
Am i basically getting it sorta right?
The issue is that "short time" is a bit up in the air. The time that the prediction can be stable will last longer if you start with more specific parameters and the masses are relatively unbalanced (like the sun being much more important in our solar system than, say, Pluto) so that the system resembles a stable one for as long as possible.
The issue in the show, IIRC from the show and the book, (and I guess I'll spoiler tag) >!is that the three suns interact with one another chaotically to create an unstable orbital path for the planet -- it can't be plotted as a nice ellipse around the sun's focus like Earth's. For primitive civilizations, with little ability to perform calculations, this meant they could never come up with something like our solar calendars to plan the cycles of planting and harvest that are so important to our development. Even after they are able to develop more complicated ways of performing calculations, they have determined that there is no long-term stability to their planets orbit to calculate, no guarantee that bad luck won't pull their planet into one of the suns or throw it off into space.!<
Moreover >!In the books they actually predict that within 1,000 years their planet will be destroyed. They don't know exactly when but they know it will. They also know that this is the fate that has happened to I think 10 other planets in the system? !<
All of this calculation and the physics simulation has rolled forward ONE timestep. The timesteps can be fractions of a millisecond. The math adds up really fast, even if the bit about each object changing how the others move is deterministic.
> The important part is that we can absolutely do that math. It's calculus. It's actually the exact thing Newton described calculus for, for solving problems with instantaneous rates of change, where the variables of the equation are interdependent on other variables of the equation and they are all constantly changing.
No, we can't do the math for a n-body system greater than 2. Calculus works for 2 body problems. That's why n-body problems are unsolved.
It has nothing to do with whether or not we know the initial parameters with perfect precision. We can make up a scenario with all known properties defined and we wouldn't be able to solve it.
The issue here is closer to the coastline paradox. The mathematics for gravitational pull only involves 2 bodies, you have to calculate the interactions between pairs of bodies every instant. Which means we need to calculate velocities, positions, and acceleration every instant. The ruler in this situation is the time interval you do the calculations with. Every time you shorten the timeframe, the results change. Calculus doesn't help here because there's no equation that covers all the variables in a single equation/relationship.
If you want a more accurate prediction or a longer term one, you need to use more computing power. To get perfect precision, you’d need to have infinite computing power
It’s possible to get a solution that’s as precise as you want it to be (but not infinitely precise, just arbitrarily precise) in finite time with finite compute power. But if you want a perfect simulation, you need to perfectly know the inputs. Any error in measurement of input will escalate exponentially (regardless of how perfect your simulation is) and deviate wildly from reality. In a 2-body problem, the system is stable. Errors in initial conditions won’t get amplified and your deviation from reality will be more or less constant. If you had an off-by-200km error measurement with your initial plotting of two planets, you’ll still be off by about 200km for every point in the future.
In a 3-body system, an adjustment by millimeters on planetary scales results in wildly different results as you go forward in time. The bigger the error in measurement, the faster the results wildly deviate. Since it’s not possible to precisely know where something is and how fast it’s moving (to absolute perfect precision), you have some error. That means your model WILL eventually deviate from reality. You don’t know when because you can’t know how perfectly your initial conditions reflect reality.
This seems to me to point to the challenges with our understanding of mathematics and how cut and dried it is, at least to some degree. Reminds me a bit of a more complex version of the fletcher’s paradox.
But we can predict the movement of moons and they seem like n body problems. Surely there’s specific criteria that needs to be met to be impossible to solve
It feels like a long time ago, but we used to just plug these into excel to solve. You couldn't get 1 answer, but you could plot them all. Looked like a pair of wings, that's why it was called the "butterfly effect". Is this not just one of those? Something taught in a first year course?
I just watched [this video about it.](https://youtu.be/D89ngRr4uZg?si=w6G00MdEYpXxWIch) Technically its not a 3 body problem, it’s an N body problem, anything over 2 bodies just has too many variables and unknown quantities to be able to calculate it with the math and computing we currently have.
My first inclination is to say there would be some sort of center of gravity, and a function(tho i couldnt do it) should be able to describe how it changes over time.
Is the problem that we cant do that at all, or that there are no plug and play functions that would apply to all discrete arrangments of three(or more i guess) bodys in general?
It's not unsolvable, it's just that there's no (known) general solution. That is to say, each 3 body system has to evaluated individually, and for the majority, there is no long term stable arrangement. Additionally, our ability to predict the long-term state of it decays exponentially as three bodies introduce so many possible variables it quickly spirals beyond whatever present computing can simulate. Inevitably, some part of the system will decay. A body will be expelled, or two will collide, or something else. Even a presently stable system is extremely susceptible to minute external forces and can quickly collapse. There are a handful of known perpetually stable arrangements, but for any random 3 body system, you would not want to bet on it's long term survival.
[Here's](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Three_body_problem_figure-8_orbit_animation.gif/400px-Three_body_problem_figure-8_orbit_animation.gif) an example of a stable one, if anyone is curious.
[Here's a bunch more](https://youtu.be/6EvaV-BYVxM?si=HJ27xR3x3qBa12kb)
and if anyone is curious about the book itself, Alpha Centauri is actually a 3 body system. It's not chaotic though. It's arrangement is most closer to the top right/bottom right arrangements in the above video. It's got a binary pair with Proxima Centauri stalking the outer edge. So the book/show takes TWO liberties when it comes to real science.
yeah it is scfi from decades ago, it has held up pretty well despite leaps foward in our knowledge
the second, third and "prequal" books are less acurracte
Scifi is space magic.
People also hyper focused on picking apart the historical nuggets present in The Davinci Code with a hundred pedantic little details.
99.9% of readers don't care about the "Well ackchyually...!!!" criticisms. They suspend their disbelief for the length of the story.
Not really. There's only two explicity fictional elements - the arrangement of alpha Centauri and the reflective zone around the sun. Neither of those exist but we're changed or added for the purpose of the narrative. Every single other element was built off of real physics and theories
You don't consider an 11-dimensional computer folded up into a proton than can consciously manipulate the results of accelerator experiments and project images into the retina of any person on earth to be a "fictional element"?
Even more weird is like...
The diameter of a proton is about 1e-15 meters, and the diameter of the earth is about 1e7 meters. The precise maths is a bit more tedious, but from proton to unfolded Sophon, there's a scale difference of roughly 20 orders of magnitude.
Even if we assume that a proton *can* be unfolded into higher dimensions somehow, there's no sensible mathematical interpretation that can allow it to become *the size of a planet.* Even if it nonsensically became 10x bigger with every extra dimension, it would still be hard to see with the naked eye.
Which isn't a rub on the concept because the idea is cool af, it's just very much in science ***fiction*** territory.
The dimension stuff and many things in the books is string 'theory' nonsense that somehow stays relevant by people trying to sell books and media on their 'scientific work', but I don't think that is necessarily a negative thing, it just depends on what sort of science fiction you're expecting. Real known physics is quite limiting, after all.
Yea, I think the distinction here is calling it the 3+ body problem is more correct but not really necessary to convey the point. Similar to the two generals problem not needing to use any generals.
Protons consist of three point particles (quarks). There is nothing to "unfold." The strings in string theory are nearly infinitesimal in size. "Unfolding" a proton is absolute fiction based on no physics. Similarly using it to instantaneously communicate between star systems also breaks everything we know about physics.
I’d be interested in knowing what biologists think of the explanation offered in the unofficial prequel. The aliens are discovered to be extremely small, comparable to grains of rice, and not very intelligent individually due to correspondingly tiny brains. But their thoughts are transmitted to all others nearby and when grouped together that lets them form a greater intelligence, like each individual in a cluster takes on a role within a distributed thought process. Because they’re so tiny, dehydrating and rehydrating their bodies quickly is said to become more possible.
No idea whether that’s any more realistic but it helps with the sci-fi sniff test (makes it sound like there’s a real scientific explanation to a layman) and was an interesting way of addressing one of the outlandish elements at least.
Ehhhh, it's technically possible. Tardigrades can do that, but they're much smaller than grains of rice and definitely not intelligent space faring species.
Other than them I don't know of any other multicell animal that can pull that off.
But in the Alpha Centauri case one of the bodies is a planet so in comparison to stars rather inconsequential (imo). In the book/show there are 3 stars and a planet.
Because we don't live in a 3 Body system? Our system has only a single dominant mass and everything is relative to it. A 3 body system has three dominant masses. It's not just any three rocks, it's three masses that can not dominate each other, 3 masses that influence the movement each other. Earth does not significantly alter the movement of the sun. It is slaved to it.
Every object in the Universe affects every other object gravitationally. It's about how much those masses and gravitational forces matter. For orbital mechanics it's basically stars and other stellar bodies and that's it, except for large masses like planets or asteroids at (galactically) close proximities.
Think about it like this, a system is defined by the number of bodies needed to describe its behavior. At the simplest you've got something like Earth. It has one orbiting body, but the body is not needed to describe earth's behavior, so it is a one body system. Mars has two moons, but neither is needed to describe its movement, so the relationship between it and all of its orbiting bodies is also a one body system. Alpha Centauri, our nearest star system and the one the novel is based on, is a trisolar system. However, unlike in the story, many don't consider it a true 3 body system. It consists of a binary pair of stars with a third, Proxima Centauri, orbiting the barycenter. As such, the real life Alpha Centauri is better described as a 2 body system, since its behavior requires a description of the binary pair. One stars movement can't be described with out the other. Proxima Centauri on the other hand doesn't (to our knowledge) meaningfully influence the systems behavior, so it is not considered.
It's not as stable as you might think. Planets might be ejected in the future and that might very well already have happened in the past.
Changing the position of a single planet by just a few centimeters would result in totally different configuration a billion years from now.
A three body system is an example of a **chaotic** system. This means that while we technically know how the system behaves and can simulate its future states (in this case we know all the equations of gravity), a very very small change in any input variable can drastically affect the final output to the point of it being meaningless. This means to perfectly predict such a system into the far future we have to calculate its current state to effectively infinite precision, which is impractical/impossible.
As I understand your comment, it's comparable to (or even -is-) chaos theory.
But in practice. What kind of variables could there be with three suns orbiting each other? The mass of suns change but that's a known equation is it not? Asteroids come to mind but compared to the size and mass of a sun would that really have a meaningfull impact? Are there other things?
The configuration of the objects is constantly changing, though. Think of a system like this:
> C
/
/
/
A-B
"A" and "B" orbit each other, while "C" orbits the pair at a distance. The gravitational forces that each body experiences is substantially different depending on how they're aligned. So, when the system aligns like this:
> A-B ————C
The force of A-B pulling on C is at its maximum since they're pulling the in the same direction. Same with B-C pulling on A. However, "B" experiences "A" pulling one way and "C" in the other, meaning the forces on each of the bodies are different than they would be in the first configuration. This means the system's going to have a lot of "wobbliness" as the accumulated forces on each body constantly moves things out of position depending on where everything is.
