The REAL Three Body Problem in Physics

this episode is sponsored by brilliant hello welcome to up and Adam 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 CRS 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 had 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 solve 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 Edmund Hy 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 road map 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 one 100% Precision measurements of position and momentum would inevitably be slightly off if even if it was imperceptible to our senses 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 hle 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 HRI ponar Rose to the challenge now I'm way too to say that without offending some French people so I'm going to get my French husband to say it he started with the 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 panare couldn't figure out the equation 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 panker wanted to zoom out and see the bigger picture if Newton's law showed us the road panare 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 poery found that the interaction of the gravitational forces can create these fixed points in space points where the force are perfectly balanced so that an object is either drawn in to 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 inac accuracy 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 panare 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

Pon discovered intersections in our road map 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 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 Pan'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 poer 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 poar did win the king's math confidence ition 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 8 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 Pan's geometric techniques not only allow us to solve problems that Newton's old school methods couldn't but they also birthed chaos theory in the field of nonlinear 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 panor 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 today's sponsor brilliant brilliant is an interactive website and app that focuses on learning by doing it has thousands of lessons in math data analysis programming and AI what I love about brilliant is one their first principles approach to Concepts and two their strong focus on problem solving the first principles approach builds your understanding from the ground up so

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