Daniel and Kelly’s Extraordinary Universe - Why is the three-body problem so hard?

Episode Date: August 12, 2021

Daniel and Jorge talk about why something so simple is one of the oldest and hardest problems in physics. Learn more about your ad-choices at https://www.iheartpodcastnetwork.comSee omnystudio.com/listener for privacy information.

This is an I-Heart podcast. December 29th, 1975, LaGuardia Airport. The holiday rush, parents hauling luggage, kids gripping their new Christmas toys. Then, everything changed. There's been a bombing at the TWA terminal. Just a chaotic, chaotic scene. In its wake, a new kind of enemy emerged, terrorism. Listen to the new season of Law and Order Criminal Justice System

On the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. My boyfriend's professor is way too friendly, and now I'm seriously suspicious. Wait a minute, Sam. Maybe her boyfriend's just looking for extra credit. Well, Dakota, luckily, it's back to school week on the OK Storytime podcast, so we'll find out soon. This person writes, my boyfriend's been hanging out with his young professor a lot. He doesn't think it's a problem, but I don't trust her. Now he's insisting we get to know each other, but I just want or gone.

Now, hold up. Isn't that against school policy? That seems inappropriate. Maybe find out how it ends by listening to the OK Storytime podcast and the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. I'm Dr. Scott Barry Kaufman, host of the Psychology Podcast. Here's a clip from an upcoming conversation about how to be a better you. When you think about emotion regulation, you're not going to choose an adaptive strategy,

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Apple Podcasts, or wherever you get your podcasts. Every case that is a cold case that has DNA. Right now in a backlog will be identified in our lifetime. On the new podcast, America's Crime Lab, every case has a story to tell. And the DNA holds the truth. He never thought he was going to get caught. and I just looked at my computer screen.

I was just like, ah, gotcha. This technology is already solving so many cases. Listen to America's Crime Lab on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. Hey, Daniel, when you are teaching, are you the kind of professor that assigns super hard homework in your classes? Like, find the motion of a banana tied to a string held by a squirrel, riding on a roller coaster, all of that in orbit around a black hole, like that kind of problem. What, are you Professor Rube Goldberg? No, that was just a joke.

I actually like to make the homework just a little bit harder than what we work on in class. You know, that's where the concepts really come together in your mind. Right, right. You're an evil professor, basically. You never assign unsolved research problems to, first year students? No, that only happens in the movies, man. Goodwill hunting is not a documentary. You're not Matt Damon? I don't have the looks for it. Yes. There's always room for improvement. Hi, I'm Jorge. I'm a cartoonist and the creator of PhD comics.

I'm Daniel, I'm a particle physicist, and I've never solved an outstanding math problem. Not yet, do you mean, right? Like, if you had solved it, it wouldn't be an outstanding math problem. That's true, yeah. There are these famous outstanding problems, and it's cool when they stand for hundreds of years. And then somebody comes along and figures them out. Wow. What do you think happens?

Like somebody just comes up with the right way to look at it, or like they see something nobody else had seen before? Or the history was just waiting for the right intellect? Yeah. Sometimes it's a slow construction of, ideas over hundreds of years and you look at the history of it and be like problem proposed in 1619 progress is made in 1814 and then 2017 Samantha figures it out and it's pretty awesome to see the stretch of history there I'm waiting for people to solve some pretty intractable

parenting problems that sometimes they have those are eternal man they will never be solved they'll never be solved it's part of being human I guess but welcome to our podcast Daniel and Jorge Explain the Universe, a production of I-Hard Radio. In which we tackle the hardest problem, which is understanding the nature of this universe we find ourselves in. How does it work? Where did it come from? Why is it the way that it is?

And is it even possible to understand it? We dive right into the biggest, hardest, deepest questions. We explain the answers and our ignorance to you. I wonder if that's the harder problem, Daniel, explaining something to other people. We do our best here, but it's pretty hard to sort of wrap your head around all the amazing and incredible stuff that is happening in the universe. And one of my favorite things about doing this podcast is exercising that part of my brain

that translates these ideas from like the cutting edge of physics to things everybody can understand. Because to do that, you have to have a really good grasp on what's going on. Oh, I see. You actually have to understand it first before explaining it to people. So what am I doing here, Daniel? Well, sometimes when you try to explain something, you realize, hold on a second. I don't really understand how this works as well as I thought it did.

Sometimes. Only sometimes, though, that never happens on our podcast. It happens to me all the time when I'm teaching and also on this podcast. And that's one reason why it's so fun. Because not only are we explaining stuff, we're also learning as we go. That is a pretty good parenting lesson also. It's good to share what you know, what you learn, what you love about this crazy, beautiful cosmos. Yeah, how does that help you with your parenting?

Well, it's just good to share, I think. Well, you have to share with your children. I think that's a law. That is a rule. But it's also good to teach it to share, you know? Everyone's more generous with what they have and their knowledge. We're all happy. Oh, I thought you were going to use the wonder and glamour of the universe

to convince your kids to do their chores. Like, take out the trash because stars are amazing. Or you are insignificant in this universe. You're a tiny speck of dust in the floating and vast vacuum of perhaps infinite space. And therefore, you should do your homework. I think that'll work against you. So then why should I bother taking out the trash if nothing matters? Because if nothing matters, children, everything matters.

I like that. I like that. Turn it philosophical. Use that PhD for something. But anyways, we do like to talk about not only what scientists know about this universe and all of the wonderful stuff in it, but also what scientists are struggling with understanding about how things work. That's right, because we have this amazing mental machinery of science that lets us build up a body of knowledge, things we do understand about the universe. It has a machinery to it and that machinery is mathematical. It's incredible to me sometimes that mathematics can describe the way the world works at all. You know, you throw a baseball and it follows a parabola, which is a very simple mathematical relationship.

So it's incredible when you can use mathematics to describe what's really very complex behavior, all sorts of zillions of particles moving through the air altogether. But sometimes it's easier than other times. Yeah. The cool thing about science is that it's always at the sort of the leading edge of human knowledge, right? Like, that's what science is. It's sort of like asking the questions, nobody's ever asked or finding the answers nobody has so far.

