#Physics #3BodyProblem #Astrophysics
Have you ever wondered why the 3 Body Problem in physics is so perplexing and seemingly unsolvable? 🤔 It’s a concept that has baffled even advanced alien races in sci-fi series like the 3 Body Problem on Netflix. Let’s break it down in simple terms and explore whether there is a solution to this intriguing puzzle.
## What is the 3 Body Problem?
The 3 Body Problem refers to a specific scenario in physics where three massive objects interact gravitationally with each other. These objects could be planets, stars, or any other celestial bodies. The challenge arises from the complexity of predicting their movements over time due to the gravitational forces at play.
In simple terms, imagine trying to predict the exact paths of three planets orbiting each other in space. The gravitational pull of each planet on the others constantly changes their trajectories, making it extremely difficult to accurately calculate their positions at any given moment.
## Why is it Unsolvable?
The reason why the 3 Body Problem is deemed unsolvable lies in the chaotic nature of gravitational interactions between multiple celestial bodies. Even the most advanced alien race in the 3 Body series found it impossible to predict the movements of three celestial bodies accurately in all situations.
This unpredictability stems from a phenomenon known as “sensitive dependence on initial conditions.” Essentially, small changes in the starting positions or velocities of the three bodies can lead to drastically different outcomes over time. This sensitivity makes it nearly impossible to determine a universal solution to the 3 Body Problem.
## Is There a Solution?
While the 3 Body Problem remains a challenging puzzle in physics, researchers have developed various computational methods and mathematical techniques to study and understand its complexities. These approaches include numerical simulations, perturbation theory, and qualitative analysis to gain insights into the behavior of systems with multiple interacting bodies.
While a precise and exact solution to the 3 Body Problem may still be elusive, scientists continue to make progress in unraveling its mysteries and expanding our understanding of gravitational dynamics in celestial systems. Perhaps one day, we may come closer to cracking the code behind this fascinating phenomenon.
In conclusion, the 3 Body Problem is a captivating enigma that continues to intrigue scientists and enthusiasts alike. Its complexity and unpredictability showcase the awe-inspiring power of gravitational interactions in the vast cosmos. So next time you gaze at the night sky, remember the intricate dance of celestial bodies that defy easy solutions and spark our curiosity endlessly. ✨
Feel free to dive deeper into the world of astrophysics and explore the wonders of the 3 Body Problem with a curious mind and a sense of wonder. Who knows what new discoveries and insights you may uncover along the way! 🌌
if you take any 2 masses and put them in space you can perfectly predict exactly where they will be at any point in time ever.
as they orbit
if you try with 3 (or more) masses, you cant (outside of certain special cases). the best you can do is simulate it, which adds increasingly more error the longer out you simulate and the faster your simulation is.
I belive it has been proven to be unsolvable (as in, there is no general equation you can put the initial positions and the current time to get out the exact positions at that time) , but im not 100% sure, proving unsolvability can get strange
It’s not unsolvable, it’s just that there’s no (known) general solution. That is to say, each 3 body system has to evaluated individually, and for the majority, there is no long term stable arrangement. Additionally, our ability to predict the long-term state of it decays exponentially as three bodies introduce so many possible variables it quickly spirals beyond whatever present computing can simulate. Inevitably, some part of the system will decay. A body will be expelled, or two will collide, or something else. Even a presently stable system is extremely susceptible to minute external forces and can quickly collapse. There are a handful of known perpetually stable arrangements, but for any random 3 body system, you would not want to bet on it’s long term survival.
Object A and B in space pull on each other because of gravity. We’ve got this one down easily – the math is relatively straightforward, and very very predictable.
Now add a third object.
We know how A and B pull on each other, easy peasy! Wait though, C is pulling on A and B, so re-do all the math because of that.
Wait, A is pulling on C too, so that changes things, so recalculate again.
Wait, B is pulling on C too, so recalculate again.
Wait, that changes how C pulls on A and B, so recalculate again.
Wait, that changes how A pulls on B and C, so recalculate again.
Wait, that changes how B pulls on A and C, so recalculate again.
Wait…
[https://en.wikipedia.org/wiki/Three-body_problem](https://en.wikipedia.org/wiki/Three-body_problem)
>In [physics](https://en.wikipedia.org/wiki/Physics) and [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics), the **three-body problem** is the problem of taking the initial positions and velocities (or [momenta](https://en.wikipedia.org/wiki/Momentum)) of three point masses and solving for their subsequent motion according to [Newton’s laws of motion](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) and [Newton’s law of universal gravitation](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation).[^([1])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) The three-body problem is a special case of the [n-body problem](https://en.wikipedia.org/wiki/N-body_problem). Unlike [two-body problems](https://en.wikipedia.org/wiki/Two-body_problem), no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_solution) exists,[^([1])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) as the resulting [dynamical system](https://en.wikipedia.org/wiki/Dynamical_system) is [chaotic](https://en.wikipedia.org/wiki/Chaos_theory) for most [initial conditions](https://en.wikipedia.org/wiki/Initial_condition), and [numerical methods](https://en.wikipedia.org/wiki/Numerical_method) are generally required.
