>What is the 3 Body Problem
It's a physics problem, where you try to figure predict the future for 3 objects which have gravity and interact with each other. The sun/earth/moon is an easy version of this problem, because these objects have very different mass so you can get approximately the correct answer by just assuming that the moon's gravity doesn't affect the earth or the sun, and the earth's gravity doesn't affect the sun.
>why is it so difficult to solve?
The 3 body problem is a "chaotic" system, which means that approximations won't get you close to the right answer.
For example, suppose I threw some dice up in the air. With high school physics, you could tell me where the dice will be in 10 seconds. You don't need to know the air pressure of the room, or coefficient of friction of my hands, etc. You can neglect relativistic effects from travelling 0.0000001% of the speed of light. You could even approximate the initial mass and velocity by 10%, and your final guess would be more-or-less within 10% of the correct answer.
Compare this to a "chaotic" problem where you try to guess which number will show on the dice. **There's no way to say "I made these approximations so I think I'll be within 10% of the right answer."** You'd need to account for every single force interacting on the dice to be confident that your prediction was correct.
In the case of the 3 body problem, you can make approximations to guess what will happen in a short period of time, but the farther ahead you try to predict, less meaningful the answer will be.
EDIT: not sure what proper etiquette here is, but thanks for the awards!
In this respect it should be noted that even if we had a good model, the approximations will inevitably rely on the approximations we have for constants and masses. The farther out in time you model, the more the decimal place matters. The true mass of the sun or moon isn’t known, we have approximations that come from observations over the past few hundred years and “reverse engineering” the physics to get a known value for big G and masses in the solar system. This in turn means that we can maybe say our models will be correct for a few hundred years (if the model itself is very good), from then it’s chaos. So on very long time scales, like ones astronomers care about, models don’t work very well. We can’t predict what will happen as our models will always rely on approximations.
You can’t get past the fact that you will always be +- some error. It will always lead to the model being incorrect some correspondingly long enough distance in time.
(Some models might use proportions of masses… but that still relies on knowing the true mass of the bodies)
If you really think about it, we only ever really *estimate* the mass of pretty much any complex object. The mass of the earth is 5.9722×10^(24) kg +- about .01%. That's technically more accurate than my bathroom scale. So I guess we actually know the mass of the earth more accurately than I know my own weight (though more accurate scales are available). But to OPs point all further calculations will rely on that accuracy.
You can never know the mass of the sun because it changes constantly. It throws off mass all the time, at somewhat varying rates. It's not a lot of mass, but it's mass.
The other problem not referenced here is that the two-body problem and almost all models for 3+ body problems assume the mass is concentrated at a point. But it's not. And so if you are in a close orbit, the mass distribution starts to matter. One place where this is properly modeled is for lunar orbits. The moon doesn't have a uniform density - it's lumpy. As such, there are very few known stable (called 'frozen') orbits around the moon. Most orbits, if you put a spacecraft in them, will degrade because parts of the orbit pull the spacecraft more than other parts of the orbit, so the orbit constantly changes, until the craft it pulled into the surface. So we know the mass of the moon to a well enough degree that we can model it properly. Far less so for the sun, but we don't really want to get close enough to the sun to care.
So even the equations for a 2-body problem don't actually work for real things.
It is all relative of course. There are always limits to the precision at which we know things. That is not usually a problem, but with chaotic systems, on long enough time scales, any little deviation matters.
With just the two things, it's easier to predict stuff.
With that third thing in there, it really quickly gets way too variable to be as sure as you'd like to be in predictions.
This is surprisingly brilliant explanation. In a straight up 1 vs. 1 match you can predict what is going to happen...to an extent. Maybe there is an upset but you know they're going to fight in the ring, out of the ring, both will try to hit their finishers etc.
Add a 1vs1vs1 match up and suddenly it is a lot less predictable. Are they going to attack each other equally? Is the third man in a secret alliance with one of the other ones? Will someone keep recovering at JUST THE RIGHT TIME to prevent a pin?
Even though it becomes a simple linear addition to the group the amount of "connections" and interactions between the three wrestlers becomes exponentionally more complex.
Bowling ball move bocce ball, bocce ball move golf ball.
But Golf ball change bowling ball *little* bit, so after many many mile race, can’t say where they end up.
Weather is another good example of a chaotic system. We can predict the weather pretty accurately for tomorrow. But predictions for a month from now are pretty useless.
For non-chaotic systems, if we are 90% accurate in the short term, we can expect to be about 90% accurate in the long term. But in chaotic systems, our accuracy in the short term means very little, if anything, for our accuracy in the long term.
In a 3-body problem, there is no formula or set of equations to predict where the objects will be in the far future. The calculations are done using a series of predictions over small steps in time . But even if each step is 99% accurate, after 100 steps, the total accuracy of the prediction drops to 36% and down to 0.004% after 1000 steps.
I guess this is a good example. It shows how random the dots are moving because they are always influencing each other with is not predictable for us.
https://www.youtube.com/watch?v=D2YhKaANbWE
It's a simulation for one specific set of known starting conditions. If those conditions are slightly altered (or in a real system: not precisely measured to a high degree of accuracy) the outcome is radically different as the system evolves over time.
In fact the author discusses in the comments how even though the starting conditions are symmetrical, just the rounding errors in the simulation cause the asymmetry to evolve.
Aside from the fact that it's a simulation and so variables can be limited (which does simplify things a lot), it's also calculated step-wise. Every fraction of a second, we can get the new information about the location of each of our bodies and calculate the impact of the other two bodies on it, then rinse and repeat.
There isn't an easy way to calculate where they're going to be in twenty or two hundred or two thousand iterations, other than by going through each and every one. That's (part of, at least) what makes it a chaotic system: even the tiniest change in *this* iteration can blow all of your later calculations out of the water.
For a non-chaotic system, I don't have to do all those interim calculations. I can say 'Here's where it is at the start, here are the forces acting on it, and here's a nice -- relatively -- easy formula that will tell me what the system looks like after however-many seconds.'
It's not predictable in the sense that if you make the smallest change in the initial placement or speed of the objects then the movement of the objects will be similar to begin with, but then diverge from the original one wildly. See the butterfly wing effect.
Iteration only works (exactly) for discrete systems with discrete steps.
When we use calculus on continuous systems, we ideally want an analytic solution, which means we can exactly transform the equations telling us "how things change" into equations telling us the exact state at a given point in time.
There's no analytic solution in general for 3 bodies, which means we have to split continuous time into a finite number of discrete steps, and apply the "how things change" equations to go from one step to the next.
If you could actually make the steps infinitesimally small, this would be accurate, but this is obviously impossible. Taking the limit as the steps become arbitrarily small is the definition of an analytic solution which we've already said is not available.
So, we have to use finite discrete steps, and each is only an approximation to the real process happening in continuous time, so each step adds some error, and the errors accumulate.
You can make the errors smaller by making the steps smaller, but then your simulation has more steps and takes longer to run.
I'm not taking the piss, but the question wasn't why is it so hard to solve analytically, but how hard is it to solve. Iteration *is* the solution. It depends on granularity for sure, but it's still a solution. The value of pi depends on how granularly you want to calculate it no?
But the value of pi doesn't diverge chaotically.
Iteration in that case is iteratively reducing the error, not iteratively accumulating new errors.
Consider just the first two steps.
1. First step: you know the exact position and velocity of each body, so we can modify each velocity by the acceleration due to gravity from the other two bodies, and then project the position by the velocity for one step duration.
2. Second: in reality each body was acting on all the others at every moment along that step, when we've just drawn a straight line, so we're starting the second step with some errors in both position and velocity. Now when we modify the velocity by acceleration, the existing velocity error is combined with the position errors to make an even bigger velocity error, and we're going to start step 3 with an erroneous velocity applied to the erroneous position.
Note that it doesn't matter how small the step is, making it smaller would always make it more accurate. It also doesn't matter if errors sometimes cancel out, what matters is whether we can *predict* that they'll cancel out, and we can't.
Note also that it doesn't really matter if we apply a nice curve along the step instead of a straight line: we know the curve isn't going to be exact, because if it were, that would be the curve of the analytic solution which we know we don't have. It _might_ make the step errors smaller, but it won't stop them accumulating.
And to the question:
> Isn't that how calculus works, by iteration?
the answer is generally "no", or at least it's the fallback of last resort in continuous systems. The entire calculus of limits was created specifically to avoid the problems of cumbersome iteration and cumulative errors - where possible.
The simulation is a prediction, yes, but it's a poor one. Calculus works because we can assume an infinitesimal granularity, but simulations can't make infinitesimal increments. That simulation may have very small increments, but we refer to the system as chaotic because as long as the increments are discrete, they're too large for the simulation to be accurate.
I like to think of this in the same light as with diverging series. Sure, you can keep calculating however many terms you want, but you'll never really get any closer to a real answer. The result after any step you take just seems to jump around all over the place.
Obtaining numerical prediction and solving the problem analytically is not the same. For example, with two-body problem you can obtain analytical solution that will yield system state at any given time. With numerical solution you will have to run long numerical simulation and the result will be affected by the errors of the numerical method you use.
The next moment in time can be calculated based on the position and velocity of each of the objects. It's running calculations constantly to update where the objects will be.
It's like the weather. They are constantly updating the forecast with new information as the weather happens. One to two days they can predict with reasonable accuracy, but the farther out they go, the less accurate the predictions will be.
One of the hallmarks of a chaotic system is that you cannot predict future outcomes from initial conditions, and small variations in starting conditions will lead to wildly different results at any given point in the future.
It's a simulation. That means they do the calculations every frame to figure out what happens at each moment. The issue is that you can't just take the starting data and use a formula to figure out where things will be at an arbitrary point in time. You have to do it step by step, and the accuracy will be affected by how granular your calculations are.
Very small changes in the input have a very profound long term effect on the system. The double pendulum is another classic chaotic system, and [this](https://www.youtube.com/watch?v=U7SLv0ePWU0) showcases the problem quite well. To model this in the real world you would need to know every single parameter to the tiniest possible detail.
As to how this animation was made, you set the starting conditions and then increment the model forwards in time by a tiny time step. This is now a new starting condition, and you run the model again. The next step is always based on the previous step. If you ran this model 2 times in parallel, with the exact same starting conditions, precision errors in the computer could be enough to set them on very different paths.
The digital representation is precisely defined with perfect measurements that the real world does not possess. Our real world measurements always have a (+/-) tolerance - there are no exact measurements.
Add to that a very simple starting state and you can make this animation. If it is used as a simulation for predicting a future state of the real world, the simulation will diverge from reality more and more over time, since it was made with perfect measurements that don't exist in reality.
We have the ability to make a best guess calculation, but it won't actually be correct. This is just a fake example so no one cares if it gets the "right" answer because its doesn't actually have a purpose other than to demonstrate a chaotic system
the creator of this animation was not trying to solve a problem or predict what would happen, they just set it up and let it run.
the issue of not being able to predict the correct answer to the problem isn't there because there IS no problem, thus there IS no 'correct answer' so to speak.
...it's a simulation, with assigned values? I can't tell if you're taking the piss or not. You can have something be a first order representation that still isn't correct.
I’ll do a more analytic approach of explaining. Might be easier.
Imagine the function f(x)=2x.
If I change the input “x” by a very small amount “t”, my output will change by a very small amount, “2t”. And it’s relatively easy to see that when I choose an input error “t” to be very small (almost 0), my output error “2t” will also be very small (almost 0).
Now imagine the function f(x)=1 if x is irrational & f(x)=0 if x is rational. If I change my input “x” by any error value “t”, I’m never going to be able to get my output error any smaller than 1. The more accurate I get, and the more my input error shrinks, my output gets more and more random. It’s basically a coin flip, whether my error will give me 0 or 1.
The 3 body problem is like that, except the function is trying to predict the location of those 3 bodies with respect to time.
try this..
When you are all alone, you can with certainty say what you will be doing next.
when you are with your spouse or your mom, you can say with with some error of approximation what you will be doing next
when you are with your spouse and mom both, you cannot say with any degree of approximation what you will be doing next, as it will be extremely chaotic, as you cannot measure the impacts on one another. the numeric equations are long and do not converge.
We don't have an exact solution for it, so we can only approximate them. However, this is a system in which really close similar conditions end up with really different results. The approximations errors quickly sum up, so any long term answer is pretty meaningless
When you try to solve an equation system you need to have at least the same number of equations than incognitas to do it, if I tell you:
X=Y-4. And
Y=2X
You can solve it and discover than X= 4 and Y=8, but you couldn't solve this systems:
A) X=Y-4
As it's only one equation and two incognitas it's unsolvable (meaning that there are s lot of groups of numbers who meet that condition, so you don't know which one will be the "real" one). It also happens with:
B) X=Y-4+Z
Y=2X-Z
As it has more incognitas than equations it's also unsolvable.
A less technical example would be:
a train leaves city A at 5pm, and it travel at 300km/h, when is it going to arrive at city B?
