Why is the three-body problem so hard?

Daniel and Jorge talk about why something so simple is one of the oldest and hardest problems in physics.

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Episode Transcript

Speaker 1 (00:08):
Hey, Daniel, when you are teaching, are you the kind
of professor that assigns super hard homework in your classes?
You mean, 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,

(00:29):
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 assigned unsolved research problems to
first year students. Know that only happens in the movies. Man,
Goodwill Hunting is not a documentary. You're not Matt Damon.

(00:51):
I don't have the looks for it. Yes, there's always
room for improvement. Hi am more hamming cartoonists and the

(01:12):
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
it wouldn't be an 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. What do you think happens?

(01:33):
Like somebody just comes up with the right way to
look at it, or like they see something nobody else
has 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. When you look at the
history of it and be like, problem proposed in sixteen nineteen,
progress has made in eighteen fourteen, and then Samantha figures

(01:55):
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. Didn't sometimes they had Those are eternal, man,
they will never be solved. They will never be solved.
It's part of being human, I guess. Welcome to our
podcast Daniel and Jorge Explain the Universe, a production of
our Heart Radio in which we tackle the hardest problem,

(02:17):
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. What if that's the harder problem,
Daniel explaining something to other people. We do our best here,

(02:40):
but um, it's pretty hard to 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,
two things everybody can understand, because to do that you
have to have a really good grasp on what's going on. See,
you actually have to understand it first before explaining into people.

(03:03):
So what am I doing here? Then? Well, sometimes when
you try to explain something, you realize whole lot of 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.
But that is a pretty good parenting lesson. Also, it's

(03:25):
good to share what you know when you learn what
you love about this crazy beautiful costmas. Yeah, how does
that help you with your parenting? Well, it's just good
to share. I think it's good. Well, you have to
share with your children. I think it's a law, that
is a rule. But it's also good to teach it
to share, you know. It's just everyone's more generous with
their what they have in their knowledge. We're all happy.
I thought you were going to use the wondering glamor
of the universe to convince your kids to do their chores,

(03:47):
like take out the trash because stars are amazing? Are
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 will work against you. So then why should
I bother taking out the trash if nothing matters? Because
if nothing matters children, everything matters. Like that that trended

(04:11):
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 has

(04:34):
a machinery to it, and that machinery is mathematical. It's
incredible in 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

(04:57):
sometimes it's easier than other times. And the cool thing
about sciences 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 that has so far.
And so sometimes you run into things that are just
really really extra hard or maybe even impossible. Yeah, and

(05:17):
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's been lots of times in history
when mathematicians have developed tools, not because they thought they
were gonna be useful for physics, but just because they
thought it was fun, and then later on physicists were
like a whold on a second. That totally helps me
solve this problem I've been struggling with for twenty years.

(05:39):
A great example is general relativity, which is built on geometry,
which was developed just ten years before. Without all that
work developing geometry, there's no way Einstein could have developed relativity.
Is 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?

(05:59):
Or like hip hop breakdancing 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

(06:19):
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, and they're like, whatever,
who cares about the real universe? I'm walking the halls
of truth. You're selling their halls with like reality and
like red and dirt, Like that's just dirt. Adams are

(06:41):
just dirt. If they cared about getting dirty, they would
have been physicists instead of mathematicians. I see, physicis are
are the down and dirty of of scientists. I think
physicists are to mathematicians what engineers are to physicists. Oh,
I see the better the better people, right, the true
heroes on the hierarchy of useless pure the hierarchy of usefulness.

(07:02):
You mean, depending on what you put in 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 are very movement through space,

(07:25):
and it affects how planets revolve around their suns, and
which we may never find the answer for. So to
be on 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?

(07:47):
I mean, I see something here at 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? Is the safe preparatory
in the heavenly bodies relationships? You know, some bodies here
on Earth are quite heavenly as well, but we're talking

(08:08):
about celestial bodies, that's right, the real stars. Alright, So
the three button. More specifically, this is kind of about
what is the three body problem at all? Because I
imagine that a lot of people have heard of him,
although it is the title of 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

(08:30):
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 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 it's
pretty dense, it's kind of thick. Yeah, there's a lot
of physics in that book, which is pretty fun for

(08:52):
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 was like a
bestseller and want all the awards right in science fiction.
M 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.