…
There is no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_expression) to the three-body problem,[^([1])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.
More history of it in the “n-body problem” page
[https://en.wikipedia.org/wiki/N-body_problem#History](https://en.wikipedia.org/wiki/N-body_problem#History)
There is an analytic solution but it’s not practical…
>That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman’s series were to be used for astronomical observations, then the computations would involve at least 10^(8,000,000) terms.
It’s a problem in modelling the behaviour of three or more objects interacting through gravity (but also electromagnetism).Â
Let’s start with the thing that isn’t a problem: Two bodies. With two bodies you can mathematically create an exact equation, with an exact solution, of how the two objects will orbit each other. The only variable left is time and you can go infinitely into to the past or future. With infinite accuracy.
When you try to do the same for three or more bodies you run into a problem. There’s more variables/unknowns than equations. In maths, this means you cannot have an exact solution where you simply go forward and backwards in time. There’s no “analytical” solution.Â
That’s not to say there’s no way to solve it at all. But it requires making some guesses and then running the maths over and over hoping it settles on an approximate solution. Then you advance a time step and do it again. And again.Â
And “approximate” and “time step” are key words here. A solution, not THE solution. An approximation. And with a resolution in time, the more resolution you want, the finer you have to make your steps, the more work you have to put in. And even then you will always eventually diverge from the true solution if you run too far into the future, as the errors accumulate.
It’s not that it’s unsolvable. We don’t have the solution because it’s extremely complex because the result is very sensitive to initial conditions.
In 2 body problem (for example, Moon orbiting around Earth), you can predict very accurately where the Moon is exactly at any given point in time because the orbit of the Moon is not affected to a significant degree by any third body.
But in a 3-body problem, body A affects body B, body B affects body C, body C affects body A, which affects body B, so the position of bodies at some future point in time is very sensitive to initial positions.
Imagine a simple pendulum. Just a simple ball hanging off a simple string. With the pendulum formula, if you know the initial position of the ball, you can predict where the ball is going to be after, say, 10 seconds of swinging. If you **slightly** change the initial position of the ball, the position of the ball after 10 seconds is also going to **slightly** change.
However, if you have a double pendulum ([Double Pendulum (youtube.com)](https://www.youtube.com/watch?v=U39RMUzCjiU), if you **slightly** change the initial position of the pendulum, the position of the pendulum after 10 seconds is going to be **very** different, which makes it very difficult to solve mathematically.
I’m just wondering why an advanced alien race needs to solve the problem at all.
They made sophans.
Why didn’t they just send those to spectate the suns and send back live data?
So why 3 suns in the show.
Isn’t 2 suns and one planet enough to be a 3 body problem?
* “Body” is just a physics word for ‘thing made of matter’.
* Things made of matter typically (always?) have mass.
* Gravity is a property of objects with mass.
* We have physics theories that allow us to apply powerful mathematics to accurately describe how objects move due to gravity.
* So, therefore, you’d *expect* that we can just consider any group of ‘bodies’, apply our mathemtatics to them, and now accurately describe how objects move.
but there is a problem here. The mathematics is hard to *exactly* solve.
* If you have 3 things with mass, it turns out that we usually cannot calculate how they’ll move. We can approximate it, but eventually the approximation will be wildly wrong.
* (In some special cases we might be able to solve it, like if you imagine them in some perfectly symetrical scenario, for instance. But in general, we cannot solve it.)
* It is possible that a reliable solution exists, but we haven’t found one, and for all we know, it might be impossible.
What actually happens in real world scenario tho? Can’t we replicate that in a closed vacuum chamber?
In short, in the show (and the books) it’s wrongly stated that it’s “not solvable” or that “it’s unpredictable”.
The 3 body problem, or rather N body problem, because it’s the same for any number greater than 2, is the problem of, how will the bodies move, if they all interact on each other with attractive force. The N(3+) body problem does not have a so called “General analytical solution” which means you cannot find a mathematical formula for the curves of your bodies. For two bodies you can do this. This means that the problem has to be solved numerically, which is an approximate solution.
Numerically solving something means you take the positions of your bodies, like a snapshot, and calculate all the forces that they interact with, then, you calculate how much they will move in a brief time period if those forces were constant. Then you use this new snapshot to calculate new forces, then again, and again.
