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#### 解决三体问题

Solving the Three Body Problem

The three body problem is famous for being impossible to solve.

But actually it’s been solved many times, and in ingenious ways.

Some of those solutions are incredibly useful,

and some are incredibly bizarre.

PBS太空
Physics, and arguably all of science changed forever in 1687

when Isaac Newton published his Principia.

Within it were equations of motion and gravity
《自然定律》描述的万有引力和运动定律
that transformed our erratic-seeming cosmos

into a perfectly tuned machine of clockwork predictability.

Given the current positions and velocities of the bodies of the solar system,

Newton’s equations could be used in principle

to calculate their locations at any distant time,

future or past.

I say “in principle”

because the reality is not so simple.

Despite the beauty of Newton’s equations,

they lead to a simple solution for planetary motion

in only one case,

when two and only two bodies orbit each other

sans any other gravitational influence in the universe.

Add just one more body

and in most cases

all motion becomes fundamentally chaotic.

There exists no simple solution.

This is the three-body problem,

and we’ve been trying to solve it for 300 years.
300年来 我们一直在尝试解决它
What does it mean to find a solution to the three-body problem?

Newton’s laws of motion and his law of universal gravitation

give us a set of differential equations.

In some cases these can be solved with Newton’s other great invention, calculus,

to give a simple equation.

Plug numbers into that equation and its solved.

Those numbers are the starting positions and velocities of your gravitating bodies,

plus a value for time.

The equations will then give you the state of the system at that time,

no matter how far in the past or future.

We call such a simple, exactly-solvable equation

an analytic expression.

That just means it can be written out with a finite number of mathematical operations and functions.

In the case of two gravitating bodies,

the solutions to Newton’s laws

are just the equations for the path traveled by the bodies,

be it a parabola of a thrown ball,

a circle or ellipse of a planetary orbit,

or the hyperbola of an interstellar comet.

In general, conic sections,

the shapes you get when you slice up a cone.

These solutions were so simple

that Johannes Kepler figured out much about the elliptical solution for planetary motion

70 years before Newton’s laws were even known.

And after the Principia was published,

many sought simple, analytic solutions for more complex systems,

with systems of three gravitating bodies being the natural next step.

But the additional influence of even a single extra body

appeared to make an exact solution impossible.

The three body problem became the obsession for many great mathematicians,

but over the following three centuries,

solutions have been found for very few specialized cases.

Why?

Well, in the late 1800s,

mathematicians Ernst Bruns and Henri Poincaré

convincingly asserted

that no general analytic solutions exists.

The reality of the three-body problem is

that the evolution of almost all starting configurations

is dominated by chaotic dynamics.

Future states are highly dependent on small changes in the initial conditions.

Orbits tend towards wild and unpredictable patterns,

and almost inevitably one of the bodies is eventually ejected from the system.

But despite the apparent hopelessness,

there was much profit in learning to predict the gravitational motion of many bodies.

For most of the three centuries since Newton,

predicting the motion of the planets and the moon

was critical for nautical navigation.

Now it’s essential to space travel.

How do we do it?

Well, just because the three body problem

for the most part has no useful analytic solution,

approximate solutions can be found.

For example, if the bodies are far enough apart,

then we can approximate a many-body system

as a series of two-body systems.

For example, each planet of our solar system

can be thought of as a separate two-body system with the Sun.

That gives you a series of simple elliptical orbits,

like those predicted by Kepler.

But those orbits eventually shift

due to the interactions between the planets.

Another useful approximation

is when one of the three bodies has a very low mass

compared to the other two.

We can ignore the minuscule gravitational influence of the smaller body

and assume that it moves within the completely solvable

two-body orbits of its larger companions.

We call this the reduced three-body problem.

It works very well for tiny things like artificial satellites around the Earth.

It can also be used to approximate the orbits of the moon relative to the Earth and Sun,

or the Earth relative to the Sun and Jupiter.

These approximate solutions are useful,

but ultimately fail to predict perfectly.

Even the smallest planetary bodies have some mass,

and the solar system as a whole has many massive constituents.

The Sun, Jupiter and Saturn alone

are automatically a three-body system with no analytic solution,

before we even add in the Earth.

But the absence of an analytic solution doesn’t mean the absence of any solution.