(09:15):
That's right, it's really old problem, and old problems are
the funniest problems because it means that like a lot
of smart people have been butting their heads against this
problem for decades or even centuries, and nobody has figured
it out. And 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

(09:36):
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
for a couple of years, and that you might solve
a famous problem. That's the real answer. That wasn't a metaphor.
That was that really happened to you, not to me. No,
there really is a Russian mathematician who worked all by

(09:57):
himself for a decade and saw the famous problem in math,
the remon conjecture. Wow. And he was in a cabin,
using 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.

(10:21):
He's well on one side of that line. But anyways,
let's talk about this problem, the three body problem. And so,
as usually we were wondering how many people out there
we 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 usually went out there

(10:42):
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

(11:03):
at Daniel and Jorge dot 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 barely
aware of what the three body problem is. I did
read as you should use three body problem trilogy. My
understanding is that it's a problem with how three bodies

(11:27):
orbit one another and how it could continue to do
that and be stable without cuestion into one another. A
lot of people spend a fair amount of time calculating
how two massive bodies interacts 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

(11:51):
determine an exact solution. And also if you make a
small change let's say in the initial positions of the bodies, um,
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 rotatue interacts, usually

(12:12):
like the Sun, the Earth and the Moon, for example,
would be that would be three bodies, and I think
you can solve two bodies. Any more than a few
more and you can't solve it. I think is it
is one of the issues. Wow, that is something I
am not sure where it is. I don't know what
the three body problem is m unless it's relating to

(12:36):
a previous question where if you have three bodies acting
on each other gravitationally, um, you haven't got sort of
one orbiting another or one with a joint orbit with
another that that will be probably quite at random implication

(12:56):
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 of four bodies. It just happens between these three
bodies for some reason, and for some reason the number

(13:20):
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 C and C B. And when

(13:40):
you have interactions of that order that 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, There not a lot of knowledge about this,
But someone did read the True Gene. Yeah, exactly three

(14:02):
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 rerund well, I can't just read one
three body problem book. I gotta 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 in

(14:22):
space and then 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,

(14:47):
which can be tricky even when there are two bodies involved,
even if everyone is open minded, it gets tricky. Al Right, Well,
let's dig into Daniel, how would you describe the three
body problems? 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

(15:09):
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, in which direction they're heading,
and their masses, you can write down a very simple

(15:30):
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, it 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

(15:50):
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 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

(16:11):
an infinite number of bodies out there in space, and
gravity works for over infinite distances, and so everything in
the universe is hugging 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 Andromeda when you're
doing in 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

(16:33):
two bodies and everything else can be ignored without changing
anything down to like, you know, the tenth decimal place
or something. Yes, so in 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

(16:54):
know where three objects are. You know, so 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 their masses. You cannot write down a simple formula
that tells you where they're going to be in a week,
or in a year or in a thousand years. Well,
it gets really complicated. Suddenly, it gets really complicated. We

(17:16):
don't have a solution. Now, we have an understanding for
what's going on, Like we know the forces 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. 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

(17:37):
people out there, which is 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, tells you like,
for example, how two objects pull on each other. It
doesn't tell you how those objects are going to move.

(17:58):
To figure out how the objects are to move, which
is what you need to predict their emotion. 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

(18:20):
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 free. 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 forces

(18:41):
pulling on it, the force of gravity, and I know
that F equals m A for example. So I can solve,
for example, for its exceloration 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
it's acceleration is going to be exactly. The acceleration just
tells you how is momentums and change in a given moment.

(19:02):
Right to know where it's going to be, you need
to then solve the equations of motion, which incorporate all
these forces and is 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 five equals ten. Right,
that's an equation that constrains X. It limits what X
can be, but it's not actually the solution. The solution

(19:23):
is X equals five. That's a very simple one, right.
You know exactly how to go from the equation X
plus five equals tend 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 forty nine.
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

(19:45):
do you find the solution? If I tell you X
square 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 us like my
parenting strategy right there. And I think what you mean is, like,
you know, in physics, you have equations to tell you,

(20:06):
for example, like the acceleration of x, which is like
how the velocity changes, which is like how the position
is changing. Like you have equations for that. But to
actually get the pocsisition, 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 how all those accelerations add

(20:27):
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

(20:50):
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
cell steel body into the mix, and what consequences it
has for our ability to predict the universe. But first,
let's take a quick break. All right, Daniel, we're talking

