Video Plus: The REAL Three Body Problem in Physics

Chapters 0:00-1:02 What is The Three Body Problem? 1:02-5:41 Newton's Dilemma 5:41- 7:00 A worthy contestant 7:00-8:22 Fixed Points 8:22-9:39 Saddle Points 9:39-11:07 Chaos is deterministic, but unpredictable 11:07-13:04 The Three Body Problem is unsolvable 13:04 Think like a scientist

What is The Three Body Problem?

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- This episode is sponsored by Brilliant.

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Hello, welcome to "Up and Atom," I'm Jade.

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The three-body problem

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is one of those rare problems in physics

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that changed our understanding of the universe.

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And I'm not exaggerating for views,

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it really was revolutionary.

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It goes like this.

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Imagine you have three masses

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or bodies in space,

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all affected by each other's gravitational pull.

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If you know each of their position

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and momentum at the present time,

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predict their position

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and momentum for some future time.

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That's it.

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Sounds simple, right?

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We know all of Newton's laws of motion

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and we know Newton's law of gravitation.

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But despite how simple it sounds,

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this question uprooted over 100 years of physics,

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caused a chasm in the landscape of science,

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and shone a light on the limits

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of what we as humans can know.

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How on Earth did a simple question about planets

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do all of this?

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To appreciate the gravity

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of the three-body problem,

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we need to go back to 1889.

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The story starts with a birthday.

Newton's Dilemma

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The King of Sweden, Oscar II was turning 60.

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And to celebrate,

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he held a maths competition.

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He was wild like that.

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The big question was,

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is our solar system stable?

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Will the planets continue to orbit the sun forever

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or will they one day collide,

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or will some fly off, killing us all?

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You can see why it was a question of interest.

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The winner of the competition

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would be awarded 2,500 crowns

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and academic fame.

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The interest in this question

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actually goes beyond just wondering

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if humanity would die a horrible death.

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It was mathematically significant.

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For over 200 years,

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brilliant mathematicians have tried

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and failed to answer it.

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Why?

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Well, why don't we ask the first guy

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who ever made a serious attempt?

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Newton.

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Imagine it's 1687,

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Isaac Newton has just changed the world

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with his three laws of motion

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and the universal law of gravitation.

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He'd unlocked the universe's underlying principles,

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offering a blueprint that explained everything

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from the falling of an apple

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to the orbits of planets.

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They were the perfect tools

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to solve the solar system problem.

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He could figure out the equations

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that governed the motion of the planets,

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solved them,

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and the solutions would tell him

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how they moved for all time.

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This was actually a fatal question to Newton.

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He was a devout Christian.

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So to him,

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the solar system was only around 6,000 years old.

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This timeline was far from guaranteeing

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it would orbit peacefully

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for the next thousand years too.

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The logical thing to do

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was start with just two bodies,

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figure out if they were stable,

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and then just keep adding more bodies one-by-one,

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and repeating the process.

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Newton's laws worked remarkably well

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for the two-body problem,

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and he quickly found that it was stable.

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It was when he added a third body

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that things took a turn.

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Newton couldn't figure it out.

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Not only could he not solve the equations,

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he couldn't even figure out

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what the equations of motion were.

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He wrote to his friend, Edmond Halley,

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"No problem has ever made my head ache

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like the problem of the earth, moon, and sun."

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This was deeply disturbing

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not just for the stability of the solar system,

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but for all of physics at the time.

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Newton's laws had ushered in an age of hope,

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a new understanding of our universe

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where every force and effect seemed predictable.

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His laws solidified the idea of determinism

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that the present state determines the future state.

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Physics had basically been reduced to a roadmap.

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The physical laws told us the shape of the road

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and the initial conditions told us where we were.

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Just plug in the numbers,

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solve the equations,

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and we could predict the exact location

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that we would be at some future point.

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Place the car in the same lane,

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and it would always end up

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at the same destination.

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Start a planet at the same point,

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with the same velocity,

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with the same forces acting on it,

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and you could predict its exact location

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at any future time.

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At the heart of Newton's determinism

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were analytical solutions,

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exact mathematical expressions

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that give rise to exact numerical results.

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Analytical solutions are still all the rage today.

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If you've ever solved

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a physics problem in high school,

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chances are it was an analytical solution.

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There was one caveat to all of this optimism.