And so sometimes you run into things that are just really, really extra hard or maybe even impossible. Yeah. And sometimes they are impossible because the physics is really hard. And sometimes they're impossible because we just don't have the mathematics yet. Like there have been lots of times in history when mathematicians have developed tools, not because they thought they were going to be useful for physics, but just because they thought it was fun.

And then later on, physicists were like, on a second, that totally helps me solve this problem I've been struggling with for 20 years. A great example is general relativity, which is built on geometry, which was developed just 10 years before. Without all that work developing geometry, there's no way Einstein could have developed relativity. It's this really fascinating dance between mathematics and physics. Yeah. What kind of dance? How would you describe that dance? Is it like a Charleston or more like a Waltz or is it like a hip-hop break dancing competition? What would you call it? The mathematicians carefully build their tools and we just sneak in and steal them.

So maybe it's more like a cat burglar dance. Oh, man, I can't wait for that, you know, interpretive dance, history of science, Broadway play that you're working on. Yeah, you know, I wish it was more back and forth. Sometimes I feel like shouldn't the mathematicians be excited when their tools actually get used to describe the real universe? But a lot of times, they don't seem to care at all. They're like, whatever.

Who cares about the real universe? I'm walking the halls of truth. You're sullying their halls with like reality and like real mud and dirt. Like that's just dirt. Adams are just dirt. Yeah. If they cared about getting dirty, they would have been physicists instead of mathematicians. I see.

Physicists are the down and dirty of scientists. I think physicists are to mathematicians, what engineers are to physicists. Oh, I see. The better people. Right. The true heroes. On the hierarchy of useless purity. The hierarchy of usefulness.

you mean. Depending on what you put at the top, yes, exactly. Right, yes. If you turn upside down, we're actually at the top, yes. It's all about your perspective. That's right. There is no up in space anyway. Well, there are interesting problems in physics, some of them, which are even intractable.

And so in this episode, we'll be talking about one such problem that maybe affects our very movement through space, and it affects how planets revolve around their suns and which we may never find the answer for. So to the end of the podcast, we'll be asking. the question. What's so hard about the three-body problem? Now, Daniel, this is not something that's not safe for work, is it? I mean, I see something here with three bodies.

Is this about, you know? No, this is not about being exploratory in your relationships at all. It's what's so mathematically difficult about three gravitationally attracting objects. It's the safe for work. In the territory in heavenly bodies' relationships. Some bodies here on Earth are quite heavenly as well, but we're talking about celestial bodies. That's right.

The real stars. All right, so the three, and more specifically, this is kind of about what is the three body problem at all? Because I imagine not a lot of people have heard of them. Although it is the title of a sort of a well-known science fiction novel out there, right? That's fairly recent. That's right. Yeah, it's like one of the biggest novels in the last few years.

It's a whole trilogy written by a fantastic, Chinese author. A lot of people are really into this book. And a lot of our listeners have written in asking us to talk about this book. But I thought first, maybe it'd be more fun to talk about like the physics problem that's at the heart of the novel. That we can talk about the actual problem itself. Yeah, I think I try to read the novel. It's pretty dense. It's kind of thick. Yeah, there's a lot of physics in that book, which is pretty fun for people who like really well thought out physics novels. And so it's a good idea to try to get an understanding for like what is the underlying problem at the core of the story.

Right. And it's like a bestseller and won all the awards, right, in science fiction? Mm-hmm. So you can check that out if you like. But the title of it refers to kind of an old and famous problem in physics about, I imagine, three bodies. That's right. It's a really old problem. And old problems are the funnest problems because it means that like a lot of smart people have been budding their heads against this problem for decades or even centuries.

And nobody's figured it out. And that doesn't mean it's impossible. There are other mathematical problems that have existed for hundreds of years. And then all of a sudden, some dude in a cabin in Russia comes out with like a hundred page proof of it. So it might be possible to be solved, but nobody's cracked this one. Yeah, just a book on Airbnb, that cabin in Russia and, you know, book it for a couple of years and you might solve a famous problem. That's the real answer. That wasn't a metaphor.

That really happened. To you? Not to me. No. There really is a Russian mathematician who worked all about. himself for a decade and solved a famous problem in math, the Riemann conjecture. Wow. And he was in a cabin? He was in a cabin. He worked all by himself and he just sent in the solution. And they tried to give him the Fields Medal for it and he wouldn't even

show up. Wow. That feels like such a fine line between like, you know, genius and, you know, socially unacceptable behavior. He's well on one side of that line. But anyways, let's talk about this problem, the three body problem. And so as usual, we were wondering how many people out there knew what this was, if they had heard of it before, beyond the novel, or how important it is to sort of predicting the movement of our planets in our solar system. So Daniel, as usual, went out there into the wilds of the internet to ask people, what is the three body problem? So while we are still pandemically shut down, I am very grateful to all of you who are willing to participate via email on the person on the virtual street.

interviews. So if you would like to participate and hear your speculation on the podcast, please don't be shy. Send us a message to questions at Danielanhorpe.com. Think about it for a second. Do you know what the three-body problem is? Here's what people have to say. I don't know what the three-body problem is. I'm afraid. So I'm fairly aware of what the three-body problem is. I did read as you shouldn't lose three-body problem trilogy. My understanding is that it's a problem with how three bodies orbit one another and how it could continue to do that and be stable without crashing into one another. A lot of people spend a fair amount of time calculating how two massive bodies interact due to the gravitational field surrounding them. But actually if you add

a third body, the system becomes unstable. It becomes chaotic. So you can't determine an exact solution and also if you make a small change let's say in the initial positions of the bodies you can't actually determine how let's say the forces between the three bodies will be affected i think that's to do when you've got three bodies that gratation interacts you like the sun the earth and the moon for example would be three bodies and i think you can solve two bodies and any more than three or more and you can't solve it, I think, is one of the issues. Oh, wow, that is something I am not sure what it is. I don't know what the three-body problem is, unless it's relating to a previous question