(21:21):
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

(21:43):
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 once you have three. It because
there's no easy solution to that problem. Yeah, there's no
easy solution. There's no like short mathematical answer when you

(22:03):
like is something like x f T or x is
the position of the object, and then a simple formula
where you can plug in the time and it will
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, where 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

(22:25):
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 three D? Is
it kind of about going to the third dimension? That
makes it hard because I imagine if you have two
bodies in space, you can just treat him as like
a two D problem, right, like you just imagine the

(22:47):
plane where these two bodies are. But once you have
three and then it's like, now it's a three D 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 though, right, three points define a plane,
so there's always a plane for three jects. I think
the problem is that when you have three objects, a
small change in their location leads to a big change

(23:07):
in where they're going to be in the future. At
least for gravitational interactions, whereas if you have two objects,
a small change and where the Earth is going to be,
it will 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 describing
chaotic situations, whereas very small change in the angle or

(23:31):
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,

(23:51):
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 an equals to it's nonlinear as
these distances go like one over radius squared, so there's
an inverse are 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,

(24:14):
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

(24:35):
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 and 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

(24:56):
trajectories for these objects in this three D 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 any equals too. With the two body problem,
there are a bunch of simplifications you can make that
separate the problem so that, for example, the distance between

(25:17):
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

(25:39):
not always possible, and when they're all entangled up with
each other, you can't always find a solution. Now, 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 the one that is solvable. Right. All

(26:02):
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's just gets
more complicate. Real equations start to like interact with each other,

(26:24):
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

(26:46):
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, gives
the Earth a little push, will probably roll back into
a 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

(27:08):
the reasons why we don't have a simple formula because
we don't have functions that describe that, like sign and
co sign 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

(27:28):
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,
use simulations. You know, we can describe chaotic systems like
you build a computer system, you put three objects in it,

(27:50):
and then what you do is you say, all right,
what happens in the first second, and you say, well,
the Earth's gonna move this way, the Sun is gonna
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 milli second
from now, then you can really simplify and say I

(28:10):
know what to do for a half second. Then you
just do that over and over and over again. That's
the way we can describe a chaotic system is like
slicing it in time and then try to move our
stimulation 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 sign wave,
or oh look it follows a logarithm of a sign wave.
We can't find a solution. We can't find a mathematical

(28:33):
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,
twenty 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 a little by little until you
get to ten years from now. Yeah, and even those

(28:54):
simulations are difficult because it's chaotic. Like if you don't
make those calculations very 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 the snowball into big mistakes. It's just like
you know, the butterfly problem. Butterfly flaps its wings in
China and that has cascading effects on the weather, which

(29:17):
causes eventually a storm in Central Park in New York.
And these things are real. They're 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 if you get something wrong
very early on, your results in ten years are nonsense.

(29:38):
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 ten years or in a hundred 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 amaze or

(29:59):
the you know, the horses 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. Exactly. You make
a bunch of simplications 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

(30:19):
a little bit because of that implication, 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

(30:41):
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
in a very very small target. Imagine firing a gun
from l A to New York and trying to hit
a tiny and the 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

(31:02):
target in New York. But I guess that. 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

(31:24):
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 storms,
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

(31:45):
in 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

(32:08):
to predict earthquakes and all 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

(32:29):
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

(32:51):
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 the cosmological time scales, not on like a year
or ten years, but unlike millions and billions of years,

(33:12):
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

(33:32):
might have been like another giant planet which was tossed
out of the Solar System by Jupiter and Saturn. So, yeah,
that sounds pretty 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

(33:53):
a long long time. But how stable are they? Something
which flies through the Solar system can perturb it a
little bit, and then things can very quickly go out
of whack. So if you have like another star and
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 orbit 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

(34:17):
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.

(34:40):
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

(35:00):
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, the Navy, your
Stokes equations and having non linear stuff that you can't solve,
and so you know like it. As an engineer you
always rely on simulations, but maybe in physics you get
more frustrated for not having like a neat, you know,

(35:21):
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

(35:41):
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

(36:01):
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 you see irvine, and you know there are

(36:21):
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 the situation, here's the solution.
And so, in teaching this class for like almost twenty
years now, I've noticed that basically every physics Hormemork problem
in every textbook is one variation on like one of
these five solvable problems. And so as soon as you

(36:42):
look at when you're like, oh, this is that one problem,
or this is a problem number four, except they're using
a squirrel instead of a ball of 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. I mean, there's an
analytical simple solutions to what Professor Whitesen is gonna have.
But on the final test, I hope students are taking notes.