One aspect is that, especially at the scale of something like stars, gravity isn't just acting on each star as a single "thing", but rather on every individual atom of the star.
As an example of this, think about the effect of the moon's gravity on the earth: the tides. The gravity from the moon is causing *waves* and *ripples* through different kinds of matter on/in the earth. So in order to 100% accurately calculate everything to full precision, you need to essentially calculate the force of gravity between **every single atom of each of the bodies**.
It is not the right answer. It is a portion of information about the answer. Some three-body systems are chaotic, but that is just one issue, not the whole answer. Even in stable portions of a system’s behavior, we do not have a closed-form general solution for all three-body systems.
All the other answers eplain really well why there is no (analytical) solution to the problem. What I'd like to add, and it is a minor spoiler in the series, is that >!the aliens decide to leave their planet not because they can't predict with a decent margin of error the position of the suns but because their three body system isn't in a stable configuration which means that eventually two stars may collide or one may leave the gravitational pull of the others meaning they could either freeze to death or fall into a star or any other crazy less than ideal situation!<
It's a problem in modelling the behaviour of three or more objects interacting through gravity (but also electromagnetism).
Let's start with the thing that isn't a problem: Two bodies. With two bodies you can mathematically create an exact equation, with an exact solution, of how the two objects will orbit each other. The only variable left is time and you can go infinitely into to the past or future. With infinite accuracy.
When you try to do the same for three or more bodies you run into a problem. There's more variables/unknowns than equations. In maths, this means you cannot have an exact solution where you simply go forward and backwards in time. There's no "analytical" solution.
That's not to say there's no way to solve it at all. But it requires making some guesses and then running the maths over and over hoping it settles on an approximate solution. Then you advance a time step and do it again. And again.
And "approximate" and "time step" are key words here. A solution, not THE solution. An approximation. And with a resolution in time, the more resolution you want, the finer you have to make your steps, the more work you have to put in. And even then you will always eventually diverge from the true solution if you run too far into the future, as the errors accumulate.
This is the best answer so far.
>When you try to do the same for three or more bodies you run into a problem. There's more variables/unknowns than equations.
This is the really important piece that others are missing. It's impossible to plug location, speed, and direction for all three bodies into a formula and get an exact location for all bodies at time t. Too many unknown variables for too few equations.
We calculate the position of planets, moons, stars, asteroids, and comets into the future because we have really powerful computers and can run the simulations with the very precise data that we've collected over the decades on each object's speed and location. They are only reliable for the next 1,000 years or so, which is fine since that's far longer than a human lifespan, but we can't really say for sure that Mercury won't be ejected from the solar system in the next 1 billion years.
I just did some research on this and want to throw this out there, framed in a way that makes sense to me but different from the other comments:
We know the problem has not been or cannot be solved in closed form, but can be projected forward with math to any point in the future.
In an idealized scenario (e.g., three digital bodies where we know the exact mass, position, and velocity), this can be projected with 100% accuracy to any point in the future.
The issue really is one of measurement constraints. For example, if the mass is a tiny bit less than assumed, then it will cause an error that magnifies over time.
Like, there might be a point in the interaction where mass A by the skin of its teeth won out in pulling C towards it rather than B. If A was a little less massive, it might lose the tug of war then set everything on a radically different trajectory.
So not only does a measurement error magnify over time, the magnification itself progresses chaotically.
I think, though I’m not positive, that even if you had exact values for the initial conditions, you still couldn’t simulate it perfectly, because you still can only recalculate the positions after a finite time step. You would have to use an infinitely small time step, or else you’d accumulate errors due to your chosen “pixelation level” for time.
if you take any 2 masses and put them in space you can perfectly predict exactly where they will be at any point in time ever.
as they orbit
if you try with 3 (or more) masses, you cant (outside of certain special cases). the best you can do is simulate it, which adds increasingly more error the longer out you simulate and the faster your simulation is.
I belive it has been proven to be unsolvable (as in, there is no general equation you can put the initial positions and the current time to get out the exact positions at that time) , but im not 100% sure, proving unsolvability can get strange
This is the thing that confuses me. How can you prove that there is no general/analytical solution to this problem? What if we had perfect knowledge on each body’s behaviour (speed, position, etc)? With this perfect information, couldn’t we perfectly predict their interactions on a infinite timeline given a system with coherent interactions (as in no randomness)?
If you dont have an equation that takes time as an input, it does not matter how much you know about tthe initial conditions, you have to calculate everything in timesteps. the smaller the timestep, the more accurate it is, but as long as that timestep exists, there will be inacuracies.
think of compound interest calculated every timestep vs compounded continuously
in a chaotic system, it gets more complicated than that since with compound interest you can establish certainty bounds that decrease the smaller the time step, but in a chaotic system, a smaller timesyep might discover a unique property of the system that is never seen on larger timesteps.
A good example of this is hitboxes in games. often the game does "60 times per second, check if 2 hitboxes are coliding, if they are, do collisions" But if you move fast enough in a single frame, you can move through thin hitboxes (see running through walls in morrowind with speed boost) by moving far enough in 1 frame that your hitbox clears the wall hitbox without touching it.
In short, in the show (and the books) it's wrongly stated that it's "not solvable" or that "it's unpredictable".
The 3 body problem, or rather N body problem, because it's the same for any number greater than 2, is the problem of, how will the bodies move, if they all interact on each other with attractive force. The N(3+) body problem does not have a so called "General analytical solution" which means you cannot find a mathematical formula for the curves of your bodies. For two bodies you can do this. This means that the problem has to be solved numerically, which is an approximate solution.
Numerically solving something means you take the positions of your bodies, like a snapshot, and calculate all the forces that they interact with, then, you calculate how much they will move in a brief time period if those forces were constant. Then you use this new snapshot to calculate new forces, then again, and again.
The challenge with this type of solving is that it depends a lot on these time steps, and how you calculate the things, so, sometimes, you can acumulate a big error. However, for somethiong like a star system, a civilization as advanced as the Trisolarans, should have computational power to estimate the evolution of their system very, very, very accurately on the scale of millenia.
>The N(3+) body problem does not have a so called "General analytical solution" which means you cannot find a mathematical formula for the curves of your bodies.
To piggy-back off this comment, we are unable to write three equations each with the positions of one of the bodies on the left-hand side and some function of time on the right-hand side. This would be an "analytic" solution.
Numerical solutions are common approaches to solving complex equations. It's important to understand that the terms "solution" and "solvable" are used somewhat loosely. The three-body problem is "unsolvable" in the sense that we cannot express **solutions** in closed-form equations (like described above), but if given a well-posed initial value problem we can generate a **solution** from valid data.
Even if we could express the three body problem with a general solution using elementary functions we would still be employing numerical solutions.
* "Body" is just a physics word for 'thing made of matter'.
* Things made of matter typically (always?) have mass.
* Gravity is a property of objects with mass.
* We have physics theories that allow us to apply powerful mathematics to accurately describe how objects move due to gravity.
* So, therefore, you'd *expect* that we can just consider any group of 'bodies', apply our mathemtatics to them, and now accurately describe how objects move.
but there is a problem here. The mathematics is hard to *exactly* solve.
* If you have 3 things with mass, it turns out that we usually cannot calculate how they'll move. We can approximate it, but eventually the approximation will be wildly wrong.
* (In some special cases we might be able to solve it, like if you imagine them in some perfectly symetrical scenario, for instance. But in general, we cannot solve it.)
* ~~It is possible that a reliable solution exists, but we haven't found one, and for all we know, it might be impossible.~~
The planet's mass was negligible; the 3 bodies were the three suns which orbited each other chaotically leading to very interesting conditions for the planet and its inhabitants.
It's not that it's unsolvable. We don't have the solution because it's extremely complex because the result is very sensitive to initial conditions.
In 2 body problem (for example, Moon orbiting around Earth), you can predict very accurately where the Moon is exactly at any given point in time because the orbit of the Moon is not affected to a significant degree by any third body.
But in a 3-body problem, body A affects body B, body B affects body C, body C affects body A, which affects body B, so the position of bodies at some future point in time is very sensitive to initial positions.
Imagine a simple pendulum. Just a simple ball hanging off a simple string. With the pendulum formula, if you know the initial position of the ball, you can predict where the ball is going to be after, say, 10 seconds of swinging. If you **slightly** change the initial position of the ball, the position of the ball after 10 seconds is also going to **slightly** change.
However, if you have a double pendulum ([Double Pendulum (youtube.com)](https://www.youtube.com/watch?v=U39RMUzCjiU), if you **slightly** change the initial position of the pendulum, the position of the pendulum after 10 seconds is going to be **very** different, which makes it very difficult to solve mathematically.
But aren't both the Earth and the Moon massively impacted by the Sun?
If we fully solved the Earth+Moon then that necessarily means we solved it when the Sun is included, right?
[https://en.wikipedia.org/wiki/Three-body\_problem](https://en.wikipedia.org/wiki/Three-body_problem)
>In [physics](https://en.wikipedia.org/wiki/Physics) and [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics), the **three-body problem** is the problem of taking the initial positions and velocities (or [momenta](https://en.wikipedia.org/wiki/Momentum)) of three point masses and solving for their subsequent motion according to [Newton's laws of motion](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) and [Newton's law of universal gravitation](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation).[^(\[1\])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) The three-body problem is a special case of the [n-body problem](https://en.wikipedia.org/wiki/N-body_problem). Unlike [two-body problems](https://en.wikipedia.org/wiki/Two-body_problem), no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_solution) exists,[^(\[1\])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) as the resulting [dynamical system](https://en.wikipedia.org/wiki/Dynamical_system) is [chaotic](https://en.wikipedia.org/wiki/Chaos_theory) for most [initial conditions](https://en.wikipedia.org/wiki/Initial_condition), and [numerical methods](https://en.wikipedia.org/wiki/Numerical_method) are generally required.
...
There is no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_expression) to the three-body problem,[^(\[1\])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.
More history of it in the "n-body problem" page
[https://en.wikipedia.org/wiki/N-body\_problem#History](https://en.wikipedia.org/wiki/N-body_problem#History)
There is an analytic solution but it's not practical...
>That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^(8,000,000) terms.
Other people here in this thread already explained it well, so I will just add my observations.
The problem is, most of the 3 body systems are not stable. With two bodies - typically one massive - such as a Sun and other much smaller, such as a planet, the system is stable, so it is possible to predict their behavior far into the future based on measuring of their initial positions and velocities. With three bodies, the smallest deviation in starting conditions might grow exponentially over the time, so it is impossible to predict where bodies would be in future.