where if you have three bodies acting on each other gravitationally, you haven't got sort of one orbiting another or one with a joint orbit, another, there'll be probably quite a sort of random implication to their orbits. I have never heard of the three-body problem before, but if I were to guess, I think it is three bodies interacting with each other and something unusual happens, like something that doesn't happen between two bodies or four bodies. It just happens between these three bodies for some reason and for some reason the number is three. Actually, I've studied physics before, so I know that the three-body problem is this problem where if you have two objects pulling on each other,

then those equations can be solved pretty easily. But if you add in a third body, now you have three different interactions between A, B, A, A, C, and C, B. And when you have interactions of that order that many interactions. It becomes sort of an unsolvable math problem. And so we don't have like good solutions for those sort of situations. We have to essentially come up with approximations and simulate it. All right. Not a lot of knowledge about this, but someone did read the trilogy. Yeah, exactly. Three books in the three body problem trilogy. It's nice. It must have been good because he read all three. Or I wonder if your completest, you know, tendencies would kick in after you rewritten. Well, I can't just read one three body problem book.

I got to read all three. Depends if they leave a cliffhanger at the end of the first novel. You should title all your trilogies with the number three in it. But it seems like most people here are guessing it has to do with bodies and space, and specifically three bodies, of course. But a lot of people are saying maybe it's about it becoming unsolvable or chaotic or unstable. Are they sort of in the right track? They are exactly on the right track.

It's really interesting. There's a problem which is easy if there's only two objects involved and then becomes basically unsolvable if you have three. objects involved. Right. Like real human relationships. Which can be tricky even when there are two bodies involved. Even if everyone is open-minded, it gets tricky. All right, well, let's dig into it, Daniel. How would you describe the three-body problem? I think the best way to describe it is to first talk about what we can do. And simply said, if you have two objects in space and you know where they are, how heavy they are, and the direction they're going in, then you can predict their

motion. You can say at some time in the future, I know where they are going to be. So, for example, imagine just the sun and the earth. These are two objects that pull on each other. There are forces involved. And if you know where the sun and the earth are at some moment in time and which direction they're heading and their masses, you can write down a very simple formula that will tell you where they will be in the future. Like you say, where will the sun be in a year or in a thousand years or in a million years? It's like a very simple mathematical expression. You plug in the time, tells you where the sun will be. So that's the two-body problem. And we have a solution for that. We can crank through the mathematics and get a very nice simple formula that tells us where they

will be at any moment in the future. Right. But you have to kind of assume that they're alone in the whole universe. Like there's nothing else in the universe pulling on them, right? That's right. Only two bodies. And as usual, you know, physics is telling a story and that story is always approximate. The reality never matches the approximate stories we try to use when we tell physics stories because in reality there's an infinite number of bodies out there in space and gravity works for over infinite distances and so everything in the universe is tugging on things all the time but usually you can get away with disregarding that you don't have to care about what's happening in indromeda when you're doing the calculation of whether your satellite is

going to go around the earth because it's basically zero contribution so here we're talking about the scenario where you have two bodies and everything else can be ignored without changing anything down to like you know the 10th decimal place or something yeah so in the the sort of simplified universe of exactly two things in your universe, you can predict the motion of two objects. All right. So then I'm guessing when you get to three bodies, it gets a little harder. When you get to three bodies, it doesn't just get a little harder.

It becomes impossible. If you know where three objects are, you know, say you have, for example, the sun, the earth, and then another object. Now you just have three objects and you know exactly where they are, what direction they're going in, and you know they're masses. You cannot write down a simple formula that tells you where they're going to be in. in a week or in a year or in a thousand years. Whoa, it gets really complicated suddenly.

It gets really complicated. We don't have a solution. Now, we have an understanding for what's going on. Like, we know the force is involved. We know what the gravity is between two objects given their distance, right? That's a pretty simple formula. Newton told us how to do that. But that doesn't mean we know how to find the solution.

It doesn't mean we can take those forces and predict the motion. Right. Well, I think this might be kind of a subtle subject for a lot of people out there, which is it like what you mean? in physics as a solution because it doesn't mean that you can't predict where they're going to be. You just don't have an easy solution to the equations to predict this, right? It means that we know what the constraints are. Like, physics tells you what the rules are.

It tells you, like, for example, how two objects pull on each other. It doesn't tell you how those objects are going to move. To figure out how the objects are going to move, which is what you need to predict their motion, you need to be able to solve all of those equations and get the answer out. So physics gives you, like, all the equations you need to solve, it doesn't mean you know how to solve the equations. Like not every equation you get is solvable or we don't necessarily have the mathematical tools

to solve an arbitrary equation. Turns out in physics, there are only like five problems we do know how to solve and everything else is intractable. Well, that probably makes for a short workday there for you. But I think what you mean is like, for example, like a ball, if I throw a ball here at my son in our backyard here, like I know that that ball,

I know the constraints on it. Like I know the force is pulling on it, the force of gravity, and I know that F equals MA, for example. So I can solve, for example, for its acceleration very easily, but maybe getting like a formula for what its position is going to be is a little tricky. It's different than knowing what its acceleration is going to be. Exactly.

The acceleration just tells you how is momentum is going to change in a given moment, right? To know where it's going to be, you need to then solve the equation of motion, which incorporate all these forces and is effective. affected by that acceleration, but it requires actually solving the equation. You know, it's like if I have an equation that says X plus 5 equals 10, right? That's an equation that constrains X. It limits what X can be, but it's not actually the solution.

The solution is X equals 5. That's a very simple one, right? You know exactly how to go from the equation X plus 5 equals 10 to the solution, but you don't necessarily know how to do that for an arbitrary equation. Take another simple example like x squared equals 49. How do you find the solution to that? You know off the top of your head that X equals seven works. You can plug it in and check it.