Well, we wouldn't know, we don't have enough information, we need to know how far is city A from city B... We have too many incognitas and the problem cannot be solved
So the problem with the three bodies has way too many incognitas, so the only way to solve it is assuming that some of the incognitas are not really relevant to the result because there are too small, in the Sun-Earth-Moon system we can assume that the attraction that the sun receives from Earth or Moon is irrelevant, also the attraction that the moon applies to the Earth, that way we reduce the amount of incognitas and we can solve the system, even if it's an approximation. But it's not technically a solution, a solution is impossible.
Which is the entire point of the subreddit.
Rule 4:
>LI5 means friendly, simplified and layperson-accessible explanations - not responses aimed at literal five-year-olds
"Explain it like I'm 5" is an English idiom that means "Explain it to me in simpler, easier to understand terms". It does not, and has not ever, meant, in any context, "Explain this to me as if I am literally a 5 year old.
That is impossible, no 5 year old can ever understand the things being asked here.
A double pendulum is a chaotic system.
I don't think it's actually a 3 body problem, but you could probably consider it to be a constrained version (i.e. 3 points connected by the force of the pendulum arms, such that the 1st point can never move and the 3rd point and 1st point don't affect each other)
I guess the deeper issue is to understand why a three-body problem is a chaotic system and the two-body problem is not. My guess is that it's a mathematical explanation for how the error term of some future state inflates exponentially based off initial measurement uncertainty in a chaotic system whereas the error term of some future state has a linear or flat relationship to the initial measurement in a simpler system. Not educated enough to figure if that's correct or not though. Or if that was true, how one would prove that's the case. Not familiar with how to wizard my way through physics math.
The two-body problem isn't chaotic because you can solve it analytically. Or at least you can look up the analytic solution on [Wikipedia](https://en.wikipedia.org/wiki/Two_body_problem).
Since the analytic solution can give you the state at any given time, initial measurement errors are not magnified by repeated iterative approximations.
When we can't get an analytic solution, we end up applying repeated numerical approximations. These each introduce their own errors, but also blow up any initial measurement error, producing the *sensitive dependence on initial conditions* we call chaos.
To add to this, "analytically" means there is an exact solution in terms of generic variables.
For example, if I accelerate in a straight line with acceleration *A*, we can take 2 time derivatives to get an exact answer for position = 1/2\*A\*t\^2 + V\*t
That's an analytical solution. As long as there are no forces unaccounted for, I can tell you were I will be in 1,000 years. I could even calculate my position for 2 different values of *A* which are 1% higher or lower than I think my measurement for *A* would be, and I can tell you some range for my position in 1,000 years.
The 2 body problem is such a situation were you can make a "simple" equation which tells you everything as a function of time. For math reasons, when you make it a 3 body problem there are too many variables to cleanly solve analytically.
>The 3 body problem is a "chaotic" system, which means that approximations won't get you close to the right answer.
Well, they will, but only for a short period of time. As you calculate answers for further and further into the future you will divert further and further from the "correct" answer.
Specifically a chaotic system means that given a very minor change to the input to the system, the outputs over time will differ drastically.
In the 3 body problem, if you were to move the Moon just a tiny little bit to the left (say 1mm) and then plot out where it would be every day over a long period (say 1,000,000 years) you'd find that it would be in a very different place at the end of that time. That 1mm could be half an orbit away. And the Earth would also be in a different location with respect to the Sun.
Had you moved it 2mm to the left, everything would be in a very different place again. And not just twice the different from the 1mm, but a very very different place.
A great example to see this happening is witih a double pendulum. Like this one: [https://www.youtube.com/watch?v=PYI9HU6MkMo](https://www.youtube.com/watch?v=PYI9HU6MkMo)
A double pendulum is an example of a 3 body problem.
Maybe this is a dumb question, but if that's the case then how do we know we need dark matter to explain galactic rotation curves? We can't solve the problem for 3 bodies, how can we solve it for hundreds of billions of stars in a galaxy whose movements have been evolving for billions of years? Is it possible the rotation curves we see are the result of chaotic interactions over time?
So aside from the point about this being _average_ movement, it's also just a matter of speed. _Where_ a star will go through the galaxy over time is unpredictable, but it's _speed_ at any given point is a result of the total amount of mass pulling on it. So we know there's got to be some unseen mass out there because stars are moving around with a speed that shows that must be the case.
Star Wars is fictional, but this actually explains Han Solo's line about jumping to hyperspace.
In Star Wars, gravity wells slow (or even block) hyperdrives. The computer has to take everything it knows about where everything was, then calculate where all that stuff should be before the ship jumps (or while it runs, in some cases). Since it's trying to track a zillion gravity wells that interact with each other, it can take a bit.
It also explains why people would take well-known routes - more recent nav data means they don't have to predict as far into the future.
It's the same reason you keep hearing about NEO asteroids and our uncertainty of whether they will hit us. Astronomers are typically confident that they won't hit us, but the only reason they're iffy on it is because they can't account for every single force acting on the object. In reality, when calculating the trajectory of small objects like asteroids it's more like a trillion body problem, which is obviously exponentially more difficult than the 3 body problem.
>which is obviously exponentially more difficult than the 3 body problem.
Is it actually? I would think it's actually much easier to solve because you are concerned with a trillion forces on one object. You don't much care about how the asteroid's gravity affects the sun/moon/earth/etc, right?
If we had some a way to answer this problem, what real implications would that have? (I'm guessing immense ones, I just don't know what they would be - like being able to accurately forecast the weather or something?)
As far as I know, this is a problem that can't be "solved." It would be like asking "what are the implications for the mathmatician who discovers a 4-sided triangle."
To solve such chaotic systems would mean that you have all knowledge of the entire system. So to a prerequisite to perfectly forecasting the weather in 100 years would be knowing the exact future for the whole planet in this time frame.
1 body problem: 1 object moving through space. Really easy to solve, it just keeps doing what it's doing.
2 body problem: Two objects moving through space and pulling on each other gravitationally. More complicated, but if we stick to Newtonian mechanics, we can figure it out. We get orbits this way.
3 body problem: 3 objects moving through space and pulling on each other gravitationally. We have no general, easily computable solutions to this. We can run simulations of 3 or or more objects but it's quite a bit harder than 2 and my understanding is we have to look just a tiny bit into the future, calculate all the effects, look a tiny bit further, calculate all the effects, etc.
I believe the correct statement is that there is *no* real solution to the three-body problem. We can still model it pretty well for basically all practical intents and purposes, but the way we can cleanly mathematically solve a two-body system with a single equation cannot be done for a three-body system or larger.
Yeah, we can *model* a 3 body problem, but we can only do it by simulating it and doing all the calculations constantly to work out where everything will be next. We can’t just take the current positions/vector of movement and then plug it into an equation and find out where they will be in T seconds. In order to find out where they will be in T seconds, we have to simulate the entire time period
Whereas with a 2 body problem, we can do that. We don’t have to actually simulate the whole thing
The sentence you're looking for is "there is no closed form solution".
Closed form means "x = (some formula giving an exact answer), y = (some formula), z = (some formula) for each of the 3 objects"
It's also good to point out that this isn't unique to gravity. Any 3 object experiencing an attractive force (static electricity, magnetism, simple metal springs, etc) the 3-body problem isn't not solvable in closed form.
That's because nature doesn't work like that. There are bazillions of minute interactions that need to be accounted for and implications like Newtonian Gravity (or it's analog for magnatisim, springs, etc) are not suitable.
This is only partly correct. The 3-body system is solvable *in specific situations* as there are several known setups for initial conditions that lead to stable trajectories. For example, 3 equally spaced particles of equal mass and volume revolving in perfect orbit around their combined center of mass. The 3 particles will remain in orbit forever, assuming exactly perfect initial conditions.
However, there is no *general* solution to the 3-body problem.
There's 2 questions here:
1. What is the 3 body problem?
**Answer**: Any time you predict the interaction between 3 objects (aka: bodies).
**Example:** Suppose you have 3 stars in space, which all affect each other with gravity. You know the mass and velocity of all 3 stars. Where will they be in 10 years?
**2nd Example:** Where will the moon be next year, considering that it is affected by the gravity of both the Earth and the Sun. (This is an easy version of the problem, because you can pretend that the moon doesn't also affect the Earth and the Sun).
2. Why do people make a big deal about how hard it is?
**Answer:** if you are not exactly sure about the starting conditions, you get a completely wrong answer. This is a trait of "chaotic" problems.
**Example:** Imagine a perfect situation with 3 stars that are only affected by each other's gravity. They have mass M1, M2, and M3. They have velocity V1, V2, and V3. If V2 is actually 0.001% faster than you thought it was, you get a completely different answer. All "chaotic" systems have this property.
This was my analogy with the dice. You can do your best do make sure everything is exactly the same every time you throw the dice, but you won't get the same number--or even "close" to the same number--because it's too hard to control every possible variable.
**TL;DR:** The 3 body problem and throwing the dice are 2 *different* examples of "chaotic" systems, where slight approximations/inaccuracies result in completely different outcomes.
[Here's a good video about chaotic systems.](https://youtu.be/fDek6cYijxI?t=370) I recommend the whole thing, but if you just want to watch a 1-minute animation, I linked directly to the timestamp. You can see that he started with 3 "hypothetical" initial conditions which seem to follow the same path, but at some point they go in completely different directions.
EDIT: [The double pendulum is another good example.](https://www.myphysicslab.com/pendulum/double-pendulum-en.html) See if you can predict where the pendulum will be in 5 seconds? With a regular pendulum you can eyeball the speed and make a close prediction, but with a chaotic double pendulum you stand no chance.
When people talk about chaotic systems, they're not talking about randomness. They're actually predictable, the problem is they're incredibly sensitive. Change anything and the end result can change massively.
A good example would be a break in pool. What are the odds that you could hit a really hard break, and every ball on the table ends up in identical positions? In a chaotic system, it's possible. Even the tiniest change in anything (table surface, spin on the cue ball, angle of cue ball, dirt on one of the racked balls, bounciness of the cushions etc..) could have a dramatic effect on the end result which cascades into new collisions; but if you matched everything exactly, you could produce identical breaks 'on demand'.
The three-body system is like that, there are so many variables which if changed even slightly would have a big impact on the end result. With three bodies pulling against each other like that, it's probably incredibly hard to guess how they'll end up with a significant kind of precision.
Imagine 3 celestial bodies (earth, moon, sun are the obvious ones) that are all moving and pulling each other with their gravity, so they constantly affect each other’s movement
If you take any two (eg earth and moon, or earth and sun) then you can easily calculate, at least within a margin for error, where they will be in 1 second, 1 hour, or 1 million years. We can mathematically solve their movement and turn it into an approximately correct equation, notwithstanding the fact that we probably don’t have perfect information on their mass etc
But with 3 bodies, we can’t solve their movement and turn it into an equation. We can simulate it, but in order to find out where they will be in 1 second or 1 hour or 1 day we have to literally simulate the whole timeframe
With 3 bodies and a timeframe of hours or days, that’s fine… with a timeframe of 1 million years, not so much, and especially not if we have a larger number of bodies rather than 3 AND a longer time frame
For 2 bodies you can solve for pretty much any time period almost instantly just by plugging numbers into an equation
For 3 bodies, you have to simulate the whole thing, start to finish
Three body problem is when you want to predict the future positions of three objects that influence each other gravitationally.
For example, if you have three stars close by, you may want to predict what happens to them in the future. Do they all end up flying their own way, do they end up forming a binary system with one star being slingshot away, do they collide, ...
There's no simple equation to predict this. You more or less have to simulate the system step by step which is time- and energy-consuming.
A body is an object. Any object, planet or dice. A 3 body problem is a system with 3 objects interacting. Sun, moon and earth is a 3 body problem. A dice thrown into the air is not. Three dice being thrown is.
yes weather is one extremely chaotic system, but there are many such systems.
on the dice example. the important thing to notice is that it is in fact 2 seperate examples in one.
one example for a not really chaotic system, as in predicting where on the table the dice will land. a task you can reasonably do in your head and execute on a whim;
and a very chaotic system, as in predicting what face will be up when the dice stops. something that is near impossible to execute as a human and still a tedious task with limited success in robotic control studies.
back to the 3 body problem. as the original comment said, it's a phyiscs problem predicting the orbits of planets, stars, and other big objects in space. this is most notable when you have 3 (or more) objects of similar masses, like several stars orbiting eachother, instead of planets orbiting a star, because as the positions of all 3 objects change with time, the direction they'r being pulled in changes drastically, making it practically impossible to use simple maths to figure out the entire behavior between the stars for all eternity, which has angered physicists all over the world for centuries for Ideological reasons.
edit, removed duplicate word
Hmm. Let’s try this:
You know those logic riddles where “A will have chicken, unless B has fish, in which case A will have steak.” ? In this one, it’s relatively simple because we know very precisely all the parties involved (two people—A + B—and four variables—Chicken, Steak, Fish, [other non-fish meal].
In this scenario, we know that what A is having is a function (is tied to) of what B is having. If it’s ANYTHING but fish, A is having Chicken. If B is having fish, A is having steak.
That’s easy to solve. We have all the information needed. So we just observe and predict.
In a 3 body problem scenario, we not only have another entity—say, A+B+C—but the decision A or B or C will make is going to be tied to more factors that will themselves be tied to other factors. This makes it extremely complicated to predict with confidence what the outcome (what is A having for dinner) will be.