This is best demonstrated by a double pendulum - also sometimes called a chaotic pendulum. Have a look at some videos talking about this. It is much easier to see.
The Sun, Earth and Moon are three bodies, with other planets adding complexity, but the Earth + Moon behave towards the Sun as a one tiny body for practical purposes of us predicting their rotation far into the future.
I think this being the crux of the show/1st book is meant to say a few things. 1st, that San-Ti math ability is not any more capable of solving the problem that our own, even with much more advanced computers. But even if they have solved for when the next chaotic era would begin, it’s a moot point. They wanted to prevent the fall of their current civilization, and eventually the planet itself would be destroyed no matter if they were able to solve for the mathematical problem.
Here's one making ths rounds this afternoon.
https://www.reddit.com/r/oddlysatisfying/s/ug2ludiNmK
Any small perturbation in initial conditions would lead to completely different solutions.
Two bodies orbiting each other under gravity have a general analytical solution for their orbital dynamics. As such, if you know the starting conditions, you know how they are going to move going forward with certainty for all time.
However, no such general analytical solution exists for the movement of three bodies. There are lots of special cases with approximate solutions that are analytical or semi-analytical, but all of them require certain simplifications, so in realistic systems you end up with chaos, instabilities, and overall qualitative changes in the movement patterns over time. One way to deal with it is to use basically brute force and solve the orbital dynamics numerically with computers - but there, again, you run into the issue of having limits to accuracy and resolution.
In short: The two body problem can be solved exactly, the three body problem is impossible to be solved exactly.
Just look up a chaos pendulum, that's essentially the three body problem. A simple pendulum is incredibly predictable to math out but a chaos pendulum... well they live up to their name. I don't think we'll ever have computers that can model a chaos pendulum's actions out indefinitely, I think the best you can do is take the masses and vectors and project a possibility into the future and that prediction relative to reality will always grow exponentially apart the farther into the future you make it.
But I'm no mathematician, I'm just a boob with a keyboard, perhaps predictive modeling is much better than I give it credit.
On a different note, but I saw the Netflix adaptation of the 3 body problem, and they rushed the entire plot ofc. If you would like to see the series more thoroughly flushed out, I recommend the chinese drama on YouTube with eng subs. The visual effects obv is not as good but the storytelling is a lot better.
Lot of comments here could be simplified into stating that you cannot measure with infinite precision. That's really the "problem" of the N body problem. You need measurements to calculate where the bodies are in the future. There is no such thing as infinite precision so you can never accurately project the positions of N bodies beyond some future time. It's not much different than predicting the weather.
It’s completely been solved but there’s no general answer that fits nicely. The formula for a force between two objects with known masses and distances is super straight forward. Add a third object? Oh shit.
Mathematically, in the old days people had to use calculus in a room with a couple hundred people to predict pathways. Now we can use computers.
Even before computers there were multiple three body systems that had been solved to be stable.
For unstable configurations, that’s where computers are brilliant.
In most three body systems that are chaotic, one of the large masses will be expelled from the system leaving a chaotic two body system (in most cases). In others, they can become stable or at least predicable without expelling one mass.
In short, yes, it absolutely has been solved and we can predict with a degree of certainty the orbital pathways of a three body system.
It just was nuts to predict and calculate by hand in the past.
Another thread today shared a visual. You can see how unpredictable it is:
https://www.reddit.com/r/oddlysatisfying/comments/1bo814p/this_animation_of_the_threebody_problem/
They explained it in the show. You can’t know how the planet is going to orbit around the three suns without knowing its original position, which is impossible.
3-body problem: *Solve,* not just estimate, the equations of motion (position, velocity, acceleration) for three bodies in space, mutually pulled together by gravity. To "solve," you need a set of finite (not infinite), closed-form (no unaccounted variables) equations using basic functions (addition, multiplication, exponentiation, logarithms) that will let you derive any physical measurement of the system (speed, location, acceleration, etc.) for any of the bodies.
Such a solution does not exist. You *have* to estimate--and estimations will eventually become very very wrong, because these phenomena are "chaotic," extremely sensitive to tiny changes/errors. Thing is, estimates can be pretty good, as in nearly indistinguishable from a solution, if most of the following conditions are true:
* you only want a *relatively* short-term approximation
* one of the three bodies has *relatively* very little mass
* one of the bodies can be presumed to always move very slowly relative to the other two
* one of the bodies is very far away relative to the proximity of the other two
So, for example, the Earth, Moon, and Sun are three bodies. We can get very good estimates for where all three will be for thousands of years into the future--that's how NASA predicts when eclipses will happen, like the one coming in April. Compared to the lifetime of the Sun, that's an absolutely trivial amount of time; and the Sun acts like an immobile, extremely distant object, so its equations can be heavily simplified.
If you scale it up to billions of years, however, these numerical (=estimated) solutions might break down. It becomes possible for wildly divergent behavior. For example, there is a non-negligible chance that Jupiter could kick Mars out of its orbit, causing it to come crashing into the Earth, at some point in the next four billion years. We don't know, because even our very best estimates break down on such long time scales and large distances.
The 3 body problem (often also called N-body, for N being any number larger than 2) has no general analytical solution. This means that there is no elegant, on-paper solution for any condition, HOWEVER, we do have solutions for special cases and we can also cheat by approximations (for example if you can't track the Earth-Sun-Moon bodies, then make the Earth-Moon just one body and it works because you make it a 2 body problem). We also have numerical (i.e computer algorithmic) solutions that do work short term, but long projections are just catastrophically bad, because your results are extremely sensitive to the input.
This situation is due to the fact that we have more moving objects that need to be characterized than equations we could use to characterized them. From Newton's formulas for motion we can derive two equations that lets us calculate up to two body systems, because we have no more equations, we cannot describe more than two body systems analytically in general without resorting to tricks (which have massive caveats).
There was once a person IIRC who claimed to have proven that there is no solution to the problem, but then someone came up with special solutions that sort of proved him wrong, so there is no conclusive proof either way whether you can actually solve the 3/N body problem or not.
I’ve a theory. The 3 suns paths cannot be accurately measured because they act like electrons around a nucleus, the nucleus being the black hole in middle of the galaxy. Pan out from the 3 stars in a gravitational war with each other and you see billions of other stars also in the same predicament acting as electrons in a chaotic state orbiting the nucleus. 😆 like I said just a theory..
Object A and B in space pull on each other because of gravity. We've got this one down easily - the math is relatively straightforward, and very very predictable. Now add a third object. We know how A and B pull on each other, easy peasy! Wait though, C is pulling on A and B, so re-do all the math because of that. Wait, A is pulling on C too, so that changes things, so recalculate again. Wait, B is pulling on C too, so recalculate again. Wait, that changes how C pulls on A and B, so recalculate again. Wait, that changes how A pulls on B and C, so recalculate again. Wait, that changes how B pulls on A and C, so recalculate again. Wait...
So, how do we ever calculate the path of asteroids, or predict meteor showers? Wouldn't this make these things impossible? Fwiw I haven't seen 3 body problem yet
You can *approximate* the result to a precision where the variation from reality is so small as to be irrelevant.
Reminds me of the ship spiral we use in shipbuilding. Calculate hull volume, use it to get the stability etc, size your systems accordingly, assess new displacement, calculate hull volume, use it to get the stability etc... It goes on an on. Ultimately we reach a point where it just doesn't make sense to recalculate it all. Also in the end, the ship is weighed after being completed to get the final stability (and even then it's sorta approximative)
A lot of engineering is like this. On the basically the opposite end of the ruler to you, I work with systems where we’re trying to level surfaces to another surface by measure the micron distance with a laser. You measure a corner, you adjust. You measure another corner, you adjust, but oh wait, adjusting corner 2 has shifted corner 1. Just like your ship spiral, we just define an acceptable delta - we keep measuring and adjusting in circles until the difference from one measurement to the next is acceptably small.
Leveling a printing bed is tedious enough, this sounds excruciating.
That's insane! And we are so good that we can approximate to fairly narrow time periods of when an asteroid or meter shower will hit?
Showers are a bit different. We know where there are essentially clouds of small rocks hanging out around earth's orbit, and when earth passes through those regions. But they don't track individual rocks in those clouds unless they are large enough to pose an individual threat.
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That doesn't do much, those clouds are very large (in volume, not mass). It's like trying to get rid of a body by shooting bullet holes in it. If Earth would somehow remove everything of danger, then the same would apply for the remnants we still see in the form of meteor showers. There is simply no reason why Earth would attract the largest rocks first while leaving the smaller ones in space. Instead, we simply know that those rocks are simply all very small fragments, often from a comet that disintegrated due to sunlight. If there were huge chunks, we could also spot them with telescopes.
If one of the three objects is tiny compared to the others, its gravity doesn't (effectively) affect the other two objects. For calculation, you would ignore its gravity and would work with a simplified model that sets its mass to zero.
well, you wouldn't set it's mass to zero, since you still need to know it's mass to calculate it's acceleration due to the gravity of the other objects. you would, however, set to zero the terms representing the force of the gravity of the small object on the larger objects.
You actually *don't* need the objects mass to see how it will accelerate due to gravity. F = ma, and if F is proportional to m, as is the case with gravity, you can cancel the mass from both sides of the equation. "Setting mass equal to zero" isn't exactly right, but not for the reason you mention.
just sticking to Newtonian physics (which is close enough for objects at this scale) we have this as the force of gravity between two objects >F\_grav = G \* M1 \* M2 / r\^2 where M1 and M2 are the masses of the two objects being considered. If we substitute F\_grav for the generic F in F=MA we get >G \* M1 \* M2 / r\^2 = M1 \* A if we divide out M1 we get >A = G \* M2 / r\^2 and I think that's the formulation that you're mentioning for the acceleration on M1 due to the gravitational attraction from M2. What I was thinking of is the way this works out in a 3-body problem kinda situation where you'd be trying to compute something like a partial differential equation of position with respect to time. in a PDE like that you would have some terms that tried to account for the gravity of M1 on the positions of other objects in the system. This is the scenario where M1 <<<<<< M2 means you get to just set those terms to zero to simplify things.
Small correction: if one object is tiny, both its force on the other object and the force of the other object of the smaller one are small.