But how do you find the solution? If I tell you X squared equals an arbitrary number, how do you find the square root of an arbitrary number? There actually is no way to do that. There is no mechanism for solving that equation other than guessing and checking. Sounds like my parenting strategy right there. And I think what you mean is like, you know, in physics you have equations that tell you, for example, like the acceleration of x which is like how the velocity is changing which is like how the position is changing like you have equations for that but to actually get the position you have to kind

of backtrack from acceleration to velocity to position and that's where the trickiness comes from right yeah because the acceleration changes through time and so to figure out like how all those accelerations add up to describe the motion of the object is not always easy and then what you want is a simple formula that describes it and that doesn't necessarily exist right Because I guess when you go from two bodies to three bodies, then the formula just get too complicated? The formula gets too complicated, exactly. The system gets really complicated because now you have these three different objects pulling on each other, and it actually becomes chaotic. All right, well, let's dig into why exactly it is so hard and how it becomes pure chaos when you add a third celestial body into the mix.

And what consequences it has for our ability to predict the universe. But first, let's take a quick break. December 29th, 1975, LaGuardia Airport. The holiday rush, parents hauling luggage, kids gripping their new Christmas toys. Then, at 6.33 p.m., everything changed. There's been a bombing at the TWA terminal. Apparently, The explosion actually impelled metal glass.

The injured were being loaded into ambulances, just a chaotic, chaotic scene. In its wake, a new kind of enemy emerged, and it was here to stay. Terrorism. Law and order, criminal justice system is back. In season two, we're turning our focus to a threat that hides in plain sight. That's harder to predict and even harder to stop. Listen to the new season of Law and Order Criminal Justice System on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts.

My boyfriend's professor is way too friendly, and now I'm seriously suspicious. Wait a minute, Sam. Maybe her boyfriend's just looking for extra credit. Well, Dakota, it's back to school week on the OK Storytime podcast, so we'll find out soon. This person writes, my boyfriend has been hanging out with his young professor a lot. He doesn't think it's a problem, but I don't. don't trust her now he's insisting we get to know each other but i just want her gone now hold up isn't that against school policy that sounds totally inappropriate well according to this person this is her boyfriend's former professor and they're the same age and it's even more likely that they're cheating

he insists there's nothing between them i mean do you believe him well he's certainly trying to get this person to believe him because he now wants them both to meet so do we find out if this person's boyfriend really cheated with his professor or not to hear the explosive finale listen to the okay Storytime Podcasts on the IHeart Radio app, Apple Podcasts, or wherever you get your podcast. I had this, like, overwhelming sensation that I had to call it right then. And I just hit call, said, you know, hey, I'm Jacob Schick. I'm the CEO of One Tribe Foundation, and I just wanted to call on and let her know. There's a lot of people battling some of the very same things you're battling.

And there is help out there. The Good Stuff Podcast, Season 2, takes a deep look into One Tribe Foundation, a non-profit fighting suicide in the veteran community. September is National Suicide Prevention Month, so join host Jacob and Ashley Schick as they bring you to the front lines of One Tribe's mission. I was married to a combat Army veteran, and he actually took his own mark to suicide. One Tribe saved my life twice. There's a lot of love that flows through this place, and it's sincere.

Now it's a personal mission. I don't have to go to any more funerals, you know. I got blown up on a React mission. I ended up having amputation below the knee of my right leg and a traumatic brain injury because I landed on my head. Welcome to Season 2 of the Good Stuff. Listen to the Good Stuff podcast on the Iheart Radio app, Apple Podcasts, or wherever you get your podcasts. A foot washed up a shoe with some bones in it. They had no idea who it was.

Most everything was burned up pretty good from the fire that not a whole lot was salvageable. These are the coldest of cold cases, but everything is about to change. Every case that is a cold case that has DNA. Right now in the backlog will be identified in our lives. lifetime. A small lab in Texas is cracking the code on DNA. Using new scientific tools, they're finding clues in evidence so tiny you might just miss it. He never thought he was going to get caught, and I just looked at my computer screen. I was just like, ah, gotcha. On America's Crime Lab, we'll learn about victims and survivors, and you'll meet the team behind the scenes

at Othrum, the Houston Lab that takes on the most hopeless cases to finally solve the unsolvable. Listen to America's Crime Lab on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. All right, Diana, we're talking about the three-body problem, and I guess we're not just talking about, like, what happens if your spouse moves to another city, right? This is more cosmic. I can't solve that problem for you. There is no equation that tells you how to live your life. That's the one body problem is already pretty hard. Now we're talking about the three body problem. One is the loneliest number. But this is not a relationship helpline and this is not a podcast about human emotions.

We are trying to solve the much easier problem of motion of objects through space. And so you're saying that when I have two objects in space, it's easy enough to sort of predict where they're going to be able to. Once you have three, there's no easy solution to that problem. Yeah, there's no easy solution. There's no like short mathematical answer. What you'd like is something like X of T, where X is the position of the object, and then a simple formula where you can plug in the time and it'll tell you exactly the position of the object. That's what you're looking for, because you'd like to be able to take that system and say,

I want to know where the moon is going to be, or I want to know where the sun is going to be in a thousand years. The problem is that there is no such simple formula. We haven't found one at least, and we suspect that it might not exist because the system of three objects is much, much more complicated than a system of just two objects. Right. And it gets really complicated because now you have three objects in 3D. Is it kind of about going to that third dimension that makes it hard? Because I imagine if you have two bodies in space, you can just treat them as like a 2D problem, right? Like you just imagine the plane where these two bodies are. But once you have three,

then it's like now it's a 3D problem. It's true that two objects in space, you can always define a plane between them. You can also put three objects on a plane now, right? Three points define a plane. So there's always a plane for three objects. I think the problem is that when you have three objects, a small change in their location leads to a big change in where they're going to be in the future, at least for gravitational interactions. Whereas if you have two objects, a small change in where the Earth is going to be, it'll mostly settle back into the same answer. And so in terms of like finding an equation that describes it, there's a whole family of equations that can describe stable solutions. We don't really have functions that are very good at describing chaotic,

situations where a very small change in the angle or the velocity of the moon means it's now suddenly over here or it's suddenly on the other side of the sun or it flies off in a completely different direction. Our equations are not good at describing chaotic mathematics. But I guess maybe the question is like what is it that about going from two to three that actually makes the equations unsolvable? Like before with two I can solve the equations but with three there's no solution for them. Does it become nonlinear? Is that what it is? It's already nonlinear right that even with n equals two it's nonlinear because these distances go like one over radius squared so there's an inverse r squared there so it's already nonlinear i think something you said earlier