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We could never know the exact conditions

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of anything to 100% precision.

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Measurements of position and momentum

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would inevitably be slightly off,

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even if it was imperceptible to our sensors.

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And of course, we now know about quantum mechanics

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in the Heisenberg uncertainty principle,

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which says that it's impossible

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to simultaneously know the position

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and momentum of a particle

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with perfect precision.

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But there was also the belief that

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that didn't matter.

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That given approximate knowledge

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of a system's initial conditions,

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we could calculate the approximate future conditions.

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Small errors in measurement

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would lead to small errors in predictability.

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This belief was well-justified.

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A tiny error in the position of Comet Halley in 1910

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would only cause a tiny error

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in predicting its arrival in 1986.

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It was thought that

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all there was left to do in science

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was measure things more accurately,

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have a better understanding of the laws of physics,

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and get more computational power

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to carry out computations,

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then all the future would lay open before our eyes.

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So with all of this hope going on,

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it was a pretty big blow

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when Newton couldn't figure out

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the motion of just three planetary bodies.

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It questioned the foundation

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of deterministic predictability,

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an idea that he had inspired.

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This was the first hint

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that this rosy picture wasn't quite right,

A worthy contestant

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which brings us back to King Oscar's maths competition.

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More than 200 years had passed since Newton.

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And with it, many new mathematical advances.

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The time was ripe to give

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the old solar system problem another crack.

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A mathematician named Henri Poincare

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rose to the challenge.

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Now, I'm way too Aussie to say that

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without offending some French people,

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so I'm gonna get my French husband to say it.

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- Henri Poincare.

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- He started with a three-body problem as well.

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He looked at an even simpler version actually.

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Two larger masses fixed in place,

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and a third massless body,

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which was affected by their gravitational pulls.

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He limited the problem

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to just figuring out

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the motion of this smaller massless body,

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how it was affected by the two larger masses.

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Like every mathematician before him,

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Poincare couldn't figure out the equations.

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But unlike any mathematician before him,

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he didn't let that stop him.

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He just invented a new way to do physics.

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Newton's method zeroed in on individual scenarios,

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but Poincare wanted to zoom out

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and see the bigger picture.

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If Newton's law showed us the road,

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Poincare wanted to study the entire map,

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understanding the overall behavior

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and patterns within the system as a whole.

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And what he saw was the key

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to cracking the three-body problem.

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To understand what he saw,

Fixed Points

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we need to get familiar with fixed points.

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Take a ball rolling on some hills.

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There are two points

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where the ball can come to a natural stop.

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Here

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and here.

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These are called fixed points,

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But there's a very important difference

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in the nature of these fixed points.

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With this one in the valley,

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if we give the ball a slight nudge,

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it will be drawn back

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to the fixed point.

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We call this a stable fixed point,

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and they act as attractors in the system.

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Objects near them will be naturally drawn to them

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and they'll tend to stay there.

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But at this fixed point,

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if we give the ball just a slight nudge,

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it rolls away fast.

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This is called an unstable fixed point.

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They have a repelling nature.

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An object in their vicinity

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will be repelled away.

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Poincare found that the interaction

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of the gravitational forces

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can create these fixed points in space,

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points where the forces are perfectly balanced

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so that an object

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is either drawn into hang out there

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or repelled away.

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But then he found a third kind of fixed point

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with a bizarre property,

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fixed points that were stable

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and unstable at the same time,

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attracting and repelling.

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How is this possible?

Saddle Points

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Well, look at this seemingly stable fixed point.

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Now let's make it 3D.

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Oh,

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the ball is naturally attracted to the fixed point

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like it's stable.

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But the slightest nudge

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can send it rolling down either side,

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like it's unstable.

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These are called saddle points.

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And what's interesting about saddle points

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is that the tiniest difference

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in the position of the ball

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can send it rolling down either side.

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Contrast this to the Newtonian way of looking at things

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where slight inaccuracies in measurement

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lead to slight inaccuracies

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in the predicted trajectory.

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A slight inaccuracy in a saddle point can lead

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to a totally different unrecognizable trajectory.

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These saddle points are the seeds

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of what would later become known as chaos.

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Poincare saw that the gravitational interactions

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between the three bodies

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create these saddle points in space,

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extremely sensitive spots

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that attract objects in,

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only to send them off wildly

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in one direction or another.