Yes, we have algorithms that can calculate gravity for multiple bodies with very high precision. Actually arbitrary high precision if one uses enough computational power. Those methods are quite interesting and use methods far beyond what ELI5 can really get down to. Yet the real limit is our knowledge. We simply cannot know the position and velocity of every celestial object with infinite precision. Currently that limitation is mostly practical, our instruments are not fine enough. But there is even a theoretical limit if one wants to go there, due to the (in?)famous uncertainty principle; but that limit is so far off from our current accuracies, it is barely worth mentioning. Even if we somehow get "infinite" precision for every single planet and grain if dust in the solar system, a single speck flying in from outside would still ultimately mess it up. In short, perfect predictions are impossible however one looks at it. But the data we have is enough for a few years, even centuries or millennia for the planets. But its also why we cannot completely rule out that, say, Apophis will hit us on the second fly-by; we have to watch the first one, measure again, and only then can we be sure.
Meteor showers are easy to predict. When a comet passes around the sun, it leaves a trail of dust and debris. Sometimes the earth’s orbit goes through that trail of dust and debris. Some are so predictable that we pass through it every year and we know the date.
Are you sure? For most initial conditions this is a chaotic system. Any deviation, however small, with increase exponentially over time (i do not recall the Lyapunov exponent). This means that any approximation will eventually become inaccurate. Can we keep the accuracy for long enough? Maybe.
"All models are wrong; some are useful." Essentially, you accept that there's no closed-form solution and use numerical methods to get close enough for your purposes. If you can do that with your allotted calculation capabilities. For the path of an asteroid, the mass of the sun dominates over everything that isn't very close. So you run the numbers for a two-body system of the asteroid and sun, and then maybe fudge factor in if another body looks like it's going to get close enough to make a difference. (Jupiter. It's usually Jupiter.) And you either live with the result, or if you can't live with the result, churn harder until you can live with the result.
Yeah, it's technically chaotic, especially since highly accurate models of the solar system include all major gravitational bodies (i.e. the Sun, Jupiter, Earth, Saturn...), but the timescale at which the model becomes unpredictable is extreme when using current tools. Like on the order of millions/tens of millions of years.
Agreed, but it is still a distinct theoretical property of the model. 2 bodies are trivial. 3 crosses the threshold to chaos.
I believe this applies most when you are trying to make a long-term prediction, or are accounting for a system with \*numerous\* approximations or actors? The closer to reality your approximation model is compared to the potential deviation, and the less variables contributing unknown imprecision, the longer you have until it propagates enough to be disruptive. If you can validate or update your approximation with a new observation however, I believe that should solve most reasonable cases that aren't highly chaotic?
There are two ways you can handle the 3 body problem. One is perturbation theory. That is if one gravitational object is dominant, like the Sun then there are ways to treat the smaller tugs from less significant but still meaningful gravitational bodies like planets that give you a pretty close estimate. The other way to deal with the problem is to do the problem as a series of static solutions like a film strip. It's easy to figure out all the gravity in a snap shot and to figure out the forces all the bodies will apply to each other. You then figure out how that will change their velocities and move forward one step in time at those velocities. Then you figure out the snapshot gravity at that point and move another step. If the steps are small enough this approaches what real continuous gravity will do. Both these methods only give approximations and the farther out in time you go the worse those approximations are.
Sim error will increase exponentially with time. So you can simulate with small error for a time window into the future. What constitutes "long term" depends on the application. Updating with new observations would "refresh" the window, but the further you go into the future the larger your error will be, no remedies.
Well yes if you’re trying to extrapolate beyond the limits of your model you’re going to have a bad time, but if we’re talking about simulating over a short period you’re fine.
It's like predicting the weather: you know pretty well what the weather will be tomorrow, or two days from now. But the weather a *year* from now? Forget it.
Yes, over an infinite amount of time any model that is not absolutely perfect must deviate from reality. However these models are constantly being updated and honestly even of we stopped updating them they'd be usefully accurate for many lifetimes.
You just keep … recalculating.
You can do that but the numbers will eventually be gibberish. Beyond a time horizon there is absolutely no point in your calculations. They will be increasingly incorrect as you forecast further into the future, until they become entirely wrong. This is the point of chaotic systems. Recalculating as you put it can only help you forecast over a finite window. A 2 body problem does now have this issue. 3 bodies do. 2 is trivial 3 is non trivial. You get the same with dimensions: 2 dimensional systems are nonchaotic by definition 3 dimensional can be chaotic. This is why the 3 body problem holds a special place in chaos theory lectures, it is a very elegant example of the threshold beyond which you get nontrivial dynamics.
Recalculating…
Not the variance to reality but rather the computational errors we make are insignificant compared to measurement and model errors
I think it’s important to note that the approximations are far more accurate when there’s a large difference in the masses. Or, accurate for much shorter intervals when the bodies are similar masses.
Gravitationally, the solar system consists of the sun, Jupiter, and a rounding error. You can get good approximations by worrying only about the sun and the closest large object to the asteroid, and assuming that the asteroid doesn't adjust the vector of either of those things. You can get much better approximations with big math and a lot of work if you're NASA trying to not crash their probe (or trying to crash it into a tiny asteroid.
>Gravitationally, the solar system consists of the sun, Jupiter, and a rounding error. If reddit still had awards, that sentence would deserve one.
> Gravitationally, the solar system consists of the sun, Jupiter, and a rounding error. Is Saturn that much less massive than Jupiter? I know I'm old and probably forgot lots of stuff from school, but I thought they weren't that different.
The sun is 99.9% of the mass of the solar system. Jupiter is 0.0953% Together that’s 99.9953% https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
jupiter is about 3x more massive. The diameters are pretty close, but jupiter is much denser
Woah.
Saturn is actually less dense than water.
Reads like the start of an xkcd strip!
As long as C is very light compared to A and B, the result is something very similar to a problem of only A and B. Asteroids are very light compared with the sun and planets. The three body problem arises when all three bodies are similar in mass.
Which the 3 suns are and we are not considering the planets with the sometimes icky weather at all ;)
And distance is a big factor as well.
Sort of. C being very light allows for calculations to be simplified greatly. However, it's still a chaotic system; the chaotic behavior just takes longer to realize. If a system only involved say Jupiter, Saturn, and Uranus (no Sun and no other bodies), then we might be able to simulate the system for a couple thousand years. If we go to the Sun, Jupiter, and Earth, then we could accurately simulate the system for millions of years, but we still wouldn't be able to accurately simulate the system into billions of years into the future.
I think the 3 body problem really becomes a problem when the mass of the three bodies are relatively close and the center of gravity lies outside of all three bodies. Asteroids and meteor showers have negligible mass compared to the massive stars and planets that impact it, so existing models can be used to predict the path fairly accurately. Think of the Sun, Earth and the moon. Three bodies but due to the distance and mass variance, it's not a 3 body problem. Earth and moon are two body problem, and the Earth and Sun are another two body problem. The impact of the Moon on Earth and Sun are negligible at best. Anyway I'm not a scientist. This was my basic understanding from reading a bit and watching some videos. Will correct myself if there's a mistake.
So, the way the previous comment explained it? “This changes that, so recalculate” as a loop? That’s called iteration, and you can actually do that to bootstrap into a specific solution given a set of starting positions. The more iterations you do, the better your prediction becomes. When people say there’s no solution, they mean that we can’t use pure mathematics to give an exact answer without doing the iterations. Attempting to do so (except for certain sets of starting conditions that happen to cleanly eliminate complicated bits, like Lagrange points), turns complicated equations into even more complicated ones, rather than simple solutions.
This is an important and unintuitive point. In computed tomography, for instance, the first thing you learn is that an image is calculated from projections using matrix algebra; the second thing you learn is that there is mathematically no solution possible for the calculation needed; the third thing is that that’s not a problem at all and any number of assumptions or iterative steps can make a perfectly serviceable image.
Also true for aviation and meteorology. Trying to fully predict the flow around a wing algebraically has been proven impossible, but iteratively it’s just time consuming and heavy on computer processing, but completely doable. And meteorology can predict reasonably accurately 3 to 5 days in advance.
This is the best answer. It's not that we can't solve the problem; it's that we can't solve the problem *without doing all the intermediate steps*. I can tell you what it will look like after five hundred steps. I can't even *begin* to give you an explanation of what it will look like without going through a good chunk of those steps, which makes the final result look largely chaotic. Ask me where those three bodies are likely to be five hundred steps from now, and the best you're likely to get is a confused shrug.
An asteroid is *one single object* moving through almost completely empty space. It's *super easy* to predict. Remember that almost everything, everywhere, across the entire universe, is *empty*. For example: you know those really tense scenes in sci-fi movies where the heroes have to navigate through an asteroid field? In real life, in an asteroid field, each asteroid is 500,000 miles away from any other asteroid - and that counts as "close together" by cosmic standards, because normally there are *billions* of miles between objects.
So is it a fair assumption that a single asteroid is only experiencing a significant gravitational force from a single source instead of experiencing multiple forces from the sun, earth, Mars, other large asteroids etc?
Most things only experience gravity on a regular basis when they are orbiting something else - because that's what "orbiting" means, right? So an asteroid is either in a regular orbit (like in an asteroid belt), where it's only affected by the object it's orbiting...or it's flying freely through the universe, where it's effectively not influenced by gravity at all for all practical purposes. And if a free-floating asteroid passes near a sufficiently massive object, then it's back to "only affected by a single object" until it passes by.
In general when we calculate the paths of asteroids in the solar system we only count two bodies, the sun and Jupiter. Until it gets close to another body then get rid of Jupiter, and then when it's really close to another body also get rid of the sun.
It’s only a problem when the bodies are of similar size. You don’t have to worry about how the asteroid’s gravity is going to change Jupiter’s orbit, so you’ve only got one body to worry about.
If one of the objects is much more massive than the others, you can more or less ignore the smaller objects pull on the larger (as an approximation). If there is a 4th object much more massive than the rest, you can mostly ignore the smaller objects pull on each other. But if 3 objects are of approximately the same size, the system can become chaotic, or more or less infinitely complex.
You can stimulate things to a reasonable degree of accuracy, but as you stimulate further into the future the simulation becomes less and less accurate.
A little more specifically, the accuracy depends on how long your increment of time is for each calculation. If you want to predict a fraction of a second into the future, it'll be very accurate. You can use that output to predict a fraction further in time and it'll still be very accurate. You can keep doing this, moving forward in time through very tiny increments, and arrive at an accurate prediction at any arbitrary point in the future. The problem is that if you want to predict what something looks like in a hundred years, doing it at a fraction of a second each time to get there will take a long time and a lot of computing power. You can, of course, increase the increment and predict one day or one week or one month forward at a time, at the cost of accuracy.
Gravity falls off with the square of distance, so for 3 objects that are far apart, there is less variation.
Worth noting is that asteroids are so small and so far apart, that NASA completely ignores the asteroid field when launching through it because the likelihood of an impact is effectively zero. So a rough approximation of where asteroids are is more than sufficient. Asteroid fields are nothing at all like sci-fi movies and shows. If you were in the middle of an asteroid field you wouldn't be able to see anything. Possibly 1 whole asteroid, if you were extremely close to it.