is really the right way to think about it we know the forces f and the mass m and we have f equals m a so we can get the acceleration that's a very simple formula but how do you go from knowing the acceleration which is how much the speed is changing to knowing the actual location what you have to do is add up the effects of lots of little accelerations over time, which means you have to integrate. You have to use calculus. But just like there aren't that many physics problems that are solvable, not every function can be integrated, at least not into a simple formula you can write down. So just because you know the force in the acceleration doesn't mean you know how to integrate it

into getting the location. And we can go a little bit further. If you look at the structure of the problem, mathematicians call these problems non-integrable, which just means that like the possible trajectories for these objects in this 3D space don't follow simple paths right like they diverge very quickly it's not like it can be easily simplified from a whole big set of possible solutions down to just a few and with n equals two with a two body problem there are a bunch of simplifications you can make that separate the problem so that for example the distance between the objects is independent of their relative angle because for two objects you know the angle doesn't really matter what only matters is really just the distance. But for three objects, you have not just the relative distances, but you also have the relative angles. And so now all the problems are still tied together. You know, when you try to solve a set of equations, sometimes it's helpful to try to separate them and solve them independently, but that's not always possible. And when they're all entangled up with each other, you can't always

find a solution. I see. There's something sort of magical about the number two that then you lose once you get more than two, right? Because it's not just three bodies that are hard. It's also four and five and six and infinite, right? Yes. You might have thought, oh, well, two bodies are solvable, so then why not three? It's actually the other direction. Two is the only one that is solvable, right? All the problems are unsolvable, except for this one magic special case of two bodies,

which we have been able to separate using this special trick and solve. So it's sort of lucky that any of them are solvable. Well, the zero body problem is solvable, too, and the one body problem, I imagine, is solvable. It's just that it just gets more complicated. Like the equations start to like interact with each other and then you can't like fit a simple formula as a solution, right? Yeah, exactly. And in addition, there's something about chaos here, right?

That's right. The results become chaotic. As we said before, if you change a little bit the initial conditions, if Earth is a little bit further away or pointing in a slightly different direction, you can get completely different outcomes. So Earth can be like tossed out of the solar system or it can fall into another orbit or something like that. Whereas if you just

have two bodies, things tend to be pretty stable. That means that if you perturb it, something comes along and gives the earth a little push, it'll probably roll back into its initial orbit. Whereas in a three-body system, things get out of hand very quickly. And that's, you know, not just like, is it complicated motion?

That's one of the reasons why we don't have a simple formula because we don't have functions that describe that, like sign and cosine and logarithm. These things are mostly well-behaved. And so it's very difficult to describe chaotic motion using the sort of mathematical language that we have developed. Oh, I see. Because there's no function that is chaotic kind of. Is that what you're saying? Like chaotic motion is not easily kind of captured in a formula.

Yeah, it's not easily captured in a formula. It's possible to describe chaotic motion, but usually our solutions there are numerical. They're approximate. We use simulations. You know, we can describe chaotic systems like you build a, computer system, you put three objects in it. And then what you do is you say, all right, what happens in the first second? And you say, well, the Earth's going to move this way, the sun's going to move that way, and the moon is going to move this other direction. And then you update everything. And then you do it again. So you slice the problem in time and you say, what if I only want to predict a half second from now or a millisecond from now? Then you can

simplify it and say, I know what to do for a half second. And then you just do that over and over and over again. That's a way we can describe a chaotic system is like slicing it in time, and then try to move our simulation forward just one time slice at a time. But that doesn't mean that we can then look at that motion and say, oh, look, it follows a sine wave or, oh, look, it follows a logarithm of a sine wave. We can't find a solution. We can't find a mathematical description of the motion, even if we can describe it in the simulation.

I see. So, like, we can maybe predict what the system is going to do, what these three bodies are going to do, but we have to do it step by step. We can't just say, like, hey, 20 years from now, this is what it's going to be. There's no formula that will tell you that you have to like simulate it little by little till you get to 10 years from now. Yeah. And even those simulations are difficult because it's chaotic. Like if you don't make those calculations very, very precise, then your simulation is going to be wrong as you try to predict further and further into the future because small mistakes really add up they snowball into big mistakes. It's just like, you know, the butterfly problem. the butterfly flaps its wings in China and that has cascading effects on the weather which causes

eventually a storm in Central Park in New York. And these things are real, the real physical systems that behave this way where if you give them a very small nudge, it can have a very big effect downstream. And that makes them very, very challenging even to simulate as we talked about because you get something wrong very early on, your results in 10 years are nonsense. We much prefer to have like a simple, we call it an analytical formula, like a very short math expression that we can just plug numbers into because it can be exact. And it can tell us exactly what's going to happen in 10 years or in 100 years. I think what you're saying is that these numerical approaches or simulations, they're just an

approximation basically, right? Like you're looking at the equations like the F equals MAs or the, you know, the forces between the three bodies and you're saying, well, let's not try to get the exact solution. Let's just pretend that for the next millisecond, everyone has the same acceleration or something like that, right? Exactly. You make a bunch of simplifications and you say, well, I only want to predict a millisecond in the future.

So can I do that? And then you just keep doing that over and over again. You're saying that if I'm wrong a little bit because of that simplification, then in a chaotic system, I could be really wrong. Yeah. And that's a big deal if you're doing something like planning a trip to the stars or sending your probe to Jupiter or whatever. You definitely want to get that right.

Right. Yeah. You don't want to be off by a few light years. Yeah. Or even if you're just flying through the solar system, if you get it wrong, you could end up crashing into the sun. or getting tossed out of the solar system in the wrong direction. You're trying to make it to Pluto from here, right?

