Name: バタフライ効果を支える科学 (The Science Behind the Butterfly Effect)
Uploaded: 2021-01-14T10:25:32.000Z
Duration: 12 min 51 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

基礎受動態

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More about last pass at the end of the show.

The butter fly effect is the idea that the tiny causes, like a flay of a butter fly's wings in Brazil,

can have huge effects, like setting off a tornado in Texas

Now that idea comes straight from the title of a scientific paper published nearly 50 years ago

and perhaps more than any other recent scientific concept, it has captured the public imagination

different movies, TV episodes, and short films with 'butterfly effect' in the title

not to mention prominent references in movies like Jurassic Park, or in songs, books, and memes.

in pop culture the butterfly effect has come to mean

that even tiny, seemingly insignificant choices you make can have huge consequences later on in your life

and I think the reason people are so fascinated by the butterfly effect is because it gets at a fundamental question

Which is, how well can we predict the future?

Now the goal of this video is to answer that question by examining the science behind the butterfly effect

so if you go back to the late 1600s, after Isaac Newton had come up with his laws of motion and universal gravitation,

I mean we could explain the motions of all the planets and moons,

we could predict eclipses and the appearances of comets with pinpoint accuracy centuries in advance

French physicist Pierre-Simon Laplace summed it up in a famous thought experiment:

he imagined a super-intelligent being, now called Laplace's demon,

that knew everything about the current state of the universe:

the positions and momenta of all the particles and how they interact

if this intellect were vast enough to submit the data to analysis, he concluded,

then the future, just like the past, would be present before its eyes

This is total determinism: the view that the future is already fixed,

We just have to wait for it to manifest itself

I think if you've studied a bit of physics, this is the natural viewpoint to come away with

I mean sure there's Heisenberg's uncertainty principle from quantum mechanics,

Pretty insignificant on the scale of people.

Virtually all the problems I studied were ones that could be solved analytically

like the motion of planets, or falling objects, or pendulums

and speaking of pendulums I want to look at a case of a simple pendulum here

to introduce an important representation of dynamical systems, which is phase space

so some people may be familiar with position-time or velocity-time graphs

but what if we wanted to make a 2d plot that represents every possible state of the pendulum?

Every possible thing it could do in one graph

well on the x-axis we can plot the angle of the pendulum,

If the pendulum has friction it will eventually slow down and stop

and this is shown in phase space by the inward spiral --

the pendulum swings slower and less far each time

and it doesn't really matter what the initial conditions are,

we know that the final state will be the pendulum at rest hanging straight down

and from the graph it looks like the system is attracted to the origin, that one fixed point

so this is called a fixed point attractor

now if the pendulum doesn't lose energy, well it swings back and forth the same way each time

the pendulum is going fastest at the bottom but the swing is in opposite directions as it goes back and forth

the closed loop tells us the motion is periodic and predictable

anytime you see an image like this in phase space,

you know that this system regularly repeats

we can swing the pendulum with different amplitudes,

but the picture in phase space is very similar, just a different sized loop

now an important thing to note is that the curves never cross in phase space

and that's because each point uniquely identifies the complete state of the system

so once you've defined the initial state, the entire future is determined

now the pendulum can be well understood using Newtonian physics,

but Newton himself was aware of problems that did not submit to his equations so easily,

so calculating the motion of the Earth around the Sun was simple enough with just those two bodies

Newton told his friend Haley that the theory of the motions of the moon made his head ache,

and kept him awake so often that he would think of it no more

the problem, as would become clear to Henri Poincaré two hundred years later,

was that there was no simple solution to the three-body problem

Poincaré had glimpsed what later became known as chaos.

Chaos really came into focus in the 1960s,

when meteorologist Ed Lorenz tried to make a basic computer simulation of the Earth's atmosphere

he had 12 equations and 12 variables, things like temperature, pressure, humidity and so on

and the computer would print out each time step as a row of 12 numbers

so you could watch how they evolved over time

now the breakthrough came when Lorenz wanted to redo a run

but as a shortcut he entered the numbers from halfway through a previous printout

he went off to get some coffee, and when he came back and saw the results,

The new run followed the old one for a short while but then it diverged

and pretty soon it was describing a totally different state of the atmosphere

Lorenz's first thought, of course, was that the computer had broken

The real reason for the difference came down to the fact that printer rounded to three decimal places

So when he entered those initial conditions,

the difference of less than one part in a thousand

created totally different weather just a short time into the future

now Lorenz tried simplifying his equations and then simplifying them some more,

down to just three equations and three variables

which represented a toy model of convection:

essentially a 2d slice of the atmosphere heated at the bottom and cooled at the top

but again, he got the same type of behavior:

if he changed the numbers just a tiny bit, results diverged dramatically.

Lorenz's system displayed what's become known as sensitive dependence on initial conditions,