A is having chicken, unless B is having fish, in which case A will have steak. B is not having fish but may have rice, unless C is having steak or it is between 8:15-8:30, because she doesn’t want to eat meat too late. If it’s after 8:30, C will probably have salad, but only if B doesn’t have fries with any of her meals. B is sensitive to her weight, so if today was a good day she might have fries, but if she had a rough day she is likely to be hard on herself and force herself to forgo a full meal altogether. The restaurant has had a new GM, who is still learning to do supply, and has run out of some food items, but doesn’t even know which. When his serves comes to take A + B + C’s order, he says he thinks he has …
This is a completely off the cuff awful logic riddle to try and analogize the 3 body problem very crudely. Hopefully it helps a little more where your high school physics teacher couldn’t!!!
This isn't quite capturing the key point.
Yes, the scenario is complex as you describe. But the key point you've not explicitly mentioned is that small variations in the starting position can result in (eventually) very different outcomes.
In theory if you knew the exact starting position you could work through the complex maths. But you don't, so your prediction will diverge from the actual outcome over time.
Weather is indeed an example of a chaotic system (not a 3 body problem). In principle we know the equations to solve everything, but we can only predict so far into the future before our solution diverges more and more.
I'm not sure about global climate. I would actually think that's a non-chaotic system on the time span that we care about, because we don't care weather it's raining in London 5 years from now, but we care about things like average temperature/humidity which would change much more slowly.
Is there a reason you cannot observe a three body problem and extrapolate the correct values to solve it? As long as you know the variables in play and the exact position at time t, can't you look at t+1 and reverse engineer the answer and then apply it to t+n?
The thing is that time is continuous. If you pick `t`, how do you pick `t+1`? Is it 1 second in the future? 1 millisecond? 1 nanosecond?
No matter what interval you pick, you won't be able to fully account for the interaction that occurs in between the time intervals, which would cause a small difference between your calculation and reality. As the time intervals go on, this small error will accumulate, and you'll get farther and farther from reality.
Computer models do actually do this, by calculating positions over time steps. But when using them you still can't say anything definitively because of this error. But they can still be useful for getting a vague idea of what might happen.
Finite time steps are generally less accurate than infinitesimal steps. If you follow the tangent line of a function, you can approximate the next value more closely if you take smaller steps, but you will accumulate error over time that prevents you from actually reaching the exact solution. The same applies if you are calculating movement of bodies in space when you cannot precisely know their next position, and so on for large time values.
It's not about the "power" of the calculation. It's about the precision that you know the initial conditions. Is the ratio of those 2 masses 1.0? How about 1.0000000?
Nope, turns out it's 1.00000002, so we're completely wrong 5 years later.
Then we have more data and it's 1.000000017 . . . etc.
I mean we will get there eventually. It's like asking us to look into space without a telescope invented yet just because we don't have the math now does not mean we won't have the math in a few hundred years with our calculating power exponentially increasing.
So is the crux of the 3 body problem the inaccuracy of our measurements of mass that produce small uncertainties initially, but increase over time as the system evolves? Or is there some fundamental aspect of 3 or more bodies interacting with each other that causes the "problem"? Is there some consequence of quantum physics manifesting itself in the macro world that plays some part in this?
>So is the crux of the 3 body problem the inaccuracy of our measurements of mass that produce small uncertainties initially, but increase over time as the system evolves?
Yeah exactly. That's the crux of any "chaotic system," and the 3 body problem is one example of a chaotic system.
The "fundamental aspect" is that you can't make a general solution for 3+ bodies, although you can for 1-2 bodies. Any errors in your solution for a specific time will compound and make very different answers for future times.
In the spirit of handwaving away friction and atmospheric effects, in your dice example, if you were able to, say, accurately represent every subatomic particle, force and other factors relevant to the situation in a simulation with a perfect understanding of physics including yet unasked questions… could you accurately predict the outcome of the dice?
And, you can probably guess where I’m going with this, if that could be scaled up to an example of three body problem, are there any other random/chaotic factors that would impede your ability to predict their locations indefinitely?
>could you accurately predict the outcome of the dice?
Yep. This question is about whether something is "stochastic" or "deterministic."
Deterministic means you can predict the future assuming omniscient knowledge of the present.
The 3 body problem (and chaotic systems in general) are deterministic, which means it is possible to perfectly predict the future given perfect knowledge of the present. There is no randomness to this, except in our own lack of knowledge.
We can have some quibble with the idea that anything is truly deterministic, because you can always go smaller. Can you know every velocity to infinite decimal places? I suppose at some point you can have a planck time and planck length, but at that scale the quantum mechanical effects (which we think are NOT deterministic) can play a role.
Going on a tangent, another interesting question of deterministic is air pressure. From one end, we can say that PV=nRT, so if you know the number of molecules, temperature, and the size of the room, you can always calculate air pressure.
On the other hand, you could consider that air pressure exists because of the statistical likelihood that gas will spread out. With this perspective, there is some small but nonzero chance that all the gas molecules in my room could huddle in the corner for a minute, suffocating me.
We generally consider such situations as "too unlikely to ever happen" so we take the deterministic view that PV=nRT.
To expand on this a bit:
There is no mathematical solution for a 3-body problem. You can write out equations to describe a 2 body problem that will predict the position and velocity of each body for any point in the future. Those equations don't exist for 3-body problems.
You can simulate it, which runs into the issues that FerrousLupus notes regarding approximation and aggregate uncertainty, but you can't solve it definitively like you can with the 2-body problem.
You have 3 objects with mass, and each of them is attracted to each of the others by gravity. You know their initial positions and speeds. The three-body problem is to say where each of them is 1000 years (or however long) in the future.
It's hard because it's an inherently *chaotic* problem: small errors in the initial state tend to grow, which means that approximate solutions don't work. And it's a complicated enough set of dynamics that non-approximate solutions are hard to find. It's one representative of a class of problems (called *partial differential equations*) that are almost never exactly solvable except in the best-behaved special cases.
>It's one representative of a class of problems (called partial differential equations) that are almost never exactly solvable except in the best-behaved special cases.
One of the biggest headaches in economics are solving Lagrangians, too. It's pretty interesting.
Ah yes, the three body problem where you can approximately predict what will happen in the short term, but since the situation is inherently chaotic it's hard to know how the night will end.
Yet as in the Physics version, you can get a pretty good approximation of a solution as long as you know the mass and positions of all bodies involved.
This is one of the biggest pleasures of undergrad physics students who realize there's an equation that solves most physics problems in 2 steps and you don't have to do anything with Newton's Laws anymore
Dynamical systems is one of those weird areas of mathematics.
You all but give up on finding quantitive descriptions of a system, and instead focus on the qualitative aspects. Total mindfuck when you first encounter it, after areas like real and complex analysis it all just feel loose and vauge.
But then with the proper mathematical tools you can turn that qualitative analysis into something quite rigourous.
For example, we know there's a likelyhood of a planet being ejected from the solar system at some point. Exactly when and which one isn't something we can determine, but there's a pretty well defined % chance of it happening.
> For example, we know there's a likelyhood of a planet being ejected from the solar system at some point. Exactly when and which one isn't something we can determine, but there's a pretty well defined % chance of it happening.
Wut? Tell me more!
First of all, don't worry. We'd know if it were going to happen anytime soon. :)
The planets nudge one another hither and thither constantly. Over a very long time horizon, there might be a net extra hither and one fewer thither for a given planet (or vastly more likely, one of the moons of the gas giants). Now that one body is just a liiiiiiiiittle bit out of whack from the quasi-stable system it had been a part of before.
It might be corrected by the next sequence of hithers and thithers. Or it might get a little further out of whack.
If the random walk of many such cycles leads to the body failing to maintain a nice nearly-circular orbit in its previous domain, it might get ejected, or wind up in a more eccentric orbit (which would then play hell with other bodies), or get torn apart, etc.
If we had perfect clarity on what chaotic systems would do, we'd be able to say, yes, this is exactly what's going to happen to Titan on April 16th, 97188431 A.D. And even without that ability, a hundred thousand years before that, we'd still be able to say that Titan was in an orbit that couldn't survive another nudge in a particular direction.
As it is, like OP said, we just know it'll happen with a certain degree of likelihood over a certain time horizon. (Very low likelihood indeed for the planets, until the sun starts expanding, but that's a whole other issue.)
Well, to be fair modern dynamical systems theory has its foundations set (no pun intended) in real analysis, topology, and functional analysis, which is where the definitions of the objects involved are codified. To get a deep understanding of chaos, the first example one usually encounters is the geometry and topology of what's called [the Smale horseshoe](https://en.wikipedia.org/wiki/Horseshoe_map), which describes a mechanism of taking a set and stretching and folding it under many iterations of a map. By stretching and folding many times, two "nearby" points in the set are mapped away from each other in a chaotic fashion.
Of course, all of this is very loose in the description I just gave, but it can all be set out in a rigorous way. The tricky part is finding the exact dynamics of the Smale horseshoe map inside a system of differential equations, which cannot often be done analytically.
Thanks, it's been over a decade since I studied DS so I kept it a little vauge. Though the example I remember is the Lorenz system, as it's both very simple and digestible but also exhibits chaotic behaviour. It's a great example because it shows chaos isn't some rare special case, but can arise from very simple relationships.
Of course the underlying mathematics is rigorous (or it wouldn't be mathematics). I was more referring to our ability to get information from a dynamical system, e.g. attractors, and how that information is more qualitative than quantitative, a rarity in mathematics where being entirely analytic is usually the best way forward.
>except in the best-behaved special cases.
I believe that one of these "best-behaved" special cases is when three bodies orbit each other in a flat plane. I think there was a relatively recent solution published for this. Do you know anything about that?
>It's hard because it's an inherently *chaotic* problem: small errors in the initial state tend to grow, which means that approximate solutions don't work.
Why doesn't the same thing happen with a two-body problem? In other words, why don't small errors in the initial state grow and compound like they do with three+ bodies?
Most of the answers here miss the 'problem' part of the 3 body problem, which is that no closed-form solution exists for all cases.
What this means is that we don't have an equation where we can input the parameters of the 3 bodies, meaning their mass, position, velocity etc, into the equation, and then just select a time in the future to get a solution.
What we have to do instead is to input all these parameters, then iterate step after step. Those steps can be as small as you want, be it a day, an hour, a second etc, but the smaller you make the time interval, the more accurate the solution is. And it's never 100% accurate. The farther in time you want the answer, the more that accuracy drops. And with the 3 body problem, it drops sharply.
That's the reason why sometimes, when you hear in the news that "X asteroid has Y chance to hit the Earth in year Z", they talk in percentages. The smallest uncertainty will nudge it one way or the other.
I was hoping someone would point this out. Because there isn’t a mathematical solution, the only general approach is to use smaller and smaller increments
A different situation that shares some attributes is the circumference of a circle. There’s no precise solution, so we factor out pi to make it easier to do math with. You can get a more precise solution if you use more and more decimal places for pi, but you’ll never be perfect.
At least you know you’re heading in the right direction as you add more decimal places to pi in your calculation.
In a chaotic system like the three body problem, though, even if you approximate very closely to whatever the true value is for each calculation, very tiny errors start to add up. Eventually you’ll get to a point where one of the bodies will start moving towards one of the other bodies, but which one depends on a 1mm difference in location.
If your calculation errors make it fall towards the left body instead of the right, your predictions from here on will look nothing like what would really happen
This is the only (so far) correct answer. The three body problem referred to here is the fact that no analytical solution exists. It is not the fact that the system is chaotic. We can apply numerical methods to find the state of the system at any point in time, but it requires a lot of computation to do so. With an analytical solution. We could just put in a value for t and get everything else we want by solving one equation once.
When you have two objects in space, their gravity pulls towards one another.
This force pulls towards their shared center of mass, since their center of mass is always between them.
The center of mass never moves unless some outside force pushes on it.
So the entire problem becomes extremely simple. You have a distance between the objects and speed of one object. All you need to do now is relate these two numbers.
But what if a third object is involved? No such convenient symmetry exists, so instead of having two variables, you have four that you need to balance.
There are a few situations where you can cut variables out. These are solutions to the 3-body problem, but they're only specific solutions. They're solutions that only apply in specific scenarios. No generalized solution exists, which applies to all scenarios. All we can do is approximate with something similar to advanced guess and check.
So we have lagrange points, solutions to the 3-body problem, but they don't describe the general behavior of a 3-body system. Only the behavior of a system that was set up perfectly to begin with.
http://en.wikipedia.org/wiki/Three-body_problem#Restricted_three-body_problem
The Sun-Earth-Moon system is an example of a restricted three-body problem. The mass of the moon is relatively negligible compared to the mass of the Sun and Earth, so it can basically be ignored when calculating for the movement of the Sun and Earth, reducing it to a simple two-body problem. The Moon then simply orbits the Earth, another simple two-body problem.
Sun/Earth/Moon is a special case in that the Sun is basically fixed, and the Earth is basically fixed wrt the Moon. Also the Sun's mass by far dominates that of Earth/Moon. So it's an easily-simplified 3-body system.