Pluto is very far away and a very, very small target. Imagine firing a gun from L.A. to New York and trying to hit a tiny, tiny target. It's very difficult. They're very small. If you're off by a tiny little angle in L.A., you're definitely not going to hit that target in New York. But I guess, you know, we are pretty good these days with, you know, supercomputers. We are pretty good at simulating things and kind of predicting, you know, maybe not the storm

that comes from the butterfly wings, but, you know, the weather is, you know, predictable sort of up to like a week, right, or a couple of weeks, which is super impressive because they have to simulate all of those, you know, air molecules and pockets of hot air that are out there in the atmosphere. It's not that it makes the problem impossible. It's just makes it harder or, you know, kind of shorns how much we can predict it. Yeah, if you had an infinitely powerful computer, then we could solve these problems because we could simulate them with really high resolution. We could take like really, really short time steps in our simulation. Instead of stepping forward a millisecond, we could step forward a nanosecond and then correct.

And so if you had infinite computing resources, you could do these things very effectively. And some of the reasons why these problems, which seem to be impossible for a long time, like predicting the weather, seem to be getting easier and not because humans are getting smarter, but because our computers are getting more powerful. And so now we have a lot more computing power available to do things like predicting the weather and trying to predict earthquakes and all these. these really, really hard problems that are really important. Like today, we can predict how the whole solar system works, right?

We mostly can. And a lot of that is because mostly the solar system is a bunch of two-body problems. Like the Earth moving around the sun, technically it's, you know, it's an eight-body problem because the Earth is pulled on by the moon and Jupiter and Neptune and whatever. But mostly it's just the sun. You can ignore everything else when you're calculating the Earth to some degree. If you want to get it exactly right, then yes, you need to include effects from Mars and Venus. And then you can't use Kepler's laws.

You can't use the simple formulas that we have for a two body problem. You have to get down and dirty and do the simulations using very powerful computers. But then I guess would you say that our solar system is chaotic as well? Like, is our solar system a chaotic system? Because it seems sort of stable right now. But are you saying maybe like if you give it enough time, it is kind of a little unpredictable? Yeah, definitely. The solar system is chaotic, but on cosmological timescales, not on like a year or 10 years,

but unlike millions and billions of years. And it was more chaotic in the beginning. We sort of settled into something that's more stable. But when the solar system began, it was a big, hot mess and things were flying everywhere. Planets were colliding into each other and making new planets and throwing things out of the solar system. We probably had a different number of planets a billion or two billion years ago.

People suspect there might have been like another giant planet, which was tossed out of the solar system by Jupiter and Saturn. So, yeah, that sounds pretty case. chaotic to me. Solar system was like, you know, I have enough to deal with with the nine bodies, possibly eight, let's get someone out. But you can take a very complicated system like the solar system and find approximately stable solutions, things which will last for a long, long time. But how stable are they? Something which flies through the solar system can perturbate a little bit and then things can very quickly go out of whack. So if you have like another star that gets a little

close to our solar system, it could change the orbit of Jupiter, which could have knock on effects about changing the orbiter of Saturn, and then the asteroid belt and Mars, and pretty soon we could have craziness. All right, well, let's get into that craziness of our solar system and what the consequences are of this three-body problem and our ability to understand the rest of the cosmos. But first, let's take another quick break.

December 29th, 1975. airport. The holiday rush, parents hauling luggage, kids gripping their new Christmas toys. Then, at 6.33 p.m., everything changed. There's been a bombing at the TWA terminal. Apparently, the explosion actually impelled metal glass. The injured were being loaded into ambulances. Just a chaotic, chaotic scene.

In its wake, a new kind of enemy emerged. And it was here to stay. Terrorism. Law and order criminal justice system is back. In season two, we're turning our focus to a threat that hides in plain sight. That's harder to predict and even harder to stop. Listen to the new season of Law and Order Criminal Justice System on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. My boyfriend's professor is one.

way too friendly, and now I'm seriously suspicious. Well, wait a minute, Sam, maybe her boyfriend's just looking for extra credit. Well, Dakota, it's back to school week on the OK Storytime podcast, so we'll find out soon. This person writes, my boyfriend has been hanging out with his young professor a lot. He doesn't think it's a problem, but I don't trust her. Now, he's insisting we get to know each other, but I just want her gone. Now, hold up, isn't that against school policy? That sounds totally inappropriate.

Well, according to this person, this is her boyfriend's former professor, and they're the same age. And it's even more likely that they're cheating. He insists there's nothing between them. I mean, do you believe him? Well, he's certainly trying to get this person to believe him because he now wants them both to meet. So, do we find out if this person's boyfriend really cheated with his professor or not? To hear the explosive finale, listen to the OK Storytime podcast on the Iheart radio app,

Apple Podcasts, or wherever you get your podcast. Hey, sis, what if I could promise you you never had to listen to a condescending finance, bro, tell you how to manage your money again. Welcome to Brown Ambition. This is the hard part when you pay down those credit. cards. If you haven't gotten to the bottom of why you were racking up credit or turning to credit cards, you may just recreate the same problem a year from now. When you do feel like you are bleeding from these high interest rates, I would start shopping for a debt consolidation loan,

starting with your local credit union, shopping around online, looking for some online lenders because they tend to have fewer fees and be more affordable. Listen, I am not here to judge. It is so expensive in these streets. I 100% can see how, and just to a few months. You can have this much credit card debt when it weighs on you. It's really easy to just stick your head in the sand. It's nice and dark in the sand. Even if it's scary, it's not going to go away just because you're avoiding it. And in fact, it may get even worse. For more judgment-free money advice, listen to Brown Ambition on the IHeart Radio app, Apple Podcast, or wherever you get your podcast. I had this overwhelming sensation that I had to call it right then. And I just hit call. I said,

you know, hey, I'm Jacob Schick. I'm the CEO of One Tribe Foundation. And I just wanted to call on. and let her know there's a lot of people battling some of the very same things you're battling. And there is help out there. The Good Stuff podcast, Season 2, takes a deep look into One Tribe Foundation, a non-profit fighting suicide in the veteran community. September is National Suicide Prevention Month, so join host Jacob and Ashley Schick as they bring you to the front lines of One Tribe's mission. I was married to a combat army veteran,

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I ended up having amputation below the knee of my right leg and a traumatic brain injury because I landed on my head. Welcome to Season 2 of the Good Stuff. Listen to the Good Stuff podcast on the Iheart Radio app, Apple Podcasts, or wherever you get your podcast.