To get an accurate prediction for most three-body systems,

you need to break the motion of the system into many pieces,

and solve them one at a time.

A sufficiently small section of any gravitational trajectory

can be approximated with an exact, analytical solution.

Perhaps a straight line

or a segment of two-body path

around the center of mass of the entire system,

assuming everything else stays fixed.

If you break up the problem into tiny enough paths segments

or time-steps,

then the small motions of all bodies in the system

can be updated step by step.

This method of solving differential equations one step at a time

is called numerical integration.

And when applied to the motion of many bodies,

it’s an N-body simulation.

With modern computers,

N-body simulations can accurately predict
N体模拟能准确地预测
the motion of the planets into the distant future,

or solve for millions of objects

to simulate the formation and evolution of entire galaxies.

But these numerical solutions

didn’t begin with the invention artificial computers.

Before that, these calculations had to be done by hand,

in fact by many hands.

The limitations of approximate solutions,

the laboriousness of pre-computer numerical integration,

and also the legendary status of the three-body problem

inspired generations of physicists and mathematicians

to continue to seek exact, analytic solutions.

And some succeeded,

albeit in very specialized cases.

The first was Leonhard Euler,

who found a family of solutions for three bodies

orbiting around a mutual center of mass,

where all bodies remain in a straight line,

essentially in permanent eclipse.

Joseph-Louis Lagrange

found solutions in which the three bodies form an equilateral triangle.

In fact, for any two bodies orbiting each other,

the Euler and Lagrange’s solutions

define 5 additional orbits for a third body

that can be described with simple equations.

These are the only perfectly analytical solutions to the three body problem that exist.

Place a low-mass object on any of these 5 orbits

and it will stay there indefinitely,

tracking the Earth’s orbit around the Sun.

We now call these the Lagrange points,

and they’re useful places to park our spacecraft.

There was a bit of a gap after Euler and Lagrange,

because to discover new specialized three-body solutions,

we had to search the vast space of possible orbits

using computers.

The key was to find three-body systems that had periodic motion.

They evolve, sometimes in complex ways,

back to their starting configuration.

In the 70s, Michel Henon and Roger Broucke
20世纪70年代 米歇尔·赫农和罗杰·布鲁克
found a family of solutions

involving two masses bouncing back and forth

in the center of a third body’s orbit.

In the 90s, Cris Moore discovered a stable figure-8 orbit
90年代 Cris Moore发现了一个由三个相等质量天体
of three equal masses.

The numerical discovery of the figure-8 solution
Alain Chenciner和Richard Montgomery
was proved mathematically

by Alain Chenciner and Richard Montgomery,
8字形特解的数值发现
and insights gained from that proof

led to a boom in the discovery of new periodic three body orbits.

Some of these periodic solutions are incredibly complex,

but Montgomery came up with a fascinating way

to depict them in the absence of simple equations.

It’s called the shape-sphere,

and it works like this.

Imagine the bodies in 3-body system

are the vertices of a triangle,

whose center is the center of mass of the system.

The evolution of the system can be expressed

through the changing shape of that triangle.

We throw away certain information,

the size of the triangle and its orientation,

keeping only information about the relative lengths of the edges,

or equivalently the angles between the edges.

Now we map that information on the surface of a sphere.

We only need the 2-D surface,

because if we know 2 internal angles of the triangle

we also know the 3rd.

So, the equator of the sphere represents both angles being zero.

That’s a fully collapsed triangle.

The 3-bodies are in a straight line,
3个天体处在一条直线上
as in Euler’s solutions.

The poles are equilateral triangles.

So, Lagrange’s solutions.

All other orbits move on this sphere

as the triangle defined by the orbits evolves.

It turns out that the periodic motion on the shapesphere

appears much simpler and easier to analyze

than the motion of the bodies themselves.

Now hundreds of stable 3-body orbits are known,

although it should be noted

that besides the Euler and Lagrange solutions,

none of these are likely to occur in nature.

So their practical use may be limited.

Very recently, a new approach to solving the three-body problem has appeared,

which transforms the chaotic nature of three-body interactions

into a useful tool,

rather than a liability.