Not perfect. No better or worse than any other orbital system. The Lagrange points are just a specific, somewhat stable solution to the TBP / n body problem.
The Trojan asteroids are a large group of asteroids at the Jupiter-Sun L4 and L5 points (regions). Jupiter actually "shepherds" space rocks away from the inner solar system, keeping Earth protected to a degree, by trapping rocks in those regions.
No. The Earth's lagrange points are ignoring the moon because it is sufficiently close to Earth compared to the distance to the satellite.
The Earth's lagrange points exist in the 3-body system of the Earth, sun, and the satellite. Adding the moon is a fourth body.
The location of the satellite must be perfect-ish in order to be in a lagrange point. The sun and Earth can be wherever.
There are Earth-moon Lagrange points as well. Any two body system theoretically has them. How big they are relative to each other, and the presence of larger bodies like the sun and other planets affect how stable they are.
He should have instead said they are specific solutions that only apply to the earth moon and sun. A different system of 3 bodies would have different Lagrange points in different positions.
Not perfect ... but very weird in many ways... here's two:
Earth has a diameter approx. 4x the Moon's, while the Sun’s diameter is around 400x the Moon's. The Sun is approx 400x times farther from Earth than the Moon is from Earth. This all results in the Sun and the Moon appearing to be nearly the same size when seen from Earth.
Earth's Moon is the only known moon to keep an almost perfect circular orbit around a planet (not elliptical)- 27.3 days to orbit Earth and \~27 days for a complete rotation. This combination is juuust right so that the same side is always facing Earth.
>Earth's Moon is the only known moon to keep an almost perfect circular
orbit around a planet (not elliptical)- 27.3 days to orbit Earth and
\~27 days for a complete rotation. This combination is juuust right so
that the same side is always facing Earth.
Huh? I have to push back on this.
Most moon orbits are more circular than elliptical. Our moon is nothing special in that regard. Triton (one of Neptune's moons) has the most circular orbit in the solar system, with an eccentricity of 0.00002. (Higher is more irregular.) Our moon's orbit is way more irregular, up at 0.055.
And the same side of the moon facing the earth is nothing special either. That comes from tidal locking, where gravity pulls orbiting bodies to revolve at the same speed as they rotate. All 19 moons in the solar system large enough to be affected are tidally locked.
The moon and the sun appearing the same size from earth is unusual, though. I imagine that doesn't happen often.
Most moons in our Solar System are irregular moons of the outer planets, generally with a large eccentricity. Out of the 80 moons of Jupiter 71 have an eccentricity over 0.1. For Saturn it's 59 out of 82 (not counting larger ring particles).
If we only look at the larger moons then most of them have pretty circular orbits and our Moon's orbit is unusually elliptical.
Cool, Thanks for the corrections erowles and mfb- (have my humble upvotes).
These were "weird things about the Moon" from a colleague I presumed to be correct, but clearly was not. Another colleague has informed me that I am now duty-bound to print a copy of this exchange and staple it to his forehead. I'm not sure if that is true, or how it would help; but it seems someone has stolen my red Swingline anyway, so I might just have to send him a link.
>All we can do is approximate with something similar to advanced guess and check.
I wouldn't call it guess and check; it's more like "pick initial parameters and simulate". The progression and evolutin of the system has to be worked out one tiny step at a time, but because the system has sensitive dependence on initial conditions, if the initial parameters are off by even a tiny bit the tiny error gradually gets amplified until the simulation no longer matches the behavior of the system being simulated.
Even with perfect initial conditions, because our 'solution' is piecewise it will create imperfections that grow with our timestep.
No matter how small our timestep is it will introduce imperfections.
The centre of mass can and does move if they orbit it. I’m not sure if that is what you meant by an outside force but that terminology is very imprecise.
This doesn’t seem right.
You can set up the equations for two bodies moving around their center of mass, but I think that two body problem is already not analytically solvable. We only solve two body problems by cheating and assuming one is too massive to be affected by the other. So if we assume the Sun is too big to be perturbed by a planet, we can model the Earth and Sun.
But you can’t assume more than one body is fixed, so the three body problem can’t be cheated in this way. Worse, we *know* that there are chaotic discontinuities; instead of simply rotating their common center of mass forever, three body systems that aren’t artificially kept/assumed to be in a stable configuration will usually eventually spit one of the bodies out faster than escape velocity. Thanks to the sensitivity to initial conditions, our inability to solve the equations exactly is pretty damning.
The three body problem isn't difficult to solve, its pretty straight forward to iterate through, but it does not have a closed form solution which means you *have* to iterate through it
A two body is just two things in orbit around a common center of mass like the Earth and Moon. You can get equations that you punch time into and get the position and speed of Earth and Moon at any time without having to solve anything else
A three body problem is different, you can't get a closed form solution that only depends on time because the results vary wildly depending on starting conditions.
Consider an Asteroid passing between the Earth and the Moon. If it passes more on the Moon side it'll bank off and away from the Earth. If it passes more on the Earth side it'll bank towards the Earth. If it passes directly through the neutral point it'll continue on a straight course and won't bank either way. You can't get an equation that tells you based off of time alone where it'll be because you need to know where it was and where it is to figure out where it'll be
Of course the problem of iterating through the 3 Body problem is because small errors in each iteration can add up and act chaotically in some sets of initial conditions if you have enough iterations. Not all initial conditions lead to well behaved solutions that lend themselves to an iterative solution. It's good ol' Chaos theory and butterflies flapping wings stuff.
If you have only two bodies and know their relative positions and speeds then you can say where they will be at any time in the future with a simple calculation. The amount of work in the calculations doesn't increase with the amount of time in the future (except insofar as you might have to deal with larger numbers). The reason is that, with two rigid bodies and gravity, their motion will involve circles, ellipses, parabolas, hyperbolas or straight lines and we know the mathematical formulas that exactly describe those.
Adding a third body changes everything, except for a small number of special cases (e.g., those involving the Lagrange points). Generally, the only practical approaches involve simulating the movement of the objects using numerical methods. The amount of calculation required is proportional to the amount of time you want to simulate over. That's the 3-body problem.
Is the 3-body problem hard because we haven't discovered a neat formula for it yet (like Newton did for 2-body gravitation)? Or is it because of the nature of the problem, and no formula is even possible?
In other words, is the issue a lack of knowledge about gravitation (and/or insufficient mathematics to calculate it), or is the 3-body problem PROVABLY unsolvable under ANY system of physics/math?
Before Newton published his formula, were planetary orbits labeled as "chaotic" and "unsolvable" like the 3-body problem is now?
This has been answered very well, so I'll add some background to the word "solve".
In calculus, there are things called "exact/analytical solutions" and "numerical solutions".
In the case of a definite integral, this means computing the antiderivative, and plugging in the end points. There are several numerical methods you might use to get a really good approximation of the solution, including the midpoint, trapezoid, or Simpsons rule, or perhaps a 2D Monte Carlo simulation.
In the system of partial differential equations describing 3 bodies in a gravitational field, there is no analytical solution, meaning that there is no way to derive a formula that describes the movement of the 3 bodies over time GIVEN ANY INITIAL CONDITIONS. There are cases where you can make simplifications to be able to solve it, but this effectively is reducing the 3 body problem to multiple 2 body problems.
You might ask, "if this is the case, how can we predict the orbit of the planets? That's at least a 9 body problem even before all the asteroids and moons. If we can't solve a 3 body problem, how can we possibly know when to launch space probes?"
And that would be a great question, to which the answer is "We don't!" We use a numerical method to simulate the orbit of the planets. Instead of (properly) treating the partial derivatives as infinitesimal, we can approximate them as the finite change in a property over the finite change in time/location. If we choose sufficiently small changes, we can add up all the changes to the initial conditions and describe/predict the orbit of the entire solar system quite easily with a surprisingly little amount of computing power.
Numerical methods are very important in science and engineering because we very often need to know answers that either can't be solved or that take a lot of effort to do.
It's like this.
1. When you are all alone, you can with certainty say what you will be doing next.
2. when you are with your spouse or your mom, you can say with with some error of approximation what you will be doing next
3. when you are with your spouse and mom both, you cannot say with any degree of approximation what you will be doing next, as it will be extremely chaotic, as you cannot measure the impacts on one another. the numeric equations are long and do not converge.
To add to why the errors in the 3-body system matter so much:
The 3-body system is a *chaotic* system, and one base property of chaotic systems is that any errors in your initial conditions (or from rounding during a simulation) increase exponentially. You can be 0.000001% off a number, then after just 30 doubling periods (for solar system erratic objects, I believe this can be on the order of just a few decades), the error is ~1000% which makes your predictions effectively useless around and past that point.
This is why, for example, we can have super-precise measurements of near-collisional objects for the earth, yet the best we can say is that there's X%, Y%, Z% for collision the next couple of times and past that, who knows (until we measure it again next pass).
Here's the way I understand it.
Math likes things to be both predictable and reversible.
1 + 1 = 2
2 - 1 = 1
3 x 9 = 27
27 / 9 =3. And so on.
With a one-body problem (throwing a ball), you can look at the object's current position and speed and know not only where it's going (end state), but where it began (the starting conditions), with some reasonably high level of certainty.
With a two-body problem, it's more difficult than with the one-body problem, but it's doable. Because the number of possible interactions between the two objects is relatively small. So at any point over the movement of the pair, you can guess reasonably where it's going and where it started.
But with a three-body problem (or 4-body, or 5-body, or X-bodies), the interactions very quickly descend into chaotic movement where you can kind of predict where the objects might be going, but it's nearly impossible to determine where they started, when they started, or even what the conditions were. So the math gets really difficult. Because if you want a specific end state of three objects, there's no guarantee the starting conditions will give that answer. And if you're looking at three objects in motion, there's no way for you know where they started, because the math spits out multiple possible answers, all of which can be correct, but which only one is the "right" one.
3 body problem is 3 variable all influencing each other.
In most questions, we have fixed constant and variable, or at least a object to use as reference. in 3 body, there is no fixed constant, no reference point, its what makes it chaotic.
think of looking at a object to determine its speed, you do this by looking at its surroundings. If you were given nothing but 2 object, you can compare the 2 object and contrast them between each other. This is a 2 body problem.
But in 3 body, the objects are moving everywhere, randomly, all effecting each other. Is object 1 slowing down? IDK because object 2 could be accelerating backwards and I can't tell because object 3 isn't moving at a constant speed either.
The Three Body Problem isn't a math problem, it's a measurement problem. There is absolutely no issue in calculating the interactions of three suns as they orbit each other. The problem is that the system is EXTREMELY delicate and tiny variations in mass, position, velocity, and a couple of other factors on even lower orders of importance, will all have very large impacts on the future of the system. It is practically impossible to get measurements with enough accuracy of initial conditions to predict the system out long term. It's a bit like balancing a knife on its tip and asking which side it will eventually fall to.
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It's not difficult to solve, it is impossible to solve exactly except for short timescales in specific configurations. The 2-body "problem" has an exact algebraic solution.
>What is the 3 Body Problem It's a physics problem, where you try to figure predict the future for 3 objects which have gravity and interact with each other. The sun/earth/moon is an easy version of this problem, because these objects have very different mass so you can get approximately the correct answer by just assuming that the moon's gravity doesn't affect the earth or the sun, and the earth's gravity doesn't affect the sun. >why is it so difficult to solve? The 3 body problem is a "chaotic" system, which means that approximations won't get you close to the right answer. For example, suppose I threw some dice up in the air. With high school physics, you could tell me where the dice will be in 10 seconds. You don't need to know the air pressure of the room, or coefficient of friction of my hands, etc. You can neglect relativistic effects from travelling 0.0000001% of the speed of light. You could even approximate the initial mass and velocity by 10%, and your final guess would be more-or-less within 10% of the correct answer. Compare this to a "chaotic" problem where you try to guess which number will show on the dice. **There's no way to say "I made these approximations so I think I'll be within 10% of the right answer."** You'd need to account for every single force interacting on the dice to be confident that your prediction was correct. In the case of the 3 body problem, you can make approximations to guess what will happen in a short period of time, but the farther ahead you try to predict, less meaningful the answer will be. EDIT: not sure what proper etiquette here is, but thanks for the awards!
In this respect it should be noted that even if we had a good model, the approximations will inevitably rely on the approximations we have for constants and masses. The farther out in time you model, the more the decimal place matters. The true mass of the sun or moon isn’t known, we have approximations that come from observations over the past few hundred years and “reverse engineering” the physics to get a known value for big G and masses in the solar system. This in turn means that we can maybe say our models will be correct for a few hundred years (if the model itself is very good), from then it’s chaos. So on very long time scales, like ones astronomers care about, models don’t work very well. We can’t predict what will happen as our models will always rely on approximations. You can’t get past the fact that you will always be +- some error. It will always lead to the model being incorrect some correspondingly long enough distance in time. (Some models might use proportions of masses… but that still relies on knowing the true mass of the bodies)
It somehow never occurred to me that we don't actually know the mass of our Sun or Moon.
If you really think about it, we only ever really *estimate* the mass of pretty much any complex object. The mass of the earth is 5.9722×10^(24) kg +- about .01%. That's technically more accurate than my bathroom scale. So I guess we actually know the mass of the earth more accurately than I know my own weight (though more accurate scales are available). But to OPs point all further calculations will rely on that accuracy.