The challenge with this type of solving is that it depends a lot on these time steps, and how you calculate the things, so, sometimes, you can acumulate a big error. However, for somethiong like a star system, a civilization as advanced as the Trisolarans, should have computational power to estimate the evolution of their system very, very, very accurately on the scale of millenia.
Technically, and despite the name, I believe the problem in the series was a 4 body problem.
It was a planet within a 3 body system.
A three body system is an example of a **chaotic** system. This means that while we technically know how the system behaves and can simulate its future states (in this case we know all the equations of gravity), a very very small change in any input variable can drastically affect the final output to the point of it being meaningless. This means to perfectly predict such a system into the far future we have to calculate its current state to effectively infinite precision, which is impractical/impossible.
For a start, it is important to point out that the author of 3BP doesn’t know science, he even got the name of the problem wrong. Â
But anyway. 3 body problems are related to solving the movement of a single massless body in potential created by 2 massive bodies, for example, an asteroid in a solar system with sun and jupiter. There are no known closed form solutions (or even reasonable fast converging series) for the position of the massless body at some point in the future.Â
There are really good numerical solving methods for this problem, but that is not enough. We are dealing with a chaotic system, so the final position depends extremely strongly on small inaccuracies in the measurements
Fundamentally, solving this mathematical problem would not have changed anything. Your predictive power would be limited by accuracy of your observations just at it is without this solution.Â
I just did some research on this and want to throw this out there, framed in a way that makes sense to me but different from the other comments:
We know the problem has not been or cannot be solved in closed form, but can be projected forward with math to any point in the future.
In an idealized scenario (e.g., three digital bodies where we know the exact mass, position, and velocity), this can be projected with 100% accuracy to any point in the future.
The issue really is one of measurement constraints. For example, if the mass is a tiny bit less than assumed, then it will cause an error that magnifies over time.
Like, there might be a point in the interaction where mass A by the skin of its teeth won out in pulling C towards it rather than B. If A was a little less massive, it might lose the tug of war then set everything on a radically different trajectory.
So not only does a measurement error magnify over time, the magnification itself progresses chaotically.
All the other answers eplain really well why there is no (analytical) solution to the problem. What I’d like to add, and it is a minor spoiler in the series, is that >!the aliens decide to leave their planet not because they can’t predict with a decent margin of error the position of the suns but because their three body system isn’t in a stable configuration which means that eventually two stars may collide or one may leave the gravitational pull of the others meaning they could either freeze to death or fall into a star or any other crazy less than ideal situation!<
Other people have already explained what it is, but I wanna point out that there are actually a good few solutions in specific cases! 3 body problems become non-chaotic if you make a few assumptions, and this leads to some really useful orbits.
The easiest solutions come from 3-body systems where the mass of one of the bodies is insignificant when compared to the other two. This would be, for example, a Sun-Earth-satellite system. In this case there’s 5 specific points where the orbit of the satellite is completely stable around the other two bodies, called Lagrange points. These points are particularly interesting because they’re fixed to Earth. L1 is between the Earth and the Sun, making it so a satellite there has an uninterrupted view of the Sun. L2 has the Earth between it and the Sun, meaning the Sun is always covered by Earth. L4 and L5 are passively stable, which means perturbations will be corrected automatically without needing to expend fuel, making them absolutely full of captured asteroids ready for mining.
Other people here in this thread already explained it well, so I will just add my observations.
The problem is, most of the 3 body systems are not stable. With two bodies – typically one massive – such as a Sun and other much smaller, such as a planet, the system is stable, so it is possible to predict their behavior far into the future based on measuring of their initial positions and velocities. With three bodies, the smallest deviation in starting conditions might grow exponentially over the time, so it is impossible to predict where bodies would be in future.
This is best demonstrated by a double pendulum – also sometimes called a chaotic pendulum. Have a look at some videos talking about this. It is much easier to see.
The Sun, Earth and Moon are three bodies, with other planets adding complexity, but the Earth + Moon behave towards the Sun as a one tiny body for practical purposes of us predicting their rotation far into the future.
It’s pretty easy to use Isaac Newton’s basic laws to predict what will happen when two objects come into each other’s gravitational range. They will pull on each other. Maybe come into some kind of orbital arrangement. It just depends on their size and mass and movement.
However, just one more object introduces enough complexity that it becomes impossible to model what will happen with these same simple methods.
The “problem” has two meanings. On the one hand it’s a math problem/question which is hard to solve. On the other hand it’s a difficult situation where our tools break down and if we ever actually need to solve one of these, we’re in a bind. That’s an unacceptable risk, and that’s a problem.
Isnt there a solution atm?