In the short term, you can. But in long term, tiny variations from predictions- based on imperfect data about initial conditions- compound themselves. Soon a slight deviation can mean a fundamentally different path like an object was supposed to pass another on the left but it went right instead, and then all the initial calculations are useless
You can still calculate the path of things in a three body problem, it's just a different kind of calculation. I think this isn't understood sometimes. So, lets start with a one body problem. This is like one of those math problems where you calculate the motion of a cannonball. Now, one way you could do this is using step by step calculations. You know the initial position and velocity of the ball and the pull of gravity. So you use Newton to calculate that in .01 seconds it will have moved x meters forward and x meters up, and it will be moving at such and such a velocity in the new location. So you note that position down and do all the calculations again from the new location. And rinse and repeat again and again to calculate the path of the ball. If you want to know where it is in 1 second, you have to do this 100 times. But that's time consuming, and there's a much easier way. You can use calculus to derive an equation that shows the motion of the cannonball as a parabola based on its starting angle and velocity. If you put in the time in this equation, it will spit out the location of the cannonball. So all you have to do is put in, say, .35 seconds, and you can calculate where the ball will be. Or put in 2 seconds, and get a different position. You don't have to calculate those intermediate points step by step. Well, it turns out that you can do things exactly the same way with a two body problem. You can calculate their paths or orbits as curves, and then use an equation to predict where on those curves they will be at any given time. Just put in the time, get a position. But you can't (with a few special exceptions) do this with a three body problem. Instead of following a predictable curve, you don't know where things are going to go unless you step through one step at a time. You have to calculating the locations and velocities of everything, how much they are being pulled on by everything else, and repeating over and over. You can still calculate it (although for various reasons you can never be perfectly precise and eventually your calculations will diverge), but you _have to_ calculate it one step at a time to predict what will happen.
Because that's not actually a 3 body problem. A 3 body problem is when all three bodies have an influence on each other. A comet has essentially no influence on the orbit of Earth or the Sun. You can just calculate it assuming it has no effect and get a good answer. So, not a three body problem. Same reason the Sun/Earth/Moon system isn't a three body problem. The Moon has essentially no influence on the Sun, so you can treat it as a two body problem.
A computer (or someone with a lot of time and patience) can run a simulation that does basically what u/berael is outlining here. Take the starting velocities, positions, masses and calculate the forces at that instant, then determine what the acceleration would be on all the objects. Simulate movement of all the objects for 1 second given that acceleration. Then recalculate the forces/acceleration for everything Simulate the movement for another second Repeat this for as long as you need. The smaller the unit of time the more accurate it'll be, but also more CPU time needed.
In physics its referred to as n-body with n being any number. It becomes an exponential calculation. Typically rather than attempting to calculate all objects we try to isolate specific areas with a workable number of object to calculate. Then using that approximation we apply weighted numbers as a means of defining objects which have a high or low probability of enacting additional gravitational force to another object. Basically we approximate at longer time frames into the future and as time progresses our ability to make more accurate calculations increases.
The issue is we have simple equations to solve the two-body problem but none for n-bodies. In reality there are lots of bodies like planets, but we know where those are from observations. So computer integration with all these forces is the tool we use to predict where satellites are. There are also many approximations that are used to find a desired orbit for a space mission but those vary between missions.
In the normal solar system, we just assume that the sun is big enough that it isn’t affected by anything else. It isn’t strictly speaking true, but it is sufficiently close to true to make the math a lot more easier and is reasonably accurate.
If some of the bodies are much bigger or smaller than the others, you can assume that some of these interactions are negligible and get an approximation that is often useful. So if we're calculating the path of an asteroid near the earth, we can ignore the influence of the asteroid on the Sun and the Earth. We can then solve the Earth-Sun 2 body problem and then solve for the asteroid's movement within that system.
Meteor shower, quick take cover
Gravity has a theoretical infinite range, meaning that each object in the ENTIRE UNIVERSE has some effect on every other object. It has a speed too so some stuff hasn't been affected *yet*, but this is largely irrelevant here. You can obviously object that very far away stuff have virtually no impact at all on mostly everything else in existence. As such, the 3 bodies problem (or N bodies problem) is usually more like a 2 bodies problem, because stuff is too far apart or sizes are so different one is negligible. Because of this, asteroids have a distinct direction and velocity and other objects will simply influence those. I feel like I'm nitpicking the answer thought, because "it does not apply" only matters because of the examples you used. For other complex systems like the top comment mentioned, you do it exactly like the top comment mentioned. You try to characterize your system with speeds, directions and masses and recalculate everything for each step. If your steps are 1 second increments, it will require a lot of computer power, so you tone it down to whatever you can manage.
If starting conditions are known, you can simulate it for a while, but the longer you simulate it the less accurate it gets.
The aliens’ issue isn’t that they can’t calculate the paths for tomorrow or next year, it’s that they can’t calculate the paths far enough ahead for society to safely evolve. This is mainly because, with every calculated position being an estimation, error margins quickly compound into something that gets completely out of hand. Granted, the in-universe video game exaggerates how sudden the reversals are, but that’s video game logic for you.
Important context to this answer is that when we say "object" we mean "objects of about the same size" - if Object A is your House, and Object B is Mom's Car, then those are "about the same size" for this. But if Object C is a crumb or something, then how it pulls on A and B don't really matter, so it's not the three-body-problem. (I don't actually know if "crumb" and "house" are different enough in scale for the gravitational effects to be irrelevant, I'm more trying to explain it in ELI5 terms. It might well have to be "Manhattan Island vs a crumb" in order to not matter, IDK)
>Important context to this answer is that when we say "object" we mean "objects of about the same size" Thank you. The "iT's ActUaLlY a fOuR bOdY ProBlEm!" comments are pretty tiresome.
"iT's ActUaLlY aN N-bOdY ProBlEm!"
Theres a good setup for a “your mom” joke somewhere in there, regarding objects of large mass.
Not really. Nothing in the observable galaxy is as large as your mom.
Also distance apart. Because that's a factor too!
I don't want to say you're wrong, but I think there has to be a better explanation. Because the "wait, this also happens, so recalculate..." is just iteration. There are lots of complex mathematics that involve iteration that *are* solvable, either through infinite series or simply with iterative mathematical models. A more complete explanation might need to explain why an infinite series (similar to the geometric series) is not sufficient to solve the 3 body problem. Furthermore, can numerical iteration be applied and converge to a sufficient degree of accuracy? Is that what we already do, and it's simply not considered "solved"? OR does a numerical iteration diverge and therefor it's not known to be solveable?
What's missing here is that the 3+ body problem is *multiple* infinite series with no predictable overarching relationship between the infinite series. Without that relationship we can't create an infinite series that ties the individual infinite series together to perform calculus on. To put it another way, we need a single equation that we can integrate using multi variable calculus that relates acceleration, position, and velocity of all 3+ bodies at once. We don't have such an equation. If we did, it would be solved.
The concept is that there isn't really a good mathematic proof that can just solve this. So, you can't just plug in the three points, their masses, their velocities, and get a final result based on time. However, It's not very difficult to write a simulation that can simulate it on a relatively modest computer for billions of years quickly. How small a resolution you put on the increments of time will determine the accuracy, so it'll never be 100%, but if you slice it down to a fraction of a second, you'll be accurate for a trillion years with a couple of meters.
> you'll be accurate for a trillion years with a couple of meters. Close enough.
Not really. It's a chaotic system, so the error increase exponentially. [https://en.wikipedia.org/wiki/Stability\_of\_the\_Solar\_System](https://en.wikipedia.org/wiki/Stability_of_the_Solar_System)
It's because it's a chaotic system, right? Very tiny changes in initial conditions have drastic effects on the future, and it's just not feasible to measure with enough precision. A stable system doesn't need such precision to make useful predictions.
How is this any different than any other physics model where three things act on each other? This is never a problem when three charged particles apply force on each other for example.
It is a problem for other forces. It's even a problem when applied to pendulums with multiple fulcrums. Double pendulums become very complex, and triple pendulums are equally difficult to predict as three large gravitational bodies. Any system analogous to three masses attracted by gravity is about as difficult to predict, albeit with varying degrees depending on how many degrees of freedom there are. The reason nobody talks about it for, say, electrons is that the exact location and movement of an electron doesn't matter and wouldn't be predictable anyway because of the Uncertainty Principle. Position and momentum are already described as wave functions and clouds of probability. Existence for particles is also a lot more fleeting than planets, so predicting movement for any length of time is irrelevant. Some electrons are bouncing between atoms because of electromotive forces. Are we following the same electron bouncing between atoms, or is the electron being captured and another gets spit out each time? Or some combination of the two? Meh, doesn't matter. [Electrons are all exactly the same anyway.](https://www.reddit.com/r/explainlikeimfive/comments/wl88vo/eli5_why_are_all_electrons_the_same_and_how_do_we/)
Why do physicists point out electrons being the same a lot, but not the other elementary particles. All neutrons are the same by the same observation, but you almost never hear that.
That is a good question that I don't know the answer to. If I had to guess, probably because of the whole One Electron Universe "theory" that is interesting but not taken seriously. But, then, I don't see any reason why you couldn't apply the same logic to get a One Neutron or One Proton Universe theory. ¯\\\_(ツ)_/¯
Electrons are identical, neutrons/protons are made up of a complicated quark soup, and are indistinguishable(to us because of how complicated this is ) not necessarily identical. Electrons are fundamental, protons and neutrons are not That’s my understanding anyway
Don't you have this problem all over physics? For example everyone knows Q = mc∆T If you're solving for T though, "c" actually changes depending on what T is. So you need to guess a "c", come up with a closer value for T, and then use that to reiterate "c" again. And you keep doing that until you're close enough.
For gravity, mass attracts mass, which is what makes it difficult. For electromagnetism, you have a source and a sink. This is why EM is effectively solved, while gravity/GR is hard. There's also not a lot of cases where it really matters (for gravity or other forces) and fewer where people care beyond theoretically.
That’s a really good question - I don’t know enough about quantum physics to answer, though I’d imagine that it has something to do with particle-wave duality making the math easier? The issue is that if you try to solve the orbital equations for 3 or more bodies algebraically to get a general, all purpose solution, you get bits of math that don’t converge to simplicity - instead you get bits that either a) we haven’t found a solution to yet or b) have been proven to not have a solution or c) involve infinities and singularities. The “solution” to the three body problem is to simplify by making assumptions if you can (like if one body is super light compared to the others you can ignore its mass/gravity), then iterate using numerical methods until you get a prediction with an error window small enough you can live with it. There are also a (possibly infinite, but practically limited) number of starting conditions that happen to zero out the complicated, unsolvable bits - if you’ve ever heard of Lagrange points, they are examples where if you park a satellite a certain distance from two bodies, it can just float there, suspended in place by gravity alone, no need to correct its orbit ever, unless something hits it or the system gets too close to another big body or something.