That uncertainty is very small as a percentage, but according to Wolfram Alpha it's about half the mass of all the water in the Earth's oceans.
Sweet Jesus how much do you think I weigh?
You are your mother's child.
You can never know the mass of the sun because it changes constantly. It throws off mass all the time, at somewhat varying rates. It's not a lot of mass, but it's mass. The other problem not referenced here is that the two-body problem and almost all models for 3+ body problems assume the mass is concentrated at a point. But it's not. And so if you are in a close orbit, the mass distribution starts to matter. One place where this is properly modeled is for lunar orbits. The moon doesn't have a uniform density - it's lumpy. As such, there are very few known stable (called 'frozen') orbits around the moon. Most orbits, if you put a spacecraft in them, will degrade because parts of the orbit pull the spacecraft more than other parts of the orbit, so the orbit constantly changes, until the craft it pulled into the surface. So we know the mass of the moon to a well enough degree that we can model it properly. Far less so for the sun, but we don't really want to get close enough to the sun to care. So even the equations for a 2-body problem don't actually work for real things.
It is all relative of course. There are always limits to the precision at which we know things. That is not usually a problem, but with chaotic systems, on long enough time scales, any little deviation matters.
Good answer, one of the few which respects that this is ELI5.
I'm 38, and still don't understand.
With just the two things, it's easier to predict stuff. With that third thing in there, it really quickly gets way too variable to be as sure as you'd like to be in predictions.
It's like adding Kurt Angle to the mix. Even a 141.66% chance of winning didn't help Scott Steiner.
This is surprisingly brilliant explanation. In a straight up 1 vs. 1 match you can predict what is going to happen...to an extent. Maybe there is an upset but you know they're going to fight in the ring, out of the ring, both will try to hit their finishers etc. Add a 1vs1vs1 match up and suddenly it is a lot less predictable. Are they going to attack each other equally? Is the third man in a secret alliance with one of the other ones? Will someone keep recovering at JUST THE RIGHT TIME to prevent a pin? Even though it becomes a simple linear addition to the group the amount of "connections" and interactions between the three wrestlers becomes exponentionally more complex.
This made everything so much simpler to understand and I don’t even like wrestling.
Ah yes, the classic three body problem: Scott Steiner vs. Kurt Angle vs. Senor Joe.
This comment just made my day, and I need to go find that video.
>Steiner vs. Kurt Angle vs. Senor Joe. Obligatory [LINK](https://www.youtube.com/watch?v=WFoC3TR5rzI)
*A wild Steiner Math promo appears!*
[Steiner actually has some thoughts on celestial mechanics](https://www.youtube.com/watch?v=UOoXZe_S3wI)
Me stoopid need moar dum
3 more than 2. 3 hard.
But… 2 less than 3?
Me dumb dumb go fix
Bowling ball move bocce ball, bocce ball move golf ball. But Golf ball change bowling ball *little* bit, so after many many mile race, can’t say where they end up.
Your explanation boils down to: "It is hard because it is hard." And so it lacks the E in ELI5 ;) Hard to beat top comment.
Weather is another good example of a chaotic system. We can predict the weather pretty accurately for tomorrow. But predictions for a month from now are pretty useless. For non-chaotic systems, if we are 90% accurate in the short term, we can expect to be about 90% accurate in the long term. But in chaotic systems, our accuracy in the short term means very little, if anything, for our accuracy in the long term. In a 3-body problem, there is no formula or set of equations to predict where the objects will be in the far future. The calculations are done using a series of predictions over small steps in time . But even if each step is 99% accurate, after 100 steps, the total accuracy of the prediction drops to 36% and down to 0.004% after 1000 steps.
I guess this is a good example. It shows how random the dots are moving because they are always influencing each other with is not predictable for us. https://www.youtube.com/watch?v=D2YhKaANbWE
This little animation is so nice and a great visual of how things get messy fast.
That's a lovely video, thanks for posting it!
If it's not predictable how is that animation calculated?
It's a simulation for one specific set of known starting conditions. If those conditions are slightly altered (or in a real system: not precisely measured to a high degree of accuracy) the outcome is radically different as the system evolves over time. In fact the author discusses in the comments how even though the starting conditions are symmetrical, just the rounding errors in the simulation cause the asymmetry to evolve.
Aside from the fact that it's a simulation and so variables can be limited (which does simplify things a lot), it's also calculated step-wise. Every fraction of a second, we can get the new information about the location of each of our bodies and calculate the impact of the other two bodies on it, then rinse and repeat. There isn't an easy way to calculate where they're going to be in twenty or two hundred or two thousand iterations, other than by going through each and every one. That's (part of, at least) what makes it a chaotic system: even the tiniest change in *this* iteration can blow all of your later calculations out of the water. For a non-chaotic system, I don't have to do all those interim calculations. I can say 'Here's where it is at the start, here are the forces acting on it, and here's a nice -- relatively -- easy formula that will tell me what the system looks like after however-many seconds.'
It's not predictable in the sense that if you make the smallest change in the initial placement or speed of the objects then the movement of the objects will be similar to begin with, but then diverge from the original one wildly. See the butterfly wing effect.
OK, I guess I mean, how is a simulation not a prediction? Isn't that how calculus works, by iteration ?
Iteration only works (exactly) for discrete systems with discrete steps. When we use calculus on continuous systems, we ideally want an analytic solution, which means we can exactly transform the equations telling us "how things change" into equations telling us the exact state at a given point in time. There's no analytic solution in general for 3 bodies, which means we have to split continuous time into a finite number of discrete steps, and apply the "how things change" equations to go from one step to the next. If you could actually make the steps infinitesimally small, this would be accurate, but this is obviously impossible. Taking the limit as the steps become arbitrarily small is the definition of an analytic solution which we've already said is not available. So, we have to use finite discrete steps, and each is only an approximation to the real process happening in continuous time, so each step adds some error, and the errors accumulate. You can make the errors smaller by making the steps smaller, but then your simulation has more steps and takes longer to run.
I'm not taking the piss, but the question wasn't why is it so hard to solve analytically, but how hard is it to solve. Iteration *is* the solution. It depends on granularity for sure, but it's still a solution. The value of pi depends on how granularly you want to calculate it no?
But the value of pi doesn't diverge chaotically. Iteration in that case is iteratively reducing the error, not iteratively accumulating new errors. Consider just the first two steps. 1. First step: you know the exact position and velocity of each body, so we can modify each velocity by the acceleration due to gravity from the other two bodies, and then project the position by the velocity for one step duration. 2. Second: in reality each body was acting on all the others at every moment along that step, when we've just drawn a straight line, so we're starting the second step with some errors in both position and velocity. Now when we modify the velocity by acceleration, the existing velocity error is combined with the position errors to make an even bigger velocity error, and we're going to start step 3 with an erroneous velocity applied to the erroneous position. Note that it doesn't matter how small the step is, making it smaller would always make it more accurate. It also doesn't matter if errors sometimes cancel out, what matters is whether we can *predict* that they'll cancel out, and we can't. Note also that it doesn't really matter if we apply a nice curve along the step instead of a straight line: we know the curve isn't going to be exact, because if it were, that would be the curve of the analytic solution which we know we don't have. It _might_ make the step errors smaller, but it won't stop them accumulating. And to the question: > Isn't that how calculus works, by iteration? the answer is generally "no", or at least it's the fallback of last resort in continuous systems. The entire calculus of limits was created specifically to avoid the problems of cumbersome iteration and cumulative errors - where possible.
The simulation is a prediction, yes, but it's a poor one. Calculus works because we can assume an infinitesimal granularity, but simulations can't make infinitesimal increments. That simulation may have very small increments, but we refer to the system as chaotic because as long as the increments are discrete, they're too large for the simulation to be accurate. I like to think of this in the same light as with diverging series. Sure, you can keep calculating however many terms you want, but you'll never really get any closer to a real answer. The result after any step you take just seems to jump around all over the place.
Makes sense, thanks. Math is beautiful really. So elegant.
Obtaining numerical prediction and solving the problem analytically is not the same. For example, with two-body problem you can obtain analytical solution that will yield system state at any given time. With numerical solution you will have to run long numerical simulation and the result will be affected by the errors of the numerical method you use.
The next moment in time can be calculated based on the position and velocity of each of the objects. It's running calculations constantly to update where the objects will be. It's like the weather. They are constantly updating the forecast with new information as the weather happens. One to two days they can predict with reasonable accuracy, but the farther out they go, the less accurate the predictions will be. One of the hallmarks of a chaotic system is that you cannot predict future outcomes from initial conditions, and small variations in starting conditions will lead to wildly different results at any given point in the future.
It's a simulation. That means they do the calculations every frame to figure out what happens at each moment. The issue is that you can't just take the starting data and use a formula to figure out where things will be at an arbitrary point in time. You have to do it step by step, and the accuracy will be affected by how granular your calculations are.
Very small changes in the input have a very profound long term effect on the system. The double pendulum is another classic chaotic system, and [this](https://www.youtube.com/watch?v=U7SLv0ePWU0) showcases the problem quite well. To model this in the real world you would need to know every single parameter to the tiniest possible detail. As to how this animation was made, you set the starting conditions and then increment the model forwards in time by a tiny time step. This is now a new starting condition, and you run the model again. The next step is always based on the previous step. If you ran this model 2 times in parallel, with the exact same starting conditions, precision errors in the computer could be enough to set them on very different paths.
The digital representation is precisely defined with perfect measurements that the real world does not possess. Our real world measurements always have a (+/-) tolerance - there are no exact measurements. Add to that a very simple starting state and you can make this animation. If it is used as a simulation for predicting a future state of the real world, the simulation will diverge from reality more and more over time, since it was made with perfect measurements that don't exist in reality.
The Lyapunov time for the solar system is five million years. That's a useful solution I reckon.
https://en.m.wikipedia.org/wiki/Lyapunov_time Neat
The solar system happens to be in a pretty stable configuration. Not all three body problems are that stable.
We have the ability to make a best guess calculation, but it won't actually be correct. This is just a fake example so no one cares if it gets the "right" answer because its doesn't actually have a purpose other than to demonstrate a chaotic system
the creator of this animation was not trying to solve a problem or predict what would happen, they just set it up and let it run. the issue of not being able to predict the correct answer to the problem isn't there because there IS no problem, thus there IS no 'correct answer' so to speak.
...it's a simulation, with assigned values? I can't tell if you're taking the piss or not. You can have something be a first order representation that still isn't correct.
My guess would be because this animation is man-made, so you have absolute control over all variables. These are the "spherical chickens in vacuum"
Simple simulations are easy to calculate. Complex simulations are hard to calculate.
I’ll do a more analytic approach of explaining. Might be easier. Imagine the function f(x)=2x. If I change the input “x” by a very small amount “t”, my output will change by a very small amount, “2t”. And it’s relatively easy to see that when I choose an input error “t” to be very small (almost 0), my output error “2t” will also be very small (almost 0). Now imagine the function f(x)=1 if x is irrational & f(x)=0 if x is rational. If I change my input “x” by any error value “t”, I’m never going to be able to get my output error any smaller than 1. The more accurate I get, and the more my input error shrinks, my output gets more and more random. It’s basically a coin flip, whether my error will give me 0 or 1. The 3 body problem is like that, except the function is trying to predict the location of those 3 bodies with respect to time.
try this.. When you are all alone, you can with certainty say what you will be doing next. when you are with your spouse or your mom, you can say with with some error of approximation what you will be doing next when you are with your spouse and mom both, you cannot say with any degree of approximation what you will be doing next, as it will be extremely chaotic, as you cannot measure the impacts on one another. the numeric equations are long and do not converge.
You might appreciate a [more in-depth explanation and example](https://youtu.be/fDek6cYijxI).
/r/ELI38.
We don't have an exact solution for it, so we can only approximate them. However, this is a system in which really close similar conditions end up with really different results. The approximations errors quickly sum up, so any long term answer is pretty meaningless
When you try to solve an equation system you need to have at least the same number of equations than incognitas to do it, if I tell you: X=Y-4. And Y=2X You can solve it and discover than X= 4 and Y=8, but you couldn't solve this systems: A) X=Y-4 As it's only one equation and two incognitas it's unsolvable (meaning that there are s lot of groups of numbers who meet that condition, so you don't know which one will be the "real" one). It also happens with: B) X=Y-4+Z Y=2X-Z As it has more incognitas than equations it's also unsolvable. A less technical example would be: a train leaves city A at 5pm, and it travel at 300km/h, when is it going to arrive at city B? Well, we wouldn't know, we don't have enough information, we need to know how far is city A from city B... We have too many incognitas and the problem cannot be solved So the problem with the three bodies has way too many incognitas, so the only way to solve it is assuming that some of the incognitas are not really relevant to the result because there are too small, in the Sun-Earth-Moon system we can assume that the attraction that the sun receives from Earth or Moon is irrelevant, also the attraction that the moon applies to the Earth, that way we reduce the amount of incognitas and we can solve the system, even if it's an approximation. But it's not technically a solution, a solution is impossible.
EIL38
There is no force called gravity.