(37:03):
I say, if you take my class for twenty years,
it becomes pretty easy. I guess even physics professors are predictable.
Is that what you're saying 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

(37:25):
wrote down the equations for how space curves when masses 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 masses around. Those
are the solutions to the Einstein equation. And he couldn't
solve his own equations. Like he developed general relativity and

(37:45):
he's like, here the equations, I don't know how to
solve this. He wasn't even the first person to solve
the Einstein equation that was short styled. Because these equations
are like famously impossible to solve. Now, if you have
a solution, you can check it. You can say I
think space ends in this way when there's massed around.
You can plug it into the equations, and if it works,
you're like, cool, I found it. But again, just because

(38:06):
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,

(38:29):
a universe that's smoothly filled with matter like no lumps
at all, or a black hole. Basically everything else is unsolvable.
Then that's why before Sheeld found right like Evan 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 One. What

(38:51):
was he like fighting in a cabin in Russia. Also,
never fight a land war in Russian man, especially while
you're trying to solve question. 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 short Starts solved

(39:12):
this problem. I see. So it's not he solved general
relativity for all time in all cases. He just found
a solution for general relativity in this special case of
a simple black hole. Yeah, of 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,

(39:34):
I can do that? Or if I assume the universe
is like filled smoothly with matter, can I do that?
But like, 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

(39:55):
not even like the two body problem has a solution
in general relativity. Yeah, that's right. General auctivity 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 like write down simple

(40:17):
formulas to tell you like how black holes orbit each
other and collapse. Oh, I see. All this time we've
been talking about the two body probably 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, like what we actually know what's going

(40:38):
on 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
somebody what they want for dinner, and everybody says, I
want a salad, or I want pizza, or I want

(40:58):
hot dogs, Like those are the con straints. 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 data. You're trying to events, Yes, I'm
looking for a restaurant the search salads and hotdogs and pizza.
Let me know if you find one that's not even

(41:21):
the general relativity solution. Like if you add relatives to
this card, right, 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

(41:41):
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

(42:05):
them will eject the third one out into the universe.
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

(42:25):
like three stars of about the same size, right, Yeah,
three stars about the same size and about the same
distance from each other at 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

(42:46):
put them far away from the third star, and 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, we have a

(43:07):
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's acidly 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, I feel

(43:27):
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 it would make a big difference on our trajectory

(43:50):
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 plan that's
whizzing all around right through them in a very crazy,
unstable orbit. And so not only does it have like
really weird night and day patterns, but it has a
very chaotic trajectory, and so you can't necessarily predict exactly

(44:12):
where it's going to be. It's kind of like a
real couples, I guess, you know, like from a distance,
you can sort of assume they think and act as one,
but once you get go up close to them easily,
there's a lot of disagreement there. But you never want
to get between them exactly. That's right. It soundstable. Yeah,
you don't want to be the third body there, you
definitely don't. You can get tossed out of your solar
system or maybe eject the one of the others. But then, yes,

(44:38):
it sounds like we're writing a romcom 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 been out of control, or they a settle into

(45:01):
things that are more stable, like to 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 twenty years or in
fifty years, we have like a a new basic function,
you know, like we have signed and coson. These were
invented functions by human mathematicians. Somebody might come up with

(45:21):
a new function which turns out to be really useful
to describing three body motion and and allows us to
find some expression. A lot of mathematicians are skeptical because
they can sort of express these solutions is like an
infinite series, and they showed 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

(45:43):
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 cosap we
don't have, you know, chaos sign or something that describes
chaos motion. Yeah, exactly, And maybe somewhere some mathematicians is
developing the tools and they don't even realize how it's
going to useful. I love those stories of mathematicians developing
these ideas and then them later being co opted by physicists.

(46:06):
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, but it's written
down on a frozen sheet of paper in that cabin.

(46:27):
It exists. But anyways, I guess the good news is
that 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.

(46:49):
Thanks for listening, and remember that Daniel and Jorge explained
the universe is a production of I heart Radio. For
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