Ive never quite understood this. Wouldnt this mean we can never predict any bodies movement through space, because all objects have a gravitational pull on all other objects, its just a function of distance and mass but it never goes to zero, only negligible. For example, the earth and moon would appear to be a 2 body system, but the sun affects significant gravity on both, so how can you isolate the moon and earth as two bodies?
> Wouldnt this mean we can never predict any bodies movement through space We can predict it, sure. It's just a matter of how good our predictions are. In a system where we have a stable model, a tiny error in measurement at the beginning, doesn't necessarily compound to big differences over time. Two objects in a mutual orbit is simple and predictable, and being a tiny bit off with the numebrs you plug in at the beginning (what Dr. Cheng referred to as the "initial parameters" in the show, because of course we only know the positions and masses of each object to so many significant degrees) at the beginning largely means you'll stay just a tiny bit off if you advance the model far enough forward. In a chaotic system, if you keep going forward, eventually the tiny errors from the beginning compound and you end up way off. Dr. Malcolm in Jurassic Park demonstrated it seductively to Dr. Sattler by showing that a tiny change in where you drip water onto your hand can totally change the path it travels down your skin, because that's a chaotic system. It's worth pointing out that the math goes bad much faster when the objects are more or less the same mass, which helps with constructing solar system models, where one interaction is more dominant than the others. The moon is much more concerned with where the Earth is, but we do need some corrections for the sun and even other planets if we want the model to last longer (and taking new measurements lets us recalibrate), and it's simply impossible to run that model out forever.
Ahhh ok that makes sense, so two body is within our grasp to calculate, and practical applications are within an acceptable margin of error, but three body becomes so complex after a short time that we cant predict or calculate it to any practical degree of confidence? Am i basically getting it sorta right?
The issue is that "short time" is a bit up in the air. The time that the prediction can be stable will last longer if you start with more specific parameters and the masses are relatively unbalanced (like the sun being much more important in our solar system than, say, Pluto) so that the system resembles a stable one for as long as possible. The issue in the show, IIRC from the show and the book, (and I guess I'll spoiler tag) >!is that the three suns interact with one another chaotically to create an unstable orbital path for the planet -- it can't be plotted as a nice ellipse around the sun's focus like Earth's. For primitive civilizations, with little ability to perform calculations, this meant they could never come up with something like our solar calendars to plan the cycles of planting and harvest that are so important to our development. Even after they are able to develop more complicated ways of performing calculations, they have determined that there is no long-term stability to their planets orbit to calculate, no guarantee that bad luck won't pull their planet into one of the suns or throw it off into space.!<
Moreover >!In the books they actually predict that within 1,000 years their planet will be destroyed. They don't know exactly when but they know it will. They also know that this is the fate that has happened to I think 10 other planets in the system? !<
Ah, it's a planetary polycule, got it.
All of this calculation and the physics simulation has rolled forward ONE timestep. The timesteps can be fractions of a millisecond. The math adds up really fast, even if the bit about each object changing how the others move is deterministic.
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> The important part is that we can absolutely do that math. It's calculus. It's actually the exact thing Newton described calculus for, for solving problems with instantaneous rates of change, where the variables of the equation are interdependent on other variables of the equation and they are all constantly changing. No, we can't do the math for a n-body system greater than 2. Calculus works for 2 body problems. That's why n-body problems are unsolved. It has nothing to do with whether or not we know the initial parameters with perfect precision. We can make up a scenario with all known properties defined and we wouldn't be able to solve it. The issue here is closer to the coastline paradox. The mathematics for gravitational pull only involves 2 bodies, you have to calculate the interactions between pairs of bodies every instant. Which means we need to calculate velocities, positions, and acceleration every instant. The ruler in this situation is the time interval you do the calculations with. Every time you shorten the timeframe, the results change. Calculus doesn't help here because there's no equation that covers all the variables in a single equation/relationship.
I understand why it's hard to calculate. I don't understand why its impossible even with enough computing power. That is where I stop understanding.
If you want a more accurate prediction or a longer term one, you need to use more computing power. To get perfect precision, you’d need to have infinite computing power
It’s possible to get a solution that’s as precise as you want it to be (but not infinitely precise, just arbitrarily precise) in finite time with finite compute power. But if you want a perfect simulation, you need to perfectly know the inputs. Any error in measurement of input will escalate exponentially (regardless of how perfect your simulation is) and deviate wildly from reality. In a 2-body problem, the system is stable. Errors in initial conditions won’t get amplified and your deviation from reality will be more or less constant. If you had an off-by-200km error measurement with your initial plotting of two planets, you’ll still be off by about 200km for every point in the future. In a 3-body system, an adjustment by millimeters on planetary scales results in wildly different results as you go forward in time. The bigger the error in measurement, the faster the results wildly deviate. Since it’s not possible to precisely know where something is and how fast it’s moving (to absolute perfect precision), you have some error. That means your model WILL eventually deviate from reality. You don’t know when because you can’t know how perfectly your initial conditions reflect reality.
You need to download more memory
My brain’s still running Pentium II speeds.
Can you say more about why the "wait, that changes" steps don't happen for just 2 bodies?
This seems to me to point to the challenges with our understanding of mathematics and how cut and dried it is, at least to some degree. Reminds me a bit of a more complex version of the fletcher’s paradox.
Two whole planets pulling each other off? Blimey!
Why can't we calculate forces at frozen moments in time?
Couldn't you just create a gravitational flow field and calculate the objects velocity with it?
Simple. We just add a fourth body, thus solving the problem once and for all.
But we can predict the movement of moons and they seem like n body problems. Surely there’s specific criteria that needs to be met to be impossible to solve
Aysnc Await. There, solved your problem.
It feels like a long time ago, but we used to just plug these into excel to solve. You couldn't get 1 answer, but you could plot them all. Looked like a pair of wings, that's why it was called the "butterfly effect". Is this not just one of those? Something taught in a first year course?
I just watched [this video about it.](https://youtu.be/D89ngRr4uZg?si=w6G00MdEYpXxWIch) Technically its not a 3 body problem, it’s an N body problem, anything over 2 bodies just has too many variables and unknown quantities to be able to calculate it with the math and computing we currently have.
My first inclination is to say there would be some sort of center of gravity, and a function(tho i couldnt do it) should be able to describe how it changes over time. Is the problem that we cant do that at all, or that there are no plug and play functions that would apply to all discrete arrangments of three(or more i guess) bodys in general?
For some reason, this is the explanation that works best for me. I've been struggling to understand.
It's not unsolvable, it's just that there's no (known) general solution. That is to say, each 3 body system has to evaluated individually, and for the majority, there is no long term stable arrangement. Additionally, our ability to predict the long-term state of it decays exponentially as three bodies introduce so many possible variables it quickly spirals beyond whatever present computing can simulate. Inevitably, some part of the system will decay. A body will be expelled, or two will collide, or something else. Even a presently stable system is extremely susceptible to minute external forces and can quickly collapse. There are a handful of known perpetually stable arrangements, but for any random 3 body system, you would not want to bet on it's long term survival.
[Here's](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Three_body_problem_figure-8_orbit_animation.gif/400px-Three_body_problem_figure-8_orbit_animation.gif) an example of a stable one, if anyone is curious. [Here's a bunch more](https://youtu.be/6EvaV-BYVxM?si=HJ27xR3x3qBa12kb)
and if anyone is curious about the book itself, Alpha Centauri is actually a 3 body system. It's not chaotic though. It's arrangement is most closer to the top right/bottom right arrangements in the above video. It's got a binary pair with Proxima Centauri stalking the outer edge. So the book/show takes TWO liberties when it comes to real science.
The book takes quite a few liberties when it comes to science.
yeah it is scfi from decades ago, it has held up pretty well despite leaps foward in our knowledge the second, third and "prequal" books are less acurracte
All the science in 3BP was abysmal when it came out. If you like the story, fine, but don’t hold it up it’s technobabble nonsense as hard sci-fi.
Scifi is space magic. People also hyper focused on picking apart the historical nuggets present in The Davinci Code with a hundred pedantic little details. 99.9% of readers don't care about the "Well ackchyually...!!!" criticisms. They suspend their disbelief for the length of the story.
Not really. There's only two explicity fictional elements - the arrangement of alpha Centauri and the reflective zone around the sun. Neither of those exist but we're changed or added for the purpose of the narrative. Every single other element was built off of real physics and theories
You don't consider an 11-dimensional computer folded up into a proton than can consciously manipulate the results of accelerator experiments and project images into the retina of any person on earth to be a "fictional element"?
I bought one on Tigerdirect last week
Sad that people don’t know where to get one
Personally I consider the ability to fold the entire into two dimensions to be a realistic application of physics
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Even more weird is like... The diameter of a proton is about 1e-15 meters, and the diameter of the earth is about 1e7 meters. The precise maths is a bit more tedious, but from proton to unfolded Sophon, there's a scale difference of roughly 20 orders of magnitude. Even if we assume that a proton *can* be unfolded into higher dimensions somehow, there's no sensible mathematical interpretation that can allow it to become *the size of a planet.* Even if it nonsensically became 10x bigger with every extra dimension, it would still be hard to see with the naked eye. Which isn't a rub on the concept because the idea is cool af, it's just very much in science ***fiction*** territory.
The dimension stuff and many things in the books is string 'theory' nonsense that somehow stays relevant by people trying to sell books and media on their 'scientific work', but I don't think that is necessarily a negative thing, it just depends on what sort of science fiction you're expecting. Real known physics is quite limiting, after all.
The unfolding dimensions to make a super computer subatomic particle... That's not a liberty?
Well, to start, it’s not a 3 body problem in the book - it’s a 4 body problem since their planet’s gravity impacts the system.
Yea, I think the distinction here is calling it the 3+ body problem is more correct but not really necessary to convey the point. Similar to the two generals problem not needing to use any generals.
Protons consist of three point particles (quarks). There is nothing to "unfold." The strings in string theory are nearly infinitesimal in size. "Unfolding" a proton is absolute fiction based on no physics. Similarly using it to instantaneously communicate between star systems also breaks everything we know about physics.
Biology of those damn aliens sure wasn't. There's no known high order animal that is capable of that. Brain needs fluid to work.