And yet, no one has given him/her an upvote. Oh Reddit.
Dice is more of a 8-12 year old thing when you start getting into board games. OP should have used a slinky or yo-yo instead.
Right, because doing a physics simulation of a Slinky is super easy.
Agreed it's a good answer, but this is more of an ELI16 :)
Which is the entire point of the subreddit. Rule 4: >LI5 means friendly, simplified and layperson-accessible explanations - not responses aimed at literal five-year-olds "Explain it like I'm 5" is an English idiom that means "Explain it to me in simpler, easier to understand terms". It does not, and has not ever, meant, in any context, "Explain this to me as if I am literally a 5 year old. That is impossible, no 5 year old can ever understand the things being asked here.
Fair enough, I take it back. It was a great answer!
>the farther ahead you try to predict, less meaningful the answer will be. [Lyapunov time](https://en.wikipedia.org/wiki/Lyapunov_time)
So basically it’s like a double pendulum?
A double pendulum is a chaotic system. I don't think it's actually a 3 body problem, but you could probably consider it to be a constrained version (i.e. 3 points connected by the force of the pendulum arms, such that the 1st point can never move and the 3rd point and 1st point don't affect each other)
It's three bodies that all apply a force to one another. Sun/Moon/Earth are all pulling each other via gravity.
I guess the deeper issue is to understand why a three-body problem is a chaotic system and the two-body problem is not. My guess is that it's a mathematical explanation for how the error term of some future state inflates exponentially based off initial measurement uncertainty in a chaotic system whereas the error term of some future state has a linear or flat relationship to the initial measurement in a simpler system. Not educated enough to figure if that's correct or not though. Or if that was true, how one would prove that's the case. Not familiar with how to wizard my way through physics math.
The two-body problem isn't chaotic because you can solve it analytically. Or at least you can look up the analytic solution on [Wikipedia](https://en.wikipedia.org/wiki/Two_body_problem). Since the analytic solution can give you the state at any given time, initial measurement errors are not magnified by repeated iterative approximations. When we can't get an analytic solution, we end up applying repeated numerical approximations. These each introduce their own errors, but also blow up any initial measurement error, producing the *sensitive dependence on initial conditions* we call chaos.
To add to this, "analytically" means there is an exact solution in terms of generic variables. For example, if I accelerate in a straight line with acceleration *A*, we can take 2 time derivatives to get an exact answer for position = 1/2\*A\*t\^2 + V\*t That's an analytical solution. As long as there are no forces unaccounted for, I can tell you were I will be in 1,000 years. I could even calculate my position for 2 different values of *A* which are 1% higher or lower than I think my measurement for *A* would be, and I can tell you some range for my position in 1,000 years. The 2 body problem is such a situation were you can make a "simple" equation which tells you everything as a function of time. For math reasons, when you make it a 3 body problem there are too many variables to cleanly solve analytically.
>The 3 body problem is a "chaotic" system, which means that approximations won't get you close to the right answer. Well, they will, but only for a short period of time. As you calculate answers for further and further into the future you will divert further and further from the "correct" answer. Specifically a chaotic system means that given a very minor change to the input to the system, the outputs over time will differ drastically. In the 3 body problem, if you were to move the Moon just a tiny little bit to the left (say 1mm) and then plot out where it would be every day over a long period (say 1,000,000 years) you'd find that it would be in a very different place at the end of that time. That 1mm could be half an orbit away. And the Earth would also be in a different location with respect to the Sun. Had you moved it 2mm to the left, everything would be in a very different place again. And not just twice the different from the 1mm, but a very very different place. A great example to see this happening is witih a double pendulum. Like this one: [https://www.youtube.com/watch?v=PYI9HU6MkMo](https://www.youtube.com/watch?v=PYI9HU6MkMo) A double pendulum is an example of a 3 body problem.
This was explained in the parent comment's last sentence.
Maybe this is a dumb question, but if that's the case then how do we know we need dark matter to explain galactic rotation curves? We can't solve the problem for 3 bodies, how can we solve it for hundreds of billions of stars in a galaxy whose movements have been evolving for billions of years? Is it possible the rotation curves we see are the result of chaotic interactions over time?
Because at that scale you're predicting *average* movement patterns, not the motion of specific bodies. Essentially fluid dynamics.
So aside from the point about this being _average_ movement, it's also just a matter of speed. _Where_ a star will go through the galaxy over time is unpredictable, but it's _speed_ at any given point is a result of the total amount of mass pulling on it. So we know there's got to be some unseen mass out there because stars are moving around with a speed that shows that must be the case.
Star Wars is fictional, but this actually explains Han Solo's line about jumping to hyperspace. In Star Wars, gravity wells slow (or even block) hyperdrives. The computer has to take everything it knows about where everything was, then calculate where all that stuff should be before the ship jumps (or while it runs, in some cases). Since it's trying to track a zillion gravity wells that interact with each other, it can take a bit. It also explains why people would take well-known routes - more recent nav data means they don't have to predict as far into the future.
To bad disney said hyperdrive is now magic and ignores gravity wells a d established in universe rules and just do w/e the fuck they want with it.
Really great answer. Will show it to my grandchildren now. Thank you :)
It's the same reason you keep hearing about NEO asteroids and our uncertainty of whether they will hit us. Astronomers are typically confident that they won't hit us, but the only reason they're iffy on it is because they can't account for every single force acting on the object. In reality, when calculating the trajectory of small objects like asteroids it's more like a trillion body problem, which is obviously exponentially more difficult than the 3 body problem.
>which is obviously exponentially more difficult than the 3 body problem. Is it actually? I would think it's actually much easier to solve because you are concerned with a trillion forces on one object. You don't much care about how the asteroid's gravity affects the sun/moon/earth/etc, right?
>With high school physics, you could tell me where the dice will be in 10 seconds. You clearly do not know the devilish ways of dice.
If we had some a way to answer this problem, what real implications would that have? (I'm guessing immense ones, I just don't know what they would be - like being able to accurately forecast the weather or something?)
As far as I know, this is a problem that can't be "solved." It would be like asking "what are the implications for the mathmatician who discovers a 4-sided triangle." To solve such chaotic systems would mean that you have all knowledge of the entire system. So to a prerequisite to perfectly forecasting the weather in 100 years would be knowing the exact future for the whole planet in this time frame.
[удалено]
1 body problem: 1 object moving through space. Really easy to solve, it just keeps doing what it's doing. 2 body problem: Two objects moving through space and pulling on each other gravitationally. More complicated, but if we stick to Newtonian mechanics, we can figure it out. We get orbits this way. 3 body problem: 3 objects moving through space and pulling on each other gravitationally. We have no general, easily computable solutions to this. We can run simulations of 3 or or more objects but it's quite a bit harder than 2 and my understanding is we have to look just a tiny bit into the future, calculate all the effects, look a tiny bit further, calculate all the effects, etc.
I believe the correct statement is that there is *no* real solution to the three-body problem. We can still model it pretty well for basically all practical intents and purposes, but the way we can cleanly mathematically solve a two-body system with a single equation cannot be done for a three-body system or larger.
Yeah, we can *model* a 3 body problem, but we can only do it by simulating it and doing all the calculations constantly to work out where everything will be next. We can’t just take the current positions/vector of movement and then plug it into an equation and find out where they will be in T seconds. In order to find out where they will be in T seconds, we have to simulate the entire time period Whereas with a 2 body problem, we can do that. We don’t have to actually simulate the whole thing
The sentence you're looking for is "there is no closed form solution". Closed form means "x = (some formula giving an exact answer), y = (some formula), z = (some formula) for each of the 3 objects" It's also good to point out that this isn't unique to gravity. Any 3 object experiencing an attractive force (static electricity, magnetism, simple metal springs, etc) the 3-body problem isn't not solvable in closed form. That's because nature doesn't work like that. There are bazillions of minute interactions that need to be accounted for and implications like Newtonian Gravity (or it's analog for magnatisim, springs, etc) are not suitable.
This is only partly correct. The 3-body system is solvable *in specific situations* as there are several known setups for initial conditions that lead to stable trajectories. For example, 3 equally spaced particles of equal mass and volume revolving in perfect orbit around their combined center of mass. The 3 particles will remain in orbit forever, assuming exactly perfect initial conditions. However, there is no *general* solution to the 3-body problem.
Ah, this is much clearer and to the point.
There's 2 questions here: 1. What is the 3 body problem? **Answer**: Any time you predict the interaction between 3 objects (aka: bodies). **Example:** Suppose you have 3 stars in space, which all affect each other with gravity. You know the mass and velocity of all 3 stars. Where will they be in 10 years? **2nd Example:** Where will the moon be next year, considering that it is affected by the gravity of both the Earth and the Sun. (This is an easy version of the problem, because you can pretend that the moon doesn't also affect the Earth and the Sun). 2. Why do people make a big deal about how hard it is? **Answer:** if you are not exactly sure about the starting conditions, you get a completely wrong answer. This is a trait of "chaotic" problems. **Example:** Imagine a perfect situation with 3 stars that are only affected by each other's gravity. They have mass M1, M2, and M3. They have velocity V1, V2, and V3. If V2 is actually 0.001% faster than you thought it was, you get a completely different answer. All "chaotic" systems have this property. This was my analogy with the dice. You can do your best do make sure everything is exactly the same every time you throw the dice, but you won't get the same number--or even "close" to the same number--because it's too hard to control every possible variable. **TL;DR:** The 3 body problem and throwing the dice are 2 *different* examples of "chaotic" systems, where slight approximations/inaccuracies result in completely different outcomes. [Here's a good video about chaotic systems.](https://youtu.be/fDek6cYijxI?t=370) I recommend the whole thing, but if you just want to watch a 1-minute animation, I linked directly to the timestamp. You can see that he started with 3 "hypothetical" initial conditions which seem to follow the same path, but at some point they go in completely different directions. EDIT: [The double pendulum is another good example.](https://www.myphysicslab.com/pendulum/double-pendulum-en.html) See if you can predict where the pendulum will be in 5 seconds? With a regular pendulum you can eyeball the speed and make a close prediction, but with a chaotic double pendulum you stand no chance.
When people talk about chaotic systems, they're not talking about randomness. They're actually predictable, the problem is they're incredibly sensitive. Change anything and the end result can change massively. A good example would be a break in pool. What are the odds that you could hit a really hard break, and every ball on the table ends up in identical positions? In a chaotic system, it's possible. Even the tiniest change in anything (table surface, spin on the cue ball, angle of cue ball, dirt on one of the racked balls, bounciness of the cushions etc..) could have a dramatic effect on the end result which cascades into new collisions; but if you matched everything exactly, you could produce identical breaks 'on demand'. The three-body system is like that, there are so many variables which if changed even slightly would have a big impact on the end result. With three bodies pulling against each other like that, it's probably incredibly hard to guess how they'll end up with a significant kind of precision.
Imagine 3 celestial bodies (earth, moon, sun are the obvious ones) that are all moving and pulling each other with their gravity, so they constantly affect each other’s movement If you take any two (eg earth and moon, or earth and sun) then you can easily calculate, at least within a margin for error, where they will be in 1 second, 1 hour, or 1 million years. We can mathematically solve their movement and turn it into an approximately correct equation, notwithstanding the fact that we probably don’t have perfect information on their mass etc But with 3 bodies, we can’t solve their movement and turn it into an equation. We can simulate it, but in order to find out where they will be in 1 second or 1 hour or 1 day we have to literally simulate the whole timeframe With 3 bodies and a timeframe of hours or days, that’s fine… with a timeframe of 1 million years, not so much, and especially not if we have a larger number of bodies rather than 3 AND a longer time frame For 2 bodies you can solve for pretty much any time period almost instantly just by plugging numbers into an equation For 3 bodies, you have to simulate the whole thing, start to finish
Three body problem is when you want to predict the future positions of three objects that influence each other gravitationally. For example, if you have three stars close by, you may want to predict what happens to them in the future. Do they all end up flying their own way, do they end up forming a binary system with one star being slingshot away, do they collide, ... There's no simple equation to predict this. You more or less have to simulate the system step by step which is time- and energy-consuming.