All right, we're talking about the three body problem and it's hard to find an analytical solution to it as opposed to the two body problem which you can find a nice neat formula for it but i wonder then if this is sort of like a physicist frustration because as an engineer i'm pretty much used to like things not having an analytical solution like from day one like nothing only like throwing a ball up in the air has an analytical solution everything else you have to do with numerical simulations or approximating you know name of your stokes equations and having nonlinear stuff that you can't solve And so, you know, like as an engineer, you always rely on simulations.

But maybe in physics, you get more frustrated for not having like a neat, you know, clean formula to predict the future. Well, the thing that's tantalizing is that there are a few cases when you can find a neat formula where you can start with just pencil and paper, describe the pushing and the pulling of your system, and then get out a formula that tells you where everything is going to be basically for all time. That's amazing. It's beautiful. And it's tempting. It makes you think, wow, why can't I do this for other systems? Why can't I do this for every system, right?

Because if they exist for some systems, it gives you the sense that, like, if we had the right mathematics, if we knew the right language to talk about this stuff, maybe even really complicated problems would be simpler. So it's sort of aspirational. Yeah, I can imagine that frustration. You're in your cabin in the middle of Russia, in Siberia, in the middle of nowhere, and you're like, oh, shoot, I need a computer. I didn't bring one.

Or, oh, shoot, I need to talk to somebody else. I don't have a phone. That's frustrating, right? It is frustrating. And, you know, it's something funny about teaching freshman physics. I teach mechanics often here at UC Irvine. And, you know, there are not a lot of problems that really are solvable. Like, very few problems can you actually sit down with pencil and paper and say, here's a situation, here's the solution.

And so in teaching this class for like almost 20 years now, I've noticed that basically every physics homework problem in every textbook is one variation on like one of these five solvable problems. And so as soon as you look at one, you're like, oh, this is that one problem. Or this is problem number four, except they're using a squirrel instead of a ball rolling down a plane or something. And so it all boils down to like a few solvable problems because there are only a few that can actually be solved. You mean there's an analytical simple solution to what Professor Whiteson is going to put on the final test? I hope students are taking notes. I say if you take my class for 20 years, it becomes pretty easy. I guess even physics professors are predictable.

Is that what you're saying? Yes, exactly. You can predict what they're going to do? It's hard to invent new solvable problems in physics. And, you know, it's not just like motion of two objects. There are lots of places in physics where the problems are not solvable. Einstein developed general relativity, right? Which means he wrote down the equations for how space curves when mass is around.

He wrote down the equations, which means those are the constraints that space has to follow. It doesn't mean he can tell you how space behaves when mass is around. Those are the solutions to the Einstein equation. And he couldn't solve his own equations. Like he developed general relativity. He's like, here are the equations. I don't know how to solve this. He wasn't even the first person to solve the Einstein equation.

That was Schwarzschild, because these equations are like famously impossible to solve. Now, if you have a solution, you can check it. You can say, hmm, I think space bends in this way when there's mass around. You can plug it into the equations. And if it works, you're like, cool, I found it. But again, just because you have the equations doesn't mean you know how to find the solution. Anybody who's done differential equations knows that's true. We have like no general mechanism for saying, here's a differential equation. I can go from the equation to finding the

solution. And so there's lots of places in physics where we just don't know how to solve these things. Even still for general relativity, we only know how to solve it for a few cases, like an empty universe, a universe that's smoothly filled with matter, like no lumps at all, or a black hole. Basically everything else is unsolvable. And that's what Shoreshield found, right? Right? He found the solution for general relativity in the case of a simple black hole. Yeah, exactly. He was the first person to ever solve these equations. And he actually did it while he was a soldier in World War I. What? Was he fighting in a cabin in Russia also? Never fight a land war in Russia, man. Especially all you're trying to solve physics questions.

Yeah, it's like extra difficulty points. Unless he was fighting for the Russians. Maybe, I don't know, maybe he's Russian. And then he had a lot of time because the other team was doomed. No, but it's a great story. You should look up how Schwartz had solved this problem. I see. So it's not how he solved general relativity for all time and all cases. He just found a solution for general relativity in this special case of a simple black hole. Yeah, a universe that has nothing but a black hole in it. He figured out the solution, how space bends in that scenario.

And then later people figured out, okay, well, if I assume that the universe is totally empty, can I solve the equations? Oh, I can do that? Or if I assume the universe is like filled smoothly with matter, can I do that? but nobody has solved general relativity for like our solar system or even just for like the sun and the earth together. It's too complicated. Nobody has figured out how to go from those equations to say here's how space has to bend in this situation.

Oh, wait. So not even like the two body problem has a solution in general relativity. Yeah, that's right. General relativity much, much harder than Newtonian mechanics. We can do things like numerical relativity. Like we can describe how black holes orbit each other and collide and generally, gravitational waves because we can do it numerically. We can use computers to do approximate solutions to these things,

but nobody can write down simple formulas to tell you how black holes orbit each other and collapse. Oh, I see. All this time we've been talking about the two-body problem being solvable. It's only solvable in the Newtonian case, right? Like if you assume the simplest or the simple physics of Newton, then you can find a solution, but not for three. But if you assume what we actually know what's going on in general relativity,

then it's, we can't even start. Like, there's no solution. Yeah, exactly. You know, Einstein lays out the equations, the constraints, but he doesn't tell you and he doesn't know how to go from the constraints to a solution. You know, it's sort of like if you're driving down the highway with your family and you ask everybody to everybody says, I want a salad or I want pizza or I want hot dogs. Like, those are the constraints.

It doesn't necessarily mean you know how to find a restaurant that satisfies those constraints, right? Having the constraints doesn't tell you how to find a solution. Wow. It sounds like something from personal experience with Danny, you're trying to vent, perhaps. Yes, I'm looking for a restaurant that search. Salads and hot dogs and pizza. Let me know if you find one. That's not even the general relativity solution.