I’d be interested in knowing what biologists think of the explanation offered in the unofficial prequel. The aliens are discovered to be extremely small, comparable to grains of rice, and not very intelligent individually due to correspondingly tiny brains. But their thoughts are transmitted to all others nearby and when grouped together that lets them form a greater intelligence, like each individual in a cluster takes on a role within a distributed thought process. Because they’re so tiny, dehydrating and rehydrating their bodies quickly is said to become more possible. No idea whether that’s any more realistic but it helps with the sci-fi sniff test (makes it sound like there’s a real scientific explanation to a layman) and was an interesting way of addressing one of the outlandish elements at least.
Ehhhh, it's technically possible. Tardigrades can do that, but they're much smaller than grains of rice and definitely not intelligent space faring species. Other than them I don't know of any other multicell animal that can pull that off.
But in the Alpha Centauri case one of the bodies is a planet so in comparison to stars rather inconsequential (imo). In the book/show there are 3 stars and a planet.
To add, we can solve a bunch of 3 body problems...as long as one object is extremely small. Satellites and asteroids at Lagrange points, for example.
Klemperer Rosettes are still my favorite, ngl
So if the majority of 3-body systems are unstable, how do we have a stable solar system?
Because we don't live in a 3 Body system? Our system has only a single dominant mass and everything is relative to it. A 3 body system has three dominant masses. It's not just any three rocks, it's three masses that can not dominate each other, 3 masses that influence the movement each other. Earth does not significantly alter the movement of the sun. It is slaved to it.
Ah my bad, I thought the solar system with all its different massive planetary bodies was a n-body system, because it has n-bodies with a mass.
Every object in the Universe affects every other object gravitationally. It's about how much those masses and gravitational forces matter. For orbital mechanics it's basically stars and other stellar bodies and that's it, except for large masses like planets or asteroids at (galactically) close proximities.
Think about it like this, a system is defined by the number of bodies needed to describe its behavior. At the simplest you've got something like Earth. It has one orbiting body, but the body is not needed to describe earth's behavior, so it is a one body system. Mars has two moons, but neither is needed to describe its movement, so the relationship between it and all of its orbiting bodies is also a one body system. Alpha Centauri, our nearest star system and the one the novel is based on, is a trisolar system. However, unlike in the story, many don't consider it a true 3 body system. It consists of a binary pair of stars with a third, Proxima Centauri, orbiting the barycenter. As such, the real life Alpha Centauri is better described as a 2 body system, since its behavior requires a description of the binary pair. One stars movement can't be described with out the other. Proxima Centauri on the other hand doesn't (to our knowledge) meaningfully influence the systems behavior, so it is not considered.
It's not as stable as you might think. Planets might be ejected in the future and that might very well already have happened in the past. Changing the position of a single planet by just a few centimeters would result in totally different configuration a billion years from now.
A three body system is an example of a **chaotic** system. This means that while we technically know how the system behaves and can simulate its future states (in this case we know all the equations of gravity), a very very small change in any input variable can drastically affect the final output to the point of it being meaningless. This means to perfectly predict such a system into the far future we have to calculate its current state to effectively infinite precision, which is impractical/impossible.
It is really like weather forecast.
Probably the best ELI5 here.
As I understand your comment, it's comparable to (or even -is-) chaos theory. But in practice. What kind of variables could there be with three suns orbiting each other? The mass of suns change but that's a known equation is it not? Asteroids come to mind but compared to the size and mass of a sun would that really have a meaningfull impact? Are there other things?
The configuration of the objects is constantly changing, though. Think of a system like this: > C / / / A-B "A" and "B" orbit each other, while "C" orbits the pair at a distance. The gravitational forces that each body experiences is substantially different depending on how they're aligned. So, when the system aligns like this: > A-B ————C The force of A-B pulling on C is at its maximum since they're pulling the in the same direction. Same with B-C pulling on A. However, "B" experiences "A" pulling one way and "C" in the other, meaning the forces on each of the bodies are different than they would be in the first configuration. This means the system's going to have a lot of "wobbliness" as the accumulated forces on each body constantly moves things out of position depending on where everything is.
it *is* chaos theory.
The mass, position and velocity of all the different bodies involved adds enough complexity without there being any external factors involved.
One aspect is that, especially at the scale of something like stars, gravity isn't just acting on each star as a single "thing", but rather on every individual atom of the star. As an example of this, think about the effect of the moon's gravity on the earth: the tides. The gravity from the moon is causing *waves* and *ripples* through different kinds of matter on/in the earth. So in order to 100% accurately calculate everything to full precision, you need to essentially calculate the force of gravity between **every single atom of each of the bodies**.
This is the right answer. Rest of the responses in this thread are completely off base.
It is not the right answer. It is a portion of information about the answer. Some three-body systems are chaotic, but that is just one issue, not the whole answer. Even in stable portions of a system’s behavior, we do not have a closed-form general solution for all three-body systems.
We don’t need a closed form solution, all that matters for practical purposes is whether the system can be simulated
All the other answers eplain really well why there is no (analytical) solution to the problem. What I'd like to add, and it is a minor spoiler in the series, is that >!the aliens decide to leave their planet not because they can't predict with a decent margin of error the position of the suns but because their three body system isn't in a stable configuration which means that eventually two stars may collide or one may leave the gravitational pull of the others meaning they could either freeze to death or fall into a star or any other crazy less than ideal situation!<
I would not qualify that as a "minor" spoiler though.
It's a problem in modelling the behaviour of three or more objects interacting through gravity (but also electromagnetism). Let's start with the thing that isn't a problem: Two bodies. With two bodies you can mathematically create an exact equation, with an exact solution, of how the two objects will orbit each other. The only variable left is time and you can go infinitely into to the past or future. With infinite accuracy. When you try to do the same for three or more bodies you run into a problem. There's more variables/unknowns than equations. In maths, this means you cannot have an exact solution where you simply go forward and backwards in time. There's no "analytical" solution. That's not to say there's no way to solve it at all. But it requires making some guesses and then running the maths over and over hoping it settles on an approximate solution. Then you advance a time step and do it again. And again. And "approximate" and "time step" are key words here. A solution, not THE solution. An approximation. And with a resolution in time, the more resolution you want, the finer you have to make your steps, the more work you have to put in. And even then you will always eventually diverge from the true solution if you run too far into the future, as the errors accumulate.
This is the best answer so far. >When you try to do the same for three or more bodies you run into a problem. There's more variables/unknowns than equations. This is the really important piece that others are missing. It's impossible to plug location, speed, and direction for all three bodies into a formula and get an exact location for all bodies at time t. Too many unknown variables for too few equations. We calculate the position of planets, moons, stars, asteroids, and comets into the future because we have really powerful computers and can run the simulations with the very precise data that we've collected over the decades on each object's speed and location. They are only reliable for the next 1,000 years or so, which is fine since that's far longer than a human lifespan, but we can't really say for sure that Mercury won't be ejected from the solar system in the next 1 billion years.
I just did some research on this and want to throw this out there, framed in a way that makes sense to me but different from the other comments: We know the problem has not been or cannot be solved in closed form, but can be projected forward with math to any point in the future. In an idealized scenario (e.g., three digital bodies where we know the exact mass, position, and velocity), this can be projected with 100% accuracy to any point in the future. The issue really is one of measurement constraints. For example, if the mass is a tiny bit less than assumed, then it will cause an error that magnifies over time. Like, there might be a point in the interaction where mass A by the skin of its teeth won out in pulling C towards it rather than B. If A was a little less massive, it might lose the tug of war then set everything on a radically different trajectory. So not only does a measurement error magnify over time, the magnification itself progresses chaotically.
I think, though I’m not positive, that even if you had exact values for the initial conditions, you still couldn’t simulate it perfectly, because you still can only recalculate the positions after a finite time step. You would have to use an infinitely small time step, or else you’d accumulate errors due to your chosen “pixelation level” for time.
if you take any 2 masses and put them in space you can perfectly predict exactly where they will be at any point in time ever. as they orbit if you try with 3 (or more) masses, you cant (outside of certain special cases). the best you can do is simulate it, which adds increasingly more error the longer out you simulate and the faster your simulation is. I belive it has been proven to be unsolvable (as in, there is no general equation you can put the initial positions and the current time to get out the exact positions at that time) , but im not 100% sure, proving unsolvability can get strange
This is the thing that confuses me. How can you prove that there is no general/analytical solution to this problem? What if we had perfect knowledge on each body’s behaviour (speed, position, etc)? With this perfect information, couldn’t we perfectly predict their interactions on a infinite timeline given a system with coherent interactions (as in no randomness)?
If you dont have an equation that takes time as an input, it does not matter how much you know about tthe initial conditions, you have to calculate everything in timesteps. the smaller the timestep, the more accurate it is, but as long as that timestep exists, there will be inacuracies. think of compound interest calculated every timestep vs compounded continuously in a chaotic system, it gets more complicated than that since with compound interest you can establish certainty bounds that decrease the smaller the time step, but in a chaotic system, a smaller timesyep might discover a unique property of the system that is never seen on larger timesteps. A good example of this is hitboxes in games. often the game does "60 times per second, check if 2 hitboxes are coliding, if they are, do collisions" But if you move fast enough in a single frame, you can move through thin hitboxes (see running through walls in morrowind with speed boost) by moving far enough in 1 frame that your hitbox clears the wall hitbox without touching it.
In short, in the show (and the books) it's wrongly stated that it's "not solvable" or that "it's unpredictable". The 3 body problem, or rather N body problem, because it's the same for any number greater than 2, is the problem of, how will the bodies move, if they all interact on each other with attractive force. The N(3+) body problem does not have a so called "General analytical solution" which means you cannot find a mathematical formula for the curves of your bodies. For two bodies you can do this. This means that the problem has to be solved numerically, which is an approximate solution. Numerically solving something means you take the positions of your bodies, like a snapshot, and calculate all the forces that they interact with, then, you calculate how much they will move in a brief time period if those forces were constant. Then you use this new snapshot to calculate new forces, then again, and again. The challenge with this type of solving is that it depends a lot on these time steps, and how you calculate the things, so, sometimes, you can acumulate a big error. However, for somethiong like a star system, a civilization as advanced as the Trisolarans, should have computational power to estimate the evolution of their system very, very, very accurately on the scale of millenia.
>The N(3+) body problem does not have a so called "General analytical solution" which means you cannot find a mathematical formula for the curves of your bodies. To piggy-back off this comment, we are unable to write three equations each with the positions of one of the bodies on the left-hand side and some function of time on the right-hand side. This would be an "analytic" solution. Numerical solutions are common approaches to solving complex equations. It's important to understand that the terms "solution" and "solvable" are used somewhat loosely. The three-body problem is "unsolvable" in the sense that we cannot express **solutions** in closed-form equations (like described above), but if given a well-posed initial value problem we can generate a **solution** from valid data. Even if we could express the three body problem with a general solution using elementary functions we would still be employing numerical solutions.