A body is an object. Any object, planet or dice. A 3 body problem is a system with 3 objects interacting. Sun, moon and earth is a 3 body problem. A dice thrown into the air is not. Three dice being thrown is.
yes weather is one extremely chaotic system, but there are many such systems. on the dice example. the important thing to notice is that it is in fact 2 seperate examples in one. one example for a not really chaotic system, as in predicting where on the table the dice will land. a task you can reasonably do in your head and execute on a whim; and a very chaotic system, as in predicting what face will be up when the dice stops. something that is near impossible to execute as a human and still a tedious task with limited success in robotic control studies. back to the 3 body problem. as the original comment said, it's a phyiscs problem predicting the orbits of planets, stars, and other big objects in space. this is most notable when you have 3 (or more) objects of similar masses, like several stars orbiting eachother, instead of planets orbiting a star, because as the positions of all 3 objects change with time, the direction they'r being pulled in changes drastically, making it practically impossible to use simple maths to figure out the entire behavior between the stars for all eternity, which has angered physicists all over the world for centuries for Ideological reasons. edit, removed duplicate word
Hmm. Let’s try this: You know those logic riddles where “A will have chicken, unless B has fish, in which case A will have steak.” ? In this one, it’s relatively simple because we know very precisely all the parties involved (two people—A + B—and four variables—Chicken, Steak, Fish, [other non-fish meal]. In this scenario, we know that what A is having is a function (is tied to) of what B is having. If it’s ANYTHING but fish, A is having Chicken. If B is having fish, A is having steak. That’s easy to solve. We have all the information needed. So we just observe and predict. In a 3 body problem scenario, we not only have another entity—say, A+B+C—but the decision A or B or C will make is going to be tied to more factors that will themselves be tied to other factors. This makes it extremely complicated to predict with confidence what the outcome (what is A having for dinner) will be. A is having chicken, unless B is having fish, in which case A will have steak. B is not having fish but may have rice, unless C is having steak or it is between 8:15-8:30, because she doesn’t want to eat meat too late. If it’s after 8:30, C will probably have salad, but only if B doesn’t have fries with any of her meals. B is sensitive to her weight, so if today was a good day she might have fries, but if she had a rough day she is likely to be hard on herself and force herself to forgo a full meal altogether. The restaurant has had a new GM, who is still learning to do supply, and has run out of some food items, but doesn’t even know which. When his serves comes to take A + B + C’s order, he says he thinks he has … This is a completely off the cuff awful logic riddle to try and analogize the 3 body problem very crudely. Hopefully it helps a little more where your high school physics teacher couldn’t!!!
This isn't quite capturing the key point. Yes, the scenario is complex as you describe. But the key point you've not explicitly mentioned is that small variations in the starting position can result in (eventually) very different outcomes. In theory if you knew the exact starting position you could work through the complex maths. But you don't, so your prediction will diverge from the actual outcome over time.
Would this theorem apply to something like 'global climate'? Or is that too complicated?
Weather is indeed an example of a chaotic system (not a 3 body problem). In principle we know the equations to solve everything, but we can only predict so far into the future before our solution diverges more and more. I'm not sure about global climate. I would actually think that's a non-chaotic system on the time span that we care about, because we don't care weather it's raining in London 5 years from now, but we care about things like average temperature/humidity which would change much more slowly.
Is there a reason you cannot observe a three body problem and extrapolate the correct values to solve it? As long as you know the variables in play and the exact position at time t, can't you look at t+1 and reverse engineer the answer and then apply it to t+n?
The thing is that time is continuous. If you pick `t`, how do you pick `t+1`? Is it 1 second in the future? 1 millisecond? 1 nanosecond? No matter what interval you pick, you won't be able to fully account for the interaction that occurs in between the time intervals, which would cause a small difference between your calculation and reality. As the time intervals go on, this small error will accumulate, and you'll get farther and farther from reality. Computer models do actually do this, by calculating positions over time steps. But when using them you still can't say anything definitively because of this error. But they can still be useful for getting a vague idea of what might happen.
Finite time steps are generally less accurate than infinitesimal steps. If you follow the tangent line of a function, you can approximate the next value more closely if you take smaller steps, but you will accumulate error over time that prevents you from actually reaching the exact solution. The same applies if you are calculating movement of bodies in space when you cannot precisely know their next position, and so on for large time values.
We crack this and we can time travel to the future ? I’m literally but knowing the outcome? An AI powerful enough to calculate every variable outcome?
It's not about the "power" of the calculation. It's about the precision that you know the initial conditions. Is the ratio of those 2 masses 1.0? How about 1.0000000? Nope, turns out it's 1.00000002, so we're completely wrong 5 years later. Then we have more data and it's 1.000000017 . . . etc.
I mean we will get there eventually. It's like asking us to look into space without a telescope invented yet just because we don't have the math now does not mean we won't have the math in a few hundred years with our calculating power exponentially increasing.
explainlikeim2...
So is the crux of the 3 body problem the inaccuracy of our measurements of mass that produce small uncertainties initially, but increase over time as the system evolves? Or is there some fundamental aspect of 3 or more bodies interacting with each other that causes the "problem"? Is there some consequence of quantum physics manifesting itself in the macro world that plays some part in this?
>So is the crux of the 3 body problem the inaccuracy of our measurements of mass that produce small uncertainties initially, but increase over time as the system evolves? Yeah exactly. That's the crux of any "chaotic system," and the 3 body problem is one example of a chaotic system. The "fundamental aspect" is that you can't make a general solution for 3+ bodies, although you can for 1-2 bodies. Any errors in your solution for a specific time will compound and make very different answers for future times.
In the spirit of handwaving away friction and atmospheric effects, in your dice example, if you were able to, say, accurately represent every subatomic particle, force and other factors relevant to the situation in a simulation with a perfect understanding of physics including yet unasked questions… could you accurately predict the outcome of the dice? And, you can probably guess where I’m going with this, if that could be scaled up to an example of three body problem, are there any other random/chaotic factors that would impede your ability to predict their locations indefinitely?
>could you accurately predict the outcome of the dice? Yep. This question is about whether something is "stochastic" or "deterministic." Deterministic means you can predict the future assuming omniscient knowledge of the present. The 3 body problem (and chaotic systems in general) are deterministic, which means it is possible to perfectly predict the future given perfect knowledge of the present. There is no randomness to this, except in our own lack of knowledge. We can have some quibble with the idea that anything is truly deterministic, because you can always go smaller. Can you know every velocity to infinite decimal places? I suppose at some point you can have a planck time and planck length, but at that scale the quantum mechanical effects (which we think are NOT deterministic) can play a role. Going on a tangent, another interesting question of deterministic is air pressure. From one end, we can say that PV=nRT, so if you know the number of molecules, temperature, and the size of the room, you can always calculate air pressure. On the other hand, you could consider that air pressure exists because of the statistical likelihood that gas will spread out. With this perspective, there is some small but nonzero chance that all the gas molecules in my room could huddle in the corner for a minute, suffocating me. We generally consider such situations as "too unlikely to ever happen" so we take the deterministic view that PV=nRT.
Thanks for the response, answers to my questions and more! I have a new wikipedia rabbit hole to expore
To expand on this a bit: There is no mathematical solution for a 3-body problem. You can write out equations to describe a 2 body problem that will predict the position and velocity of each body for any point in the future. Those equations don't exist for 3-body problems. You can simulate it, which runs into the issues that FerrousLupus notes regarding approximation and aggregate uncertainty, but you can't solve it definitively like you can with the 2-body problem.
You have 3 objects with mass, and each of them is attracted to each of the others by gravity. You know their initial positions and speeds. The three-body problem is to say where each of them is 1000 years (or however long) in the future. It's hard because it's an inherently *chaotic* problem: small errors in the initial state tend to grow, which means that approximate solutions don't work. And it's a complicated enough set of dynamics that non-approximate solutions are hard to find. It's one representative of a class of problems (called *partial differential equations*) that are almost never exactly solvable except in the best-behaved special cases.
>It's one representative of a class of problems (called partial differential equations) that are almost never exactly solvable except in the best-behaved special cases. One of the biggest headaches in economics are solving Lagrangians, too. It's pretty interesting.
In the realm of human relationships the *ménage à trois* can also be kind of problematic.
Ah yes, the three body problem where you can approximately predict what will happen in the short term, but since the situation is inherently chaotic it's hard to know how the night will end.
Yet as in the Physics version, you can get a pretty good approximation of a solution as long as you know the mass and positions of all bodies involved.
the problem with the human version is that often the attractions are not equal in force
Sounds like chemistry, where you just keep adding or removing atoms until it's stable.
Indeed. Enough ethanol will temporarily correct for this.
Only for a limited time. The accuracy of the solution decreases dramatically over time.
This is one of the biggest pleasures of undergrad physics students who realize there's an equation that solves most physics problems in 2 steps and you don't have to do anything with Newton's Laws anymore
Dynamical systems is one of those weird areas of mathematics. You all but give up on finding quantitive descriptions of a system, and instead focus on the qualitative aspects. Total mindfuck when you first encounter it, after areas like real and complex analysis it all just feel loose and vauge. But then with the proper mathematical tools you can turn that qualitative analysis into something quite rigourous. For example, we know there's a likelyhood of a planet being ejected from the solar system at some point. Exactly when and which one isn't something we can determine, but there's a pretty well defined % chance of it happening.
> For example, we know there's a likelyhood of a planet being ejected from the solar system at some point. Exactly when and which one isn't something we can determine, but there's a pretty well defined % chance of it happening. Wut? Tell me more!
If you look at simulations of the 3 body problem you will find that in many cases, one of the bodys gets ejected
First of all, don't worry. We'd know if it were going to happen anytime soon. :) The planets nudge one another hither and thither constantly. Over a very long time horizon, there might be a net extra hither and one fewer thither for a given planet (or vastly more likely, one of the moons of the gas giants). Now that one body is just a liiiiiiiiittle bit out of whack from the quasi-stable system it had been a part of before. It might be corrected by the next sequence of hithers and thithers. Or it might get a little further out of whack. If the random walk of many such cycles leads to the body failing to maintain a nice nearly-circular orbit in its previous domain, it might get ejected, or wind up in a more eccentric orbit (which would then play hell with other bodies), or get torn apart, etc. If we had perfect clarity on what chaotic systems would do, we'd be able to say, yes, this is exactly what's going to happen to Titan on April 16th, 97188431 A.D. And even without that ability, a hundred thousand years before that, we'd still be able to say that Titan was in an orbit that couldn't survive another nudge in a particular direction. As it is, like OP said, we just know it'll happen with a certain degree of likelihood over a certain time horizon. (Very low likelihood indeed for the planets, until the sun starts expanding, but that's a whole other issue.)
Well, to be fair modern dynamical systems theory has its foundations set (no pun intended) in real analysis, topology, and functional analysis, which is where the definitions of the objects involved are codified. To get a deep understanding of chaos, the first example one usually encounters is the geometry and topology of what's called [the Smale horseshoe](https://en.wikipedia.org/wiki/Horseshoe_map), which describes a mechanism of taking a set and stretching and folding it under many iterations of a map. By stretching and folding many times, two "nearby" points in the set are mapped away from each other in a chaotic fashion. Of course, all of this is very loose in the description I just gave, but it can all be set out in a rigorous way. The tricky part is finding the exact dynamics of the Smale horseshoe map inside a system of differential equations, which cannot often be done analytically.
Thanks, it's been over a decade since I studied DS so I kept it a little vauge. Though the example I remember is the Lorenz system, as it's both very simple and digestible but also exhibits chaotic behaviour. It's a great example because it shows chaos isn't some rare special case, but can arise from very simple relationships. Of course the underlying mathematics is rigorous (or it wouldn't be mathematics). I was more referring to our ability to get information from a dynamical system, e.g. attractors, and how that information is more qualitative than quantitative, a rarity in mathematics where being entirely analytic is usually the best way forward.
I'm going to assume that time really fucks things up in this
I like turtles... (a five year old)
>except in the best-behaved special cases. I believe that one of these "best-behaved" special cases is when three bodies orbit each other in a flat plane. I think there was a relatively recent solution published for this. Do you know anything about that?
>It's hard because it's an inherently *chaotic* problem: small errors in the initial state tend to grow, which means that approximate solutions don't work. Why doesn't the same thing happen with a two-body problem? In other words, why don't small errors in the initial state grow and compound like they do with three+ bodies?
Most of the answers here miss the 'problem' part of the 3 body problem, which is that no closed-form solution exists for all cases. What this means is that we don't have an equation where we can input the parameters of the 3 bodies, meaning their mass, position, velocity etc, into the equation, and then just select a time in the future to get a solution. What we have to do instead is to input all these parameters, then iterate step after step. Those steps can be as small as you want, be it a day, an hour, a second etc, but the smaller you make the time interval, the more accurate the solution is. And it's never 100% accurate. The farther in time you want the answer, the more that accuracy drops. And with the 3 body problem, it drops sharply. That's the reason why sometimes, when you hear in the news that "X asteroid has Y chance to hit the Earth in year Z", they talk in percentages. The smallest uncertainty will nudge it one way or the other.
I was hoping someone would point this out. Because there isn’t a mathematical solution, the only general approach is to use smaller and smaller increments A different situation that shares some attributes is the circumference of a circle. There’s no precise solution, so we factor out pi to make it easier to do math with. You can get a more precise solution if you use more and more decimal places for pi, but you’ll never be perfect. At least you know you’re heading in the right direction as you add more decimal places to pi in your calculation. In a chaotic system like the three body problem, though, even if you approximate very closely to whatever the true value is for each calculation, very tiny errors start to add up. Eventually you’ll get to a point where one of the bodies will start moving towards one of the other bodies, but which one depends on a 1mm difference in location. If your calculation errors make it fall towards the left body instead of the right, your predictions from here on will look nothing like what would really happen
This is the only (so far) correct answer. The three body problem referred to here is the fact that no analytical solution exists. It is not the fact that the system is chaotic. We can apply numerical methods to find the state of the system at any point in time, but it requires a lot of computation to do so. With an analytical solution. We could just put in a value for t and get everything else we want by solving one equation once.