Like if you add relatives to this card ride, then it gets impossible, right? Because then you have all these relative dynamics. Exactly. Very chaotic, very quickly. Well, I think what's interesting is that this is not just difficult for us as physicists to, like, predict these things. and kind of like know what's going to happen. But it's also kind of hard for the universe to know what's going to happen, right?

Like if something is chaotic, it also means that things are kind of unpredictable in general. Like crazy things can happen in our solar system. Yes, systems with three objects don't last very long because they are chaotic. They don't tend to fall into stable patterns and survive for very long. So if you have like three stars orbiting each other, then pretty quickly two of them will eject the third one out into the universe because because there are not very many stable solutions to the three body problem. And this is different from like, can human mathematicians write down a simple formula to predict what will happen?

That's one question. Another question is like, how long can three stars orbit each other before two of them kick out the other one? I guess you mean like three stars of about the same size, right? Yeah. Three stars about the same size and about the same distance from each other, right? A real like three body system.

Because the only way for that to really happen for it to become stable is to sort of turn it into a double two-body system. Take your three stars, group two of them together, make them really close and put them far away from the third star. Then what you have is like a little two-body system of two stars. And then you have that two-body system. You can treat it sort of like as a single object when you're talking about the third star,

which is now orbiting that pair. And so when we do find trinary systems out there in the universe, they're typically this like two-body system in a hierarchy. where you have a two-body system and then one of those bodies turns out to have two things inside of it. Right. And I think this hierarchy, we sort of talked about it in the last podcast, but it has to do with distance, right? Like if two of them are out here, you know, interacting and orbiting around each other, then to another body that's fairly far away, our two little bodies here feel like one. And so then that makes it more stable. Exactly. If, for example, we had two sons at the center of our solar system, if they were really close to each other and they were much closer to.

each other than we were to them, we could treat it like it was just one object. It wouldn't matter to us that it was two objects. But if we got closer to them or if we even like trying to get between them, then we make a big difference on our trajectory that there were two objects instead of one. And so for example, in that novel we talked about at the top of the episode, that's exactly what's going on. There's a solar system with two stars and a planet that's whizzing all around right through them in a very crazy unstable orbit. And so not only does it have like really weird in day patterns, but it has a very chaotic trajectory. And so you can't necessarily predict exactly where it's going to be.

It's kind of like real couples, I guess. You know, like from a distance, you can sort of assume they think and act as one. But once you, like, get close to them, you see there's a lot of disagreement there. But you never want to get between them, exactly. That's right. It's unstable. Yeah. You don't want to be the third body there.

No, you definitely don't. You can get tossed out of their solar system pretty quick. Or maybe eject them one or the other. once, but then that gets complicated. Yes, it sounds like we're writing a rom-com now involving a trip to the woods in Russia. All right. Well, this is kind of an interesting question here and an interesting problem because it doesn't just tell you that some things are hard to solve in nature, but some things are hard and

unpredictable themselves in nature, right? Like some of these things out in nature, they just don't last long. They spin out of control or they settle into things that are more stable like two-body solar systems. Yeah, and it could be that in the future, somebody events mathematics that makes it easier to describe that crazy chaotic motion and that, you know, in 20 years or in 50 years, we have like a new basic function, you know, like we have sign and cosine. These were invented functions by human mathematicians. Somebody might come up with a new function, which turns out to be really useful to describing three-body motion and allows us to find some expression. A lot of mathematicians

are skeptical because they can sort of express these solutions as like an infinite series and they show that it's very complicated and they suspect that there isn't a simple function but you know future mathematicians are usually smarter than today's mathematicians and so I hold out hope or maybe like there are aliens who have figured this out you know like they'll come to us and be like yes sign and cos up you don't have you know chaos sign or something that describes chaos motion yeah exactly and maybe somewhere some mathematician is developing the tools and they don't even realize how it's going to be useful I love those stories of mathematicians developing these ideas and then them later being co-opted by a physicist.

And so maybe those ideas exist right now. And all you have to do is go out and read the right math paper and you're like, oh, this is exactly the hammer we need to hit this physics nail. Or maybe the answer is in some cabin in Russia, but the, you know, the poor soul ran out of food or something. And it's lost to us forever. But it's written down on a frozen sheet of paper in that cabin. It exists. But anyways, I guess the good news is that it's, It's an open problem, and there could be somebody listening to this podcast right now that might solve it in the future. Maybe even you.

Well, we hope you enjoyed that. Thanks for joining us. See you next time. Thanks for listening, and remember that Daniel and Jorge Explain the Universe is a production of IHeart Radio. For more podcasts from IHeart Radio, visit the IHeart Radio app, Apple Podcasts, or wherever you listen. to your favorite shows.

December 29th, 1975, LaGuardia Airport. The holiday rush, parents hauling luggage, kids gripping their new Christmas toys. Then, everything changed. There's been a bombing at the TWA terminal, just a chaotic, chaotic scene. In its wake, a new kind of enemy emerged, terrorism.

Listen to the new season of Law and Order Criminal Justice System on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. My boyfriend's professor is way too friendly, and now I'm seriously suspicious. Wait a minute, Sam. Maybe her boyfriend's just looking for extra credit. Well, Dakota, luckily, it's back to school week on the OK Storytime podcast, so we'll find out soon. This person writes, my boyfriend's been hanging out with his young professor a lot. He doesn't think it's a problem, but I don't trust her. Now he's insisting we get to know each other, but I just want her gone.

Hold up. Isn't that against school policy? That seems inappropriate. Maybe find out how it ends by listening to the OK Storytime podcast on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. I'm Dr. Scott Barry Kaufman, host of the Psychology Podcast. Here's a clip from an upcoming conversation about how to be a better you. When you think about emotion regulation, you're not going to choose an adaptive strategy which is more effortful to use unless you think there's a good outcome. Avoidance is easier. Ignoring is easier. Denials easier. Complex problem solving takes effort. Listen to the psychology podcast on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. This is an IHeart podcast.