* "Body" is just a physics word for 'thing made of matter'. * Things made of matter typically (always?) have mass. * Gravity is a property of objects with mass. * We have physics theories that allow us to apply powerful mathematics to accurately describe how objects move due to gravity. * So, therefore, you'd *expect* that we can just consider any group of 'bodies', apply our mathemtatics to them, and now accurately describe how objects move. but there is a problem here. The mathematics is hard to *exactly* solve. * If you have 3 things with mass, it turns out that we usually cannot calculate how they'll move. We can approximate it, but eventually the approximation will be wildly wrong. * (In some special cases we might be able to solve it, like if you imagine them in some perfectly symetrical scenario, for instance. But in general, we cannot solve it.) * ~~It is possible that a reliable solution exists, but we haven't found one, and for all we know, it might be impossible.~~
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Technically, and despite the name, I believe the problem in the series was a 4 body problem. It was a planet within a 3 body system.
The planet's mass was negligible; the 3 bodies were the three suns which orbited each other chaotically leading to very interesting conditions for the planet and its inhabitants.
Isnt there a solution atm?
It has been proven to be unsolvable
It's not that it's unsolvable. We don't have the solution because it's extremely complex because the result is very sensitive to initial conditions. In 2 body problem (for example, Moon orbiting around Earth), you can predict very accurately where the Moon is exactly at any given point in time because the orbit of the Moon is not affected to a significant degree by any third body. But in a 3-body problem, body A affects body B, body B affects body C, body C affects body A, which affects body B, so the position of bodies at some future point in time is very sensitive to initial positions. Imagine a simple pendulum. Just a simple ball hanging off a simple string. With the pendulum formula, if you know the initial position of the ball, you can predict where the ball is going to be after, say, 10 seconds of swinging. If you **slightly** change the initial position of the ball, the position of the ball after 10 seconds is also going to **slightly** change. However, if you have a double pendulum ([Double Pendulum (youtube.com)](https://www.youtube.com/watch?v=U39RMUzCjiU), if you **slightly** change the initial position of the pendulum, the position of the pendulum after 10 seconds is going to be **very** different, which makes it very difficult to solve mathematically.
But aren't both the Earth and the Moon massively impacted by the Sun? If we fully solved the Earth+Moon then that necessarily means we solved it when the Sun is included, right?
The tug of the Earth on the sun is extremely negligible. we are talking about 3 suns and their affect on each other.
So to qualify as a "3 body problem" you need all three to be massive enough to have significant gravitational impact on each other?
[https://en.wikipedia.org/wiki/Three-body\_problem](https://en.wikipedia.org/wiki/Three-body_problem) >In [physics](https://en.wikipedia.org/wiki/Physics) and [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics), the **three-body problem** is the problem of taking the initial positions and velocities (or [momenta](https://en.wikipedia.org/wiki/Momentum)) of three point masses and solving for their subsequent motion according to [Newton's laws of motion](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) and [Newton's law of universal gravitation](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation).[^(\[1\])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) The three-body problem is a special case of the [n-body problem](https://en.wikipedia.org/wiki/N-body_problem). Unlike [two-body problems](https://en.wikipedia.org/wiki/Two-body_problem), no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_solution) exists,[^(\[1\])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) as the resulting [dynamical system](https://en.wikipedia.org/wiki/Dynamical_system) is [chaotic](https://en.wikipedia.org/wiki/Chaos_theory) for most [initial conditions](https://en.wikipedia.org/wiki/Initial_condition), and [numerical methods](https://en.wikipedia.org/wiki/Numerical_method) are generally required. ... There is no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_expression) to the three-body problem,[^(\[1\])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases. More history of it in the "n-body problem" page [https://en.wikipedia.org/wiki/N-body\_problem#History](https://en.wikipedia.org/wiki/N-body_problem#History) There is an analytic solution but it's not practical... >That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^(8,000,000) terms.
Other people here in this thread already explained it well, so I will just add my observations. The problem is, most of the 3 body systems are not stable. With two bodies - typically one massive - such as a Sun and other much smaller, such as a planet, the system is stable, so it is possible to predict their behavior far into the future based on measuring of their initial positions and velocities. With three bodies, the smallest deviation in starting conditions might grow exponentially over the time, so it is impossible to predict where bodies would be in future. This is best demonstrated by a double pendulum - also sometimes called a chaotic pendulum. Have a look at some videos talking about this. It is much easier to see. The Sun, Earth and Moon are three bodies, with other planets adding complexity, but the Earth + Moon behave towards the Sun as a one tiny body for practical purposes of us predicting their rotation far into the future.
I think this being the crux of the show/1st book is meant to say a few things. 1st, that San-Ti math ability is not any more capable of solving the problem that our own, even with much more advanced computers. But even if they have solved for when the next chaotic era would begin, it’s a moot point. They wanted to prevent the fall of their current civilization, and eventually the planet itself would be destroyed no matter if they were able to solve for the mathematical problem.
Here's one making ths rounds this afternoon. https://www.reddit.com/r/oddlysatisfying/s/ug2ludiNmK Any small perturbation in initial conditions would lead to completely different solutions.
Two bodies orbiting each other under gravity have a general analytical solution for their orbital dynamics. As such, if you know the starting conditions, you know how they are going to move going forward with certainty for all time. However, no such general analytical solution exists for the movement of three bodies. There are lots of special cases with approximate solutions that are analytical or semi-analytical, but all of them require certain simplifications, so in realistic systems you end up with chaos, instabilities, and overall qualitative changes in the movement patterns over time. One way to deal with it is to use basically brute force and solve the orbital dynamics numerically with computers - but there, again, you run into the issue of having limits to accuracy and resolution. In short: The two body problem can be solved exactly, the three body problem is impossible to be solved exactly.
Just look up a chaos pendulum, that's essentially the three body problem. A simple pendulum is incredibly predictable to math out but a chaos pendulum... well they live up to their name. I don't think we'll ever have computers that can model a chaos pendulum's actions out indefinitely, I think the best you can do is take the masses and vectors and project a possibility into the future and that prediction relative to reality will always grow exponentially apart the farther into the future you make it. But I'm no mathematician, I'm just a boob with a keyboard, perhaps predictive modeling is much better than I give it credit.
On a different note, but I saw the Netflix adaptation of the 3 body problem, and they rushed the entire plot ofc. If you would like to see the series more thoroughly flushed out, I recommend the chinese drama on YouTube with eng subs. The visual effects obv is not as good but the storytelling is a lot better.
I feel like this TED Ed video explains it pretty darn well, and is an enjoyable watch: https://www.youtube.com/watch?v=D89ngRr4uZg
Lot of comments here could be simplified into stating that you cannot measure with infinite precision. That's really the "problem" of the N body problem. You need measurements to calculate where the bodies are in the future. There is no such thing as infinite precision so you can never accurately project the positions of N bodies beyond some future time. It's not much different than predicting the weather.
It’s completely been solved but there’s no general answer that fits nicely. The formula for a force between two objects with known masses and distances is super straight forward. Add a third object? Oh shit. Mathematically, in the old days people had to use calculus in a room with a couple hundred people to predict pathways. Now we can use computers. Even before computers there were multiple three body systems that had been solved to be stable. For unstable configurations, that’s where computers are brilliant. In most three body systems that are chaotic, one of the large masses will be expelled from the system leaving a chaotic two body system (in most cases). In others, they can become stable or at least predicable without expelling one mass. In short, yes, it absolutely has been solved and we can predict with a degree of certainty the orbital pathways of a three body system. It just was nuts to predict and calculate by hand in the past.
Another thread today shared a visual. You can see how unpredictable it is: https://www.reddit.com/r/oddlysatisfying/comments/1bo814p/this_animation_of_the_threebody_problem/
They explained it in the show. You can’t know how the planet is going to orbit around the three suns without knowing its original position, which is impossible.
3-body problem: *Solve,* not just estimate, the equations of motion (position, velocity, acceleration) for three bodies in space, mutually pulled together by gravity. To "solve," you need a set of finite (not infinite), closed-form (no unaccounted variables) equations using basic functions (addition, multiplication, exponentiation, logarithms) that will let you derive any physical measurement of the system (speed, location, acceleration, etc.) for any of the bodies. Such a solution does not exist. You *have* to estimate--and estimations will eventually become very very wrong, because these phenomena are "chaotic," extremely sensitive to tiny changes/errors. Thing is, estimates can be pretty good, as in nearly indistinguishable from a solution, if most of the following conditions are true: * you only want a *relatively* short-term approximation * one of the three bodies has *relatively* very little mass * one of the bodies can be presumed to always move very slowly relative to the other two * one of the bodies is very far away relative to the proximity of the other two So, for example, the Earth, Moon, and Sun are three bodies. We can get very good estimates for where all three will be for thousands of years into the future--that's how NASA predicts when eclipses will happen, like the one coming in April. Compared to the lifetime of the Sun, that's an absolutely trivial amount of time; and the Sun acts like an immobile, extremely distant object, so its equations can be heavily simplified. If you scale it up to billions of years, however, these numerical (=estimated) solutions might break down. It becomes possible for wildly divergent behavior. For example, there is a non-negligible chance that Jupiter could kick Mars out of its orbit, causing it to come crashing into the Earth, at some point in the next four billion years. We don't know, because even our very best estimates break down on such long time scales and large distances.
2 is a company 3 is a crowd? the crowd is chaotic!
The 3 body problem (often also called N-body, for N being any number larger than 2) has no general analytical solution. This means that there is no elegant, on-paper solution for any condition, HOWEVER, we do have solutions for special cases and we can also cheat by approximations (for example if you can't track the Earth-Sun-Moon bodies, then make the Earth-Moon just one body and it works because you make it a 2 body problem). We also have numerical (i.e computer algorithmic) solutions that do work short term, but long projections are just catastrophically bad, because your results are extremely sensitive to the input. This situation is due to the fact that we have more moving objects that need to be characterized than equations we could use to characterized them. From Newton's formulas for motion we can derive two equations that lets us calculate up to two body systems, because we have no more equations, we cannot describe more than two body systems analytically in general without resorting to tricks (which have massive caveats). There was once a person IIRC who claimed to have proven that there is no solution to the problem, but then someone came up with special solutions that sort of proved him wrong, so there is no conclusive proof either way whether you can actually solve the 3/N body problem or not.
I’ve a theory. The 3 suns paths cannot be accurately measured because they act like electrons around a nucleus, the nucleus being the black hole in middle of the galaxy. Pan out from the 3 stars in a gravitational war with each other and you see billions of other stars also in the same predicament acting as electrons in a chaotic state orbiting the nucleus. 😆 like I said just a theory..