- This episode is sponsored by Brilliant. Hello, welcome to "Up and Atom," I'm Jade. The three-body problem is one of those rare problems in physics that changed our
understanding of the universe. And I'm not exaggerating for views, it really was revolutionary. It goes like this. Imagine you have three masses or bodies in space, all affected by each
other's gravitational pull. If you know each of their position and momentum at the present time, predict their position and momentum for some future time. That's it. Sounds simple, right? We know all of Newton's laws of motion and we know Newton's law of gravitation. But despite how simple it sounds, this question uprooted
over 100 years of physics, caused a chasm in the
landscape of science, and shone a light on the limits of what we as humans can know. How on Earth did a simple
question about planets do all of this? To appreciate the gravity of the three-body problem, we need to go back to 1889. The story starts with a birthday. The King of Sweden,
Oscar II was turning 60. And to celebrate, he held a maths competition. He was wild like that. The big question was, is our solar system stable? Will the planets continue
to orbit the sun forever or will they one day collide, or will some fly off, killing us all? You can see why it was
a question of interest. The winner of the competition would be awarded 2,500 crowns and academic fame. The interest in this question actually goes beyond just wondering if humanity would die a horrible death. It was mathematically significant. For over 200 years, brilliant mathematicians have tried and failed to answer it. Why? Well, why don't we ask the first guy who ever made a serious attempt? Newton. Imagine it's 1687, Isaac Newton has just changed the world with his three laws of motion and the universal law of gravitation. He'd unlocked the universe's
underlying principles, offering a blueprint
that explained everything from the falling of an apple to the orbits of planets. They were the perfect tools to solve the solar system problem. He could figure out the equations that governed the motion of the planets, solved them, and the solutions would tell him how they moved for all time. This was actually a
fatal question to Newton. He was a devout Christian. So to him, the solar system was only
around 6,000 years old. This timeline was far from guaranteeing it would orbit peacefully for the next thousand years too. The logical thing to do was start with just two bodies, figure out if they were stable, and then just keep adding
more bodies one-by-one, and repeating the process. Newton's laws worked remarkably well for the two-body problem, and he quickly found that it was stable. It was when he added a third body that things took a turn. Newton couldn't figure it out. Not only could he not solve the equations, he couldn't even figure out what the equations of motion were. He wrote to his friend, Edmond Halley, "No problem has ever made my head ache like the problem of the
earth, moon, and sun." This was deeply disturbing not just for the stability
of the solar system, but for all of physics at the time. Newton's laws had ushered
in an age of hope, a new understanding of our universe where every force and
effect seemed predictable. His laws solidified
the idea of determinism that the present state
determines the future state. Physics had basically
been reduced to a roadmap. The physical laws told
us the shape of the road and the initial conditions
told us where we were. Just plug in the numbers, solve the equations, and we could predict the exact location that we would be at some future point. Place the car in the same lane, and it would always end up at the same destination. Start a planet at the same point, with the same velocity, with the same forces acting on it, and you could predict its exact location at any future time. At the heart of Newton's determinism were analytical solutions, exact mathematical expressions that give rise to exact numerical results. Analytical solutions are
still all the rage today. If you've ever solved a physics problem in high school, chances are it was an analytical solution. There was one caveat to
all of this optimism. We could never know the exact conditions of anything to 100% precision. Measurements of position and momentum would inevitably be slightly off, even if it was imperceptible
to our sensors. And of course, we now know
about quantum mechanics in the Heisenberg uncertainty principle, which says that it's impossible to simultaneously know the position and momentum of a particle with perfect precision. But there was also the belief that that didn't matter. That given approximate knowledge of a system's initial conditions, we could calculate the
approximate future conditions. Small errors in measurement would lead to small
errors in predictability. This belief was well-justified. A tiny error in the position
of Comet Halley in 1910 would only cause a tiny error in predicting its arrival in 1986. It was thought that all there was left to do in science was measure things more accurately, have a better understanding
of the laws of physics, and get more computational power to carry out computations, then all the future would
lay open before our eyes. So with all of this hope going on, it was a pretty big blow when Newton couldn't figure out the motion of just three planetary bodies. It questioned the foundation of deterministic predictability, an idea that he had inspired. This was the first hint that this rosy picture wasn't quite right, which brings us back to King
Oscar's maths competition. More than 200 years had
passed since Newton. And with it, many new
mathematical advances. The time was ripe to give the old solar system
problem another crack. A mathematician named Henri Poincare rose to the challenge. Now, I'm way too Aussie to say that without offending some French people, so I'm gonna get my
French husband to say it. - Henri Poincare. - He started with a
three-body problem as well. He looked at an even
simpler version actually. Two larger masses fixed in place, and a third massless body, which was affected by
their gravitational pulls. He limited the problem to just figuring out the motion of this smaller massless body, how it was affected by
the two larger masses. Like every mathematician before him, Poincare couldn't figure
out the equations. But unlike any mathematician before him, he didn't let that stop him. He just invented a new way to do physics. Newton's method zeroed in
on individual scenarios, but Poincare wanted to zoom out and see the bigger picture. If Newton's law showed us the road, Poincare wanted to study the entire map, understanding the overall behavior and patterns within the system as a whole. And what he saw was the key to cracking the three-body problem. To understand what he saw, we need to get familiar with fixed points. Take a ball rolling on some hills. There are two points where the ball can come to a natural stop. Here and here. These are called fixed points, But there's a very important difference in the nature of these fixed points. With this one in the valley, if we give the ball a slight nudge, it will be drawn back to the fixed point. We call this a stable fixed point, and they act as attractors in the system. Objects near them will be
naturally drawn to them and they'll tend to stay there. But at this fixed point, if we give the ball just a slight nudge, it rolls away fast. This is called an unstable fixed point. They have a repelling nature. An object in their vicinity will be repelled away. Poincare found that the interaction of the gravitational forces can create these fixed points in space, points where the forces
are perfectly balanced so that an object is either drawn into hang out there or repelled away. But then he found a
third kind of fixed point with a bizarre property, fixed points that were stable and unstable at the same time, attracting and repelling. How is this possible? Well, look at this seemingly
stable fixed point. Now let's make it 3D. Oh, the ball is naturally
attracted to the fixed point like it's stable. But the slightest nudge can send it rolling down either side, like it's unstable. These are called saddle points. And what's interesting about saddle points is that the tiniest difference in the position of the ball can send it rolling down either side. Contrast this to the Newtonian
way of looking at things where slight inaccuracies in measurement lead to slight inaccuracies in the predicted trajectory. A slight inaccuracy in
a saddle point can lead to a totally different
unrecognizable trajectory. These saddle points are the seeds of what would later become known as chaos. Poincare saw that the
gravitational interactions between the three bodies create these saddle points in space, extremely sensitive spots that attract objects in, only to send them off wildly in one direction or another. And the slightest difference in position, force, or velocity, can catapult them into drastically
different trajectories. This is called extreme
sensitivity to initial conditions. Now, make no mistake, this
behavior isn't random. It's 100% deterministic. It's just not predictable. It's like Poincare discovered intersections in our roadmap. If a car starts in the same lane, it'll still end up at the same destination every time. But if it starts in just the next lane, it'll end up somewhere
completely different. You might be thinking, "But surely, that's not
fundamentally unpredictable. If we make better measuring devices, we can accurately determine
which lane we're in." But in reality, these lanes are infinitely
thin trajectories in a continuous space, and there are infinitely many of them. In practice, it's impossible to measure exactly which trajectory we're on, not only because of Heisenberg's
uncertainty principle, but also because tiny
disturbances miles away can impact the position
and velocity of a body. A slight gravitational
tug from a distant moon or the solar wind's subtle push. In practice, we can't measure every single tiny movement and force in the universe. We can only measure about which trajectory we're on. Which, as you can see, doesn't help us for long-term behavior. Poincare's discovery of chaos broke the Newtonian view
of a predictable universe, opening up a realm where predictability is limited not by our technologies or methods of calculation, but by the fundamental nature of the universe itself. Now, you might think that being such a revolutionary idea, this is where chaos took off. But actually, nobody
understood what Poincare did because he was so bad at drawing. He even got a zero on the entrance exam for college on the mechanical drawing part. So that was quite unfortunate. And chaos theory didn't
take off until the 1960s when it was rediscovered by this guy. But Poincare did win the
King's maths competition. Rather than solving
the three-body problem, he proved that it is
analytically unsolvable. "But wait," you say, "What about all of these articles claiming that there are solutions, or all of these videos
explaining the solutions? If the three body problem is unsolvable, what are they talking about?" Well, they generally
mean one of two things. It's true that we can't solve the three-body problem with a neat mathematical formula that works for any scenario. In other words, there
is no general solution to the three-body problem, but mathematicians have solved it for some specific cases, like when the bodies form an equilateral triangle or a stable figure-eight pattern, or for some periodic orbits, orbits that repeat over
and over indefinitely. The other thing a solution can mean is a very good approximation. In the absence of a general
analytical solution, scientists still wanted to be able to predict the orbit of three bodies, so they developed a technique called numerical integration. It works by breaking the problem down into smaller time steps. For each time step, the gravitational forces between each pair of
bodies are calculated. Using the forces and
the current velocities, an algorithm updates the velocities and positions of each body for the next time step. This process is repeated over and over, progressively calculating
the trajectory of each body. With today's computing power, numerical integration
is extremely effective. 300 years after Newton, the three-body problem continues to teach us about our world. Poincare's geometric techniques not only allow us to solve problems that Newton's old school methods couldn't, but they also birthed chaos theory and the field of non-linear dynamics. These tools have revealed chaotic systems across a diverse array of fields. From weather patterns and ocean currents to the rhythm of the human heart and the stock market, chaos is everywhere. It's fascinating how even systems governed by precise
laws can behave in ways that are fundamentally unpredictable. What I find super
inspiring about this story is how Poincare cracked
the three-body problem by looking at it a different way. I've always been fascinated by the way scientists
and mathematicians think. A lot of you don't know this about me, but I actually started
out doing a biology degree before I switched to physics. I loved biology, but I felt like I was learning things. Whereas in physics, I was learning a totally
new way of thinking. How to reason about our world, how to think logically, how to solve problems most effectively. I thought, "What could
be more mind-expanding than a totally new way of thinking?" Have you ever wanted to learn to think more scientifically? If the answer is yes, let me introduce you to
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annual premium subscription. Thanks, and I'll see
you in the next episode. Bye. (pulsing electronic music) (energetic electronic music)