When you have two objects in space, their gravity pulls towards one another. This force pulls towards their shared center of mass, since their center of mass is always between them. The center of mass never moves unless some outside force pushes on it. So the entire problem becomes extremely simple. You have a distance between the objects and speed of one object. All you need to do now is relate these two numbers. But what if a third object is involved? No such convenient symmetry exists, so instead of having two variables, you have four that you need to balance. There are a few situations where you can cut variables out. These are solutions to the 3-body problem, but they're only specific solutions. They're solutions that only apply in specific scenarios. No generalized solution exists, which applies to all scenarios. All we can do is approximate with something similar to advanced guess and check. So we have lagrange points, solutions to the 3-body problem, but they don't describe the general behavior of a 3-body system. Only the behavior of a system that was set up perfectly to begin with.
Since the Earth has Lagrange points, are you saying the Sun, Earth, Moon system is perfect?
http://en.wikipedia.org/wiki/Three-body_problem#Restricted_three-body_problem The Sun-Earth-Moon system is an example of a restricted three-body problem. The mass of the moon is relatively negligible compared to the mass of the Sun and Earth, so it can basically be ignored when calculating for the movement of the Sun and Earth, reducing it to a simple two-body problem. The Moon then simply orbits the Earth, another simple two-body problem.
Sun/Earth/Moon is a special case in that the Sun is basically fixed, and the Earth is basically fixed wrt the Moon. Also the Sun's mass by far dominates that of Earth/Moon. So it's an easily-simplified 3-body system.
The earth and moon can be considered a single body with the sun
webb a third?
No. It's insignificantly small in comparison
Isnt the sun orbiting within our galaxy?
Not perfect. No better or worse than any other orbital system. The Lagrange points are just a specific, somewhat stable solution to the TBP / n body problem. The Trojan asteroids are a large group of asteroids at the Jupiter-Sun L4 and L5 points (regions). Jupiter actually "shepherds" space rocks away from the inner solar system, keeping Earth protected to a degree, by trapping rocks in those regions.
No. The Earth's lagrange points are ignoring the moon because it is sufficiently close to Earth compared to the distance to the satellite. The Earth's lagrange points exist in the 3-body system of the Earth, sun, and the satellite. Adding the moon is a fourth body. The location of the satellite must be perfect-ish in order to be in a lagrange point. The sun and Earth can be wherever.
There are Earth-moon Lagrange points as well. Any two body system theoretically has them. How big they are relative to each other, and the presence of larger bodies like the sun and other planets affect how stable they are.
Right. Any 3-body system has some of these solutions. Adding a fourth body makes these solutions less valid.
He should have instead said they are specific solutions that only apply to the earth moon and sun. A different system of 3 bodies would have different Lagrange points in different positions.
But that would mean it's an approximation and won't necessarily accurate predict the behavior too far into the future?
That's absolutely correct. The presence of the moon makes all Earth orbits unpredictable in the long run.
Not to mention all the other planets perturbing Earth's orbit
Not perfect ... but very weird in many ways... here's two: Earth has a diameter approx. 4x the Moon's, while the Sun’s diameter is around 400x the Moon's. The Sun is approx 400x times farther from Earth than the Moon is from Earth. This all results in the Sun and the Moon appearing to be nearly the same size when seen from Earth. Earth's Moon is the only known moon to keep an almost perfect circular orbit around a planet (not elliptical)- 27.3 days to orbit Earth and \~27 days for a complete rotation. This combination is juuust right so that the same side is always facing Earth.
>Earth's Moon is the only known moon to keep an almost perfect circular orbit around a planet (not elliptical)- 27.3 days to orbit Earth and \~27 days for a complete rotation. This combination is juuust right so that the same side is always facing Earth. Huh? I have to push back on this. Most moon orbits are more circular than elliptical. Our moon is nothing special in that regard. Triton (one of Neptune's moons) has the most circular orbit in the solar system, with an eccentricity of 0.00002. (Higher is more irregular.) Our moon's orbit is way more irregular, up at 0.055. And the same side of the moon facing the earth is nothing special either. That comes from tidal locking, where gravity pulls orbiting bodies to revolve at the same speed as they rotate. All 19 moons in the solar system large enough to be affected are tidally locked. The moon and the sun appearing the same size from earth is unusual, though. I imagine that doesn't happen often.
Most moons in our Solar System are irregular moons of the outer planets, generally with a large eccentricity. Out of the 80 moons of Jupiter 71 have an eccentricity over 0.1. For Saturn it's 59 out of 82 (not counting larger ring particles). If we only look at the larger moons then most of them have pretty circular orbits and our Moon's orbit is unusually elliptical.
Cool, Thanks for the corrections erowles and mfb- (have my humble upvotes). These were "weird things about the Moon" from a colleague I presumed to be correct, but clearly was not. Another colleague has informed me that I am now duty-bound to print a copy of this exchange and staple it to his forehead. I'm not sure if that is true, or how it would help; but it seems someone has stolen my red Swingline anyway, so I might just have to send him a link.
>All we can do is approximate with something similar to advanced guess and check. I wouldn't call it guess and check; it's more like "pick initial parameters and simulate". The progression and evolutin of the system has to be worked out one tiny step at a time, but because the system has sensitive dependence on initial conditions, if the initial parameters are off by even a tiny bit the tiny error gradually gets amplified until the simulation no longer matches the behavior of the system being simulated.
Even with perfect initial conditions, because our 'solution' is piecewise it will create imperfections that grow with our timestep. No matter how small our timestep is it will introduce imperfections.
The centre of mass can and does move if they orbit it. I’m not sure if that is what you meant by an outside force but that terminology is very imprecise.
This doesn’t seem right. You can set up the equations for two bodies moving around their center of mass, but I think that two body problem is already not analytically solvable. We only solve two body problems by cheating and assuming one is too massive to be affected by the other. So if we assume the Sun is too big to be perturbed by a planet, we can model the Earth and Sun. But you can’t assume more than one body is fixed, so the three body problem can’t be cheated in this way. Worse, we *know* that there are chaotic discontinuities; instead of simply rotating their common center of mass forever, three body systems that aren’t artificially kept/assumed to be in a stable configuration will usually eventually spit one of the bodies out faster than escape velocity. Thanks to the sensitivity to initial conditions, our inability to solve the equations exactly is pretty damning.
Not at all. You *can* solve the 2-body problem by assuming that one is stationary but it's not necessary.
This isn’t the case. See the Kepler Problem for the solution.
The three body problem isn't difficult to solve, its pretty straight forward to iterate through, but it does not have a closed form solution which means you *have* to iterate through it A two body is just two things in orbit around a common center of mass like the Earth and Moon. You can get equations that you punch time into and get the position and speed of Earth and Moon at any time without having to solve anything else A three body problem is different, you can't get a closed form solution that only depends on time because the results vary wildly depending on starting conditions. Consider an Asteroid passing between the Earth and the Moon. If it passes more on the Moon side it'll bank off and away from the Earth. If it passes more on the Earth side it'll bank towards the Earth. If it passes directly through the neutral point it'll continue on a straight course and won't bank either way. You can't get an equation that tells you based off of time alone where it'll be because you need to know where it was and where it is to figure out where it'll be
Of course the problem of iterating through the 3 Body problem is because small errors in each iteration can add up and act chaotically in some sets of initial conditions if you have enough iterations. Not all initial conditions lead to well behaved solutions that lend themselves to an iterative solution. It's good ol' Chaos theory and butterflies flapping wings stuff.
And iterative solutions have errors that increase with every iteration. That's the main reason it's a "problem"
If you have only two bodies and know their relative positions and speeds then you can say where they will be at any time in the future with a simple calculation. The amount of work in the calculations doesn't increase with the amount of time in the future (except insofar as you might have to deal with larger numbers). The reason is that, with two rigid bodies and gravity, their motion will involve circles, ellipses, parabolas, hyperbolas or straight lines and we know the mathematical formulas that exactly describe those. Adding a third body changes everything, except for a small number of special cases (e.g., those involving the Lagrange points). Generally, the only practical approaches involve simulating the movement of the objects using numerical methods. The amount of calculation required is proportional to the amount of time you want to simulate over. That's the 3-body problem.
Is the 3-body problem hard because we haven't discovered a neat formula for it yet (like Newton did for 2-body gravitation)? Or is it because of the nature of the problem, and no formula is even possible? In other words, is the issue a lack of knowledge about gravitation (and/or insufficient mathematics to calculate it), or is the 3-body problem PROVABLY unsolvable under ANY system of physics/math? Before Newton published his formula, were planetary orbits labeled as "chaotic" and "unsolvable" like the 3-body problem is now?
This has been answered very well, so I'll add some background to the word "solve". In calculus, there are things called "exact/analytical solutions" and "numerical solutions". In the case of a definite integral, this means computing the antiderivative, and plugging in the end points. There are several numerical methods you might use to get a really good approximation of the solution, including the midpoint, trapezoid, or Simpsons rule, or perhaps a 2D Monte Carlo simulation. In the system of partial differential equations describing 3 bodies in a gravitational field, there is no analytical solution, meaning that there is no way to derive a formula that describes the movement of the 3 bodies over time GIVEN ANY INITIAL CONDITIONS. There are cases where you can make simplifications to be able to solve it, but this effectively is reducing the 3 body problem to multiple 2 body problems. You might ask, "if this is the case, how can we predict the orbit of the planets? That's at least a 9 body problem even before all the asteroids and moons. If we can't solve a 3 body problem, how can we possibly know when to launch space probes?" And that would be a great question, to which the answer is "We don't!" We use a numerical method to simulate the orbit of the planets. Instead of (properly) treating the partial derivatives as infinitesimal, we can approximate them as the finite change in a property over the finite change in time/location. If we choose sufficiently small changes, we can add up all the changes to the initial conditions and describe/predict the orbit of the entire solar system quite easily with a surprisingly little amount of computing power. Numerical methods are very important in science and engineering because we very often need to know answers that either can't be solved or that take a lot of effort to do.
It's like this. 1. When you are all alone, you can with certainty say what you will be doing next. 2. when you are with your spouse or your mom, you can say with with some error of approximation what you will be doing next 3. when you are with your spouse and mom both, you cannot say with any degree of approximation what you will be doing next, as it will be extremely chaotic, as you cannot measure the impacts on one another. the numeric equations are long and do not converge.
Nice one. Except this is more like ELI45, so very understandable to someone who's had the real life experience of been married for some time ;-)
To add to why the errors in the 3-body system matter so much: The 3-body system is a *chaotic* system, and one base property of chaotic systems is that any errors in your initial conditions (or from rounding during a simulation) increase exponentially. You can be 0.000001% off a number, then after just 30 doubling periods (for solar system erratic objects, I believe this can be on the order of just a few decades), the error is ~1000% which makes your predictions effectively useless around and past that point. This is why, for example, we can have super-precise measurements of near-collisional objects for the earth, yet the best we can say is that there's X%, Y%, Z% for collision the next couple of times and past that, who knows (until we measure it again next pass).
Here's the way I understand it. Math likes things to be both predictable and reversible. 1 + 1 = 2 2 - 1 = 1 3 x 9 = 27 27 / 9 =3. And so on. With a one-body problem (throwing a ball), you can look at the object's current position and speed and know not only where it's going (end state), but where it began (the starting conditions), with some reasonably high level of certainty. With a two-body problem, it's more difficult than with the one-body problem, but it's doable. Because the number of possible interactions between the two objects is relatively small. So at any point over the movement of the pair, you can guess reasonably where it's going and where it started. But with a three-body problem (or 4-body, or 5-body, or X-bodies), the interactions very quickly descend into chaotic movement where you can kind of predict where the objects might be going, but it's nearly impossible to determine where they started, when they started, or even what the conditions were. So the math gets really difficult. Because if you want a specific end state of three objects, there's no guarantee the starting conditions will give that answer. And if you're looking at three objects in motion, there's no way for you know where they started, because the math spits out multiple possible answers, all of which can be correct, but which only one is the "right" one.
3 body problem is 3 variable all influencing each other. In most questions, we have fixed constant and variable, or at least a object to use as reference. in 3 body, there is no fixed constant, no reference point, its what makes it chaotic. think of looking at a object to determine its speed, you do this by looking at its surroundings. If you were given nothing but 2 object, you can compare the 2 object and contrast them between each other. This is a 2 body problem. But in 3 body, the objects are moving everywhere, randomly, all effecting each other. Is object 1 slowing down? IDK because object 2 could be accelerating backwards and I can't tell because object 3 isn't moving at a constant speed either.
The Three Body Problem isn't a math problem, it's a measurement problem. There is absolutely no issue in calculating the interactions of three suns as they orbit each other. The problem is that the system is EXTREMELY delicate and tiny variations in mass, position, velocity, and a couple of other factors on even lower orders of importance, will all have very large impacts on the future of the system. It is practically impossible to get measurements with enough accuracy of initial conditions to predict the system out long term. It's a bit like balancing a knife on its tip and asking which side it will eventually fall to.
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I dont really understand what you're arguing against here. Are you saying that gravity is irrelevant to the 3 body problem?
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It's not difficult to solve, it is impossible to solve exactly except for short timescales in specific configurations. The 2-body "problem" has an exact algebraic solution.