Nicholas Stone and Nathan Leigh
Nicholas Stone和Nathan Leigh
published this in Nature in December 2019.
2019年12月在《自然》杂志中发表了这一观点
The thing about chaotic motion

is that the state of the system seems to get randomly shuffled over time.

The motion is actually perfectly deterministic,

defined between one instant and the next,

but can be thought of as approximately random over long intervals.

Such a pseudo-random system will, over time,

explore all possible configurations consistent with some basic properties

like the energy and angular momentum of the system.

The system explores what we call a phase space,

a space of possible arrangements of position and velocity.

Well, for a pseudo-random system,

statistical mechanics lets us calculate the probability of the system

being in any part of that phase space at any one time.

How is this useful?

Well, actually,

almost all three-body systems eject one of the bodies,

leaving a nice, stable two-body system, a binary pair.

Stone and Leigh found that they could identify
Stone和Leigh发现 他们可以识别
the regions of phase space where these ejections were likely.

And by doing so,

they could map the range of likely orbital properties

for the two objects left behind after the ejection.

This looks to be incredibly useful for understanding

the evolution of dense regions of the universe,

where three-body systems of stars or black holes

may form and then disintegrate very frequently.

One last thing about the three-body problem.

Henri Poincare

thought the general case could not be solved.

In fact he was wrong.

In 1906, not so long after Poincare stern proclamation,
1906年 就在庞加莱断言后不久
Finnish mathematician Karl Sundman

found a solution to the general three-body problem.

It was a converging infinite series

that added together an endless chain of terms

to solve the orbital calculation.

The convergence of the series,

meant that successive terms diminished to effectively nothing.

So in principle the equation could be written out on paper.

However, the convergence of Sundman’s series is so slow

that it would take something like 10^8 million terms

to converge for a typical calculation in celestial mechanics.

That is a lot of sheets of paper.

So there you have it.

The three-body problem is perfectly solved uselessly,

or for seemingly useless and bizarre orbits.

And it can be approximately solved

for all useful and practical purposes

with enough precision to work just fine.

Good to know,

next time you’re in a chaotic orbit,

trying to astronavigate around two other gravitating denizens of space time.

A few weeks ago, I invited Matt to come to Fermilab

to make an awesome crossover video on the subject of neutrinos.

He accepted and the rest,

as they say, is history.

There were some great questions in the comments

and Matt asked me to answer a few of them.
Matt让我回答其中的一部分
So here it goes.

Sanskar Jain提问
what it means for a neutrino to go with a particular lepton,

meaning electron, muon or tau.

It turns out that over short distances

and before neutrinos have a chance to oscillate,

they remember how they were made.

Neutrinos made in nuclear reactors are made with electrons,

and if they interact again, they make only electrons.

In particle beams, neutrinos are made with muons,

and can subsequently only make muons.

In fact, this observation in 1962

led to the discovery that there were different kinds of neutrinos and,

subsequently, to the 1988 Nobel Prize in physics.

Gede Ge提问
why we use Argon in our neutrino detectors,

and that’s a great question.

The answer is that we don’t always.

Neutrino detectors have been made of water, metal, dry cleaning fluid,

even baby oil doped with a chemical called scintillator.

We use Argon because it ionizes very easily.

That means when a neutrino DOES interact in the Argon,

we can see the path of the particles made in the interaction.

From that, we can reconstruct the collision

Nexus void提问
how we’ll learn if neutrinos prefer to interact with matter

or antimatter.

Actually, what we’ll do is a little different.

We want to see if matter or antimatter neutrinos

change their identity at different rates.

We do that by performing the experiment with a beam of neutrinos

and then repeating it with a beam of antineutrinos.

If they’re different, we may be on to something.

I recommend my video on Leptogenesis on the Fermilab YouTube channel.

Laura Henley notes that we look like we’re best friends.
Laura Henley留言说我们看起来像挚友
That’s, well, because we could be.

Although we hadn’t met before we started filming,

we are kindred spirits,

interested both in cutting edge science

and making videos that share the excitement with everyone.

I’m a huge fan of PBS Space Time,

and if you like them, you’ll like ours as well.

In fact, I’d like to invite you to subscribe to the Fermilab YouTube channel.

Our videos cover some of the most interesting topics in all of physics.

And that’s saying something, because physics…

and Space Time of course…is everything.

Icyyyy🌸