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And the slightest difference

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in position, force, or velocity,

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can catapult them into drastically different trajectories.

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This is called extreme sensitivity to initial conditions.

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Now, make no mistake, this behavior isn't random.

Chaos is deterministic, but unpredictable

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It's 100% deterministic.

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It's just not predictable.

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It's like Poincare discovered

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intersections in our roadmap.

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If a car starts in the same lane,

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it'll still end up

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at the same destination every time.

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But if it starts in just the next lane,

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it'll end up somewhere completely different.

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You might be thinking,

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"But surely, that's not fundamentally unpredictable.

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If we make better measuring devices,

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we can accurately determine which lane we're in."

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But in reality,

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these lanes are infinitely thin trajectories

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in a continuous space,

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and there are infinitely many of them.

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In practice, it's impossible to measure

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exactly which trajectory we're on,

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not only because of Heisenberg's uncertainty principle,

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but also because tiny disturbances miles away

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can impact the position and velocity of a body.

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A slight gravitational tug from a distant moon

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or the solar wind's subtle push.

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In practice, we can't measure

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every single tiny movement

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and force in the universe.

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We can only measure

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about which trajectory we're on.

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Which, as you can see,

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doesn't help us for long-term behavior.

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Poincare's discovery of chaos

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broke the Newtonian view of a predictable universe,

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opening up a realm

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where predictability is limited

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not by our technologies

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or methods of calculation,

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but by the fundamental nature

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of the universe itself.

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Now, you might think

The Three Body Problem is unsolvable

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that being such a revolutionary idea,

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this is where chaos took off.

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But actually, nobody understood what Poincare did

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because he was so bad at drawing.

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He even got a zero

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on the entrance exam for college

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on the mechanical drawing part.

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So that was quite unfortunate.

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And chaos theory didn't take off until the 1960s

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when it was rediscovered by this guy.

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But Poincare did win the King's maths competition.

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Rather than solving the three-body problem,

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he proved that it is analytically unsolvable.

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"But wait," you say,

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"What about all of these articles

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claiming that there are solutions,

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or all of these videos explaining the solutions?

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If the three body problem is unsolvable,

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what are they talking about?"

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Well, they generally mean one of two things.

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It's true that we can't solve

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the three-body problem

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with a neat mathematical formula

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that works for any scenario.

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In other words, there is no general solution

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to the three-body problem,

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but mathematicians have solved it

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for some specific cases,

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like when the bodies

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form an equilateral triangle

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or a stable figure-eight pattern,

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or for some periodic orbits,

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orbits that repeat over and over indefinitely.

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The other thing a solution can mean

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is a very good approximation.

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In the absence of a general analytical solution,

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scientists still wanted to be able

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to predict the orbit of three bodies,

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so they developed a technique

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called numerical integration.

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It works by breaking the problem down

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into smaller time steps.

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For each time step,

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the gravitational forces

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between each pair of bodies are calculated.

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Using the forces and the current velocities,

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an algorithm updates the velocities

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and positions of each body

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for the next time step.

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This process is repeated over and over,

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progressively calculating the trajectory of each body.

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With today's computing power,

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numerical integration is extremely effective.

Think like a scientist

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300 years after Newton,

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the three-body problem continues

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to teach us about our world.

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Poincare's geometric techniques

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not only allow us to solve problems

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that Newton's old school methods couldn't,

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but they also birthed chaos theory

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and the field of non-linear dynamics.

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These tools have revealed chaotic systems

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across a diverse array of fields.

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From weather patterns and ocean currents

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to the rhythm of the human heart

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and the stock market,

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chaos is everywhere.

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It's fascinating how even systems

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governed by precise laws can behave in ways

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that are fundamentally unpredictable.

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What I find super inspiring about this story

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is how Poincare cracked the three-body problem

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by looking at it a different way.

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I've always been fascinated

13:52

by the way scientists and mathematicians think.

13:55

A lot of you don't know this about me,

13:56

but I actually started out doing a biology degree

13:59

before I switched to physics.

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I loved biology,

14:02

but I felt like I was learning things.

14:05

Whereas in physics,

14:06

I was learning a totally new way of thinking.

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how to think logically,

14:12

how to solve problems most effectively.

14:15

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Thanks, and I'll see you in the next episode.

15:58

Bye.

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