Tbh he solved it by complicating it even more than it already was. An infinite sum is essentially a hack at a non-time dimension iteration option instead of step by step time iteration that computers use for simulations. It's still an approximation unless it converges, which you need to prove for specific boundary conditions, which is an even more difficult search than time iteration.
@@Caradorn I think there's just confusion about what he proved. A simple google search says he proved analytical solutions exist, but most converge at too many operations to calculate. Which is pretty much what we see - most three body systems have extremely complex movement that can't be represented by a simple function.
@@MasterOfYoda Thank you for clarifying. However what I was referring to more specifically was the general tendency for non mathematicians to see that an awful lot of mathematical theses are without immediate practical applications or at least an application yet to be found. Not all mathematicians can be Bernhard Riemann though. :)
Carl Sudman's solution is NOT useless in the general sense. We differential algebraists can study it and learn from it. And that may eventually lead to a practical general solution.
Sure, but also the solution itself to this problem is arbitrary because it’s humans that categorize particular configurations as being more special than others. It could in fact be that every state in the state-space is a useful conclusion because it describes what happens at that particular initial condition. Stability…instability…these are just haltings for a computation and potentially all of the unstable solutions might halt. Or they might not. If the state-space is finite then they will all halt, have a solution and what you learn in the end is that you could have chosen any initial condition and just wait long enough time for that to happen.
Pet project for me in 1990s. Newton’s two-body solution applied to each pair and averaged, single pixels for the bodies and writing directly to graphics memory - fast enough for 100 bodies. I don’t know three-body solutions but this sometimes gave patterns similar to those shown here.
That might just be the law of superposition but with extra steps. Or not exactly, but superposition does the summing, and I suppose without the averaging. Controlling the averaging, I guess that's weighting, so maybe modeling this version of chaos where simulated body ejection represents a decision being made in a chaotic system (like the human brain).
I beleive he mentions this in the video. you can compute a pretty exact estimate, but the maths is not broke down yet. you can only predict the next second then iterate
Great video! For anyone who enjoys a less mathematically but more narrative, literary approach to the idea, I humbly suggest reading "The Three Body Problem" by Liu Cixin (part one of a amazing hard-sci-fi trilogy)!
@@dankedozo there is also a Chinese adaptation (for Tencent) that has recently premiered. Not to be confused with the upcoming Netflix one by the GOT guys.
The "infinity" one is easier to find by hand because just like with the circle the three are on the same path as one another there is a similar "impossible triangle" where they all cross the center (or near it) and do a loop(classical newtonian style) but they must be exactly in sinc or the system becomes unstable and therefore you might not be able to simulate it by the step method due to floating point errors stacking up. Another one ought to be like those linear gear thingies forming a drawing similar to the gear inside gear children's drawing toy(similar to the upper right graph, but more loopy (near 4:24)), but i need more thought on that one.
3blue1brown its the channel who made the visuals, i dont know if the whole video its reuploaded from his channel because the voice its different but at least the format its his channel
Thank you for this video! I have no math skills whatsoever but I've always been fascinated by the three body problem.This was a very enjoyable and informative video! Simple and well explained!
What happens if you add even more bodies? Does it just become more and more chaotic as the number increases, and stable solutions get rarer? Also, are the animations shown all in a 2d plane, or are they 3d solutions that were simplified for the video?
@@mitchellsteindler you are right about 3 points always being in some plane or other, but 3 vectors is a whole different kettle of fish, and 3 bodies have 3 vectors.
@@mitchellsteindler I think it is relavant since, as the simulation progresses, the plane that contains all 3 will (most likely) change. It would be a special case if that plane remained constant.
Great video. It would extra nice to add the recent research on probability modeling of the three body problem. Can you give a simple explanation of recent work such as "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters"?
The reason why we haven't solved the three body problem is because we still do mathematics only from left to right ( figuratively speaking ). We have yet to discover top to bottom, diagonally and through; again *_figuratively _* .
Hey, amazing video. I was just wondering if you could upload like a 10 minute footage of just the animations to the problem, i think it's pretty neat and satisfying to watch!
When bodies get very close and very fast, the estimates here still look wrong. e.g. at 7:51 the green+blue have sharp corners and a straight line in between. One code optimization would be to use smaller delta-t for states with larger v
It might still be correct depending on the integration method. Adaptive integrators may output states at a fixed step even though internally they subdivide in order to maintain accuracy. Symplectic integrators have the additional feature that they enforce energy conservation a priori so that even inaccurate solutions still obey the conservation of energy (which explicit methods don't).
Some bad astronomy: Hierarchical triples and higher multiples: there is quite a sizeable fraction of them, so one can't say that "almost all" triple stars dissolve into binaries, that is factually inaccurate. Hypervelocity stars are a special beast from regular runaways from disrupted multiples. They come specifically from the center of the galaxy, through interacting with a supermassive black hole, not part of a primordial multiple system with it. If a star (any star, fast moving or not) were to pass by our solar system - in the vast majority of cases we wouldn't notice it even slightly. Most of the Galaxy is empty, the probability of anything coming in close enough to disrupt the planets is even a little bit is incredibly tiny - galaxies can merge with neighboring galaxies without disrupting the planetary systems.
>doesn’t happen with one body >doesn’t happen with two bodies >happens with 3+ bodies Interdasting. I know it’s obvious why it happens but it’s still kind of spooky.
pointing this out reminds me of one problem where adding some sequence of sine waves equals pi up until 14 iterations, or something like that. 3Blue1Brown made a video on it I believe.
5:53 Okay, you're right about it becoming a binary system after the third celestial object is ejected, but the simulation shows that the body being ejected is still under influence by the bodies forming that binary revolution. Eventually, that body will rejoin the system and become chaotic again, and probably result in another ejection, most likely more violent the next time. But as long as there isn't any influence by other objects in that space, the bodies will rejoin in close proximity again, which *will* result in another ejection again, having increasingly more velocity each time. It may be possible to use transformed sine and cosine relationships, along with secant deterministic boundaries, to explain why the binary system's celestials are still attracting this body, along with how much they themselves are affected by its gravity. This, as a result, could begin to predict the paths in which two bodies in a binary system will follow when affecting and being affected by an outside, oncoming celestial object, if worked backwards from points back to its start. On another note, chaos is controllable, not predictable. There will never be a single equation to describe paths that 3 celestial bodies will take, because the influence they have on each other, and are affected by as a result, also have influence on the influences that those bodies have on another different body in the system. They can only be solved when set to a uniform, or constant, start, and when that start is changed, the deterministic equation will *also* change. That is the reason why there is no uniform equation to this problem, and also why computers are able to generate the trajectory of all three bodies, and we cant. The solutions showed in the video use that exact same reasoning, in which the chaos is controlled, not predicted, by influencing the start of the simulation to a more stable system, rather than trying to solve a radically random system that does not have any control.
the chaotic paths these points take, while they cannot be confined to a single expression, or function -they CAN be described by acknowledging charge differential and polarity (of said points). Ie: the chaotic structure of natural circuits.
Me too man. I was talking with my homie just last week about this. Pooky was like, "heY! what they doin about that three bodoy problem?" I was like ''ino-know, but they takin forever huh?" Pooky be so impatient. He gonna trip when I tell him that they done solved this in 1912. I don't know how we missed that one. We always heard that it hadn't been solved too. Okay, maybe we hadn't always heard that cause who the hell talks about it but yea... crazy righ?
I'd like to see 3 non-uniform bodies of different masses (Sun - Earth - Moon) integrated / transposed with (Sun - Mercury - Venus) (Hg -V-E/m) (V-E/m-M) etc. Matrices / laplus transformation in simulation. Initial conditions are The bear but in RL a solar system always starts with the center mass of star formation.
Cool, I think I got something of that nature. I don't think it's a general solution, but cubic waves are much prettier than that. These are like gear of time.
Isn't the condition for "chaotic" motion true for any general motion? Like yes if you were to change the velocity slightly, the system would evolve differently because you changed something. Whats so "chaotic" in this?
Really well made video, I've been toying with a new method for estimating solutions to the 3 body problem, would you be able to share how you were able to simulate the movement of the 3 simulate bodies?
I've done this for purely Newtonian physics, and it really is as simple as summing the gravitational acceleration of each body on every other one, calculating the resultant positions and velocities IF that acceleration is constant over your chosen simulation "tick" period, and repeat and repeat. I think adding reletavistic physics to the system shouldn't be too hard. But... With the purely Newtonian computer model, the orbits precess, something that should only occur from considering relativity. In fact that is one of the very earliest pieces of evidence that relativity is valid.
@@User-jr7vf For the Earth, perihelion (closest to the sun) and aphelion (furthest from the sun) move year after year. The elipse of the orbit moves. If you view this orbit from a place outside (above or below) the ecliptic plane (all planets' orbits lie in a plane near enough) such that the Earth moves clockwise around the sun, the elipse will move counterclockwise. Your easiest search for more (better) information is "precession of the equinoxes"
The statement at 1:22 is not correct. The 3-body problem arises as soon as there are two massive bodies and a third that can be massless, in which case it's the restricted version of the problem, but the same problem nonetheless because it has no solutions in the general case, being chaotic.
I think that would count as how he first mentioned we can predict how moons orbit around their planet that orbits a sun, each star is so tiny and insignificant compared to the galaxy it orbits in
It's similar to our solar system how it has many planets orbiting a sun that massively outweighs it making it stable allowing perturbation theory, the galaxy with its stars and centre black hole is like our sun and asteroids
Yes, because there are other forces in the universe than gravitation. Electromagnetic interactions mean that when two bodies collide, they stay stuck. All unstable three body systems have degenerated into stable two body systems or dust disc models. I don't think we're missing anything; you don't see three body systems of equal mass out there in the galaxy because it is so unstable
Chaos theory, Heissenberg´s uncertainty principle and Newton´s three body problem are different probability approaches to one and the same certainty. Trinity. Anyway, chaos theory is beautiful...
Ok so are we not gonna talk about why a lot of the stable solutions look a lot like complex knots? Because I've seen a few videos on stable multi-body orbits and shit is SUSPICIOUS
Can't you just have the computer calculate for the two-body problem for each pair of objects. and then you just add up each force/direction applied to each object?
Doesn't have to worry with a nice Runge Kutta 4! If not, a basic solar system program I wrote in Python wouldn't have worked. So, numerical methods my integrator RK4 But, the initial position of the bodies had to be a real one, then it works fine.
Just a small tip, try mixing the background music to be even lower than your voice. It's a very common mistake among new RU-vidrs, but the purpose of the background music should be to fill silent moments, it should not be competing with your voice as it can be difficult for the viewer to make out your words. I enjoyed the rest of the video though!
