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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.
三体问题是出了名的不可解决问题
But actually it’s been solved many times, and in ingenious ways.
但实际上已经有人找出很多独特的解法了
Some of those solutions are incredibly useful,
有些方法非常有用
and some are incredibly bizarre.
有些则非常奇怪
PBS太空
Physics, and arguably all of science changed forever in 1687
自1687年艾萨克·牛顿发表了他的著作《自然定律》
when Isaac Newton published his Principia.
物理学 可以说所有科学都永远改变了
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
使得看似不规则的宇宙
into a perfectly tuned machine of clockwork predictability.
变成了发条装置可预测的完美机器
Given the current positions and velocities of the bodies of the solar system,
假定太阳系天体当前的位置和速度
Newton’s equations could be used in principle
原则上可以用牛顿的公式
to calculate their locations at any distant time,
来计算这些天体在任何时间的位置
future or past.
未来或过去都可以
I say “in principle”
我之所以说“原则上”
because the reality is not so simple.
是因为现实并非那么简单
Despite the beauty of Newton’s equations,
尽管牛顿的公式很美丽
they lead to a simple solution for planetary motion
但它们仅能用于行星运动的
in only one case,
一种情况
when two and only two bodies orbit each other
即 有且仅有两个天体围绕彼此运行
sans any other gravitational influence in the universe.
且宇宙中没有任何其他引力影响
Add just one more body
仅增加一个天体
and in most cases
大部分情况中
all motion becomes fundamentally chaotic.
所有运动 本质上都会变得混乱
There exists no simple solution.
没有简单的解决方法
This is the three-body problem,
这就是三体问题
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?
寻找三体问题的解决方法意味着什么?
Newton’s laws of motion and his law of universal gravitation
牛顿的运动定律以及万有引力定律
give us a set of differential equations.
给我们提供了一些不一样的等式
In some cases these can be solved with Newton’s other great invention, calculus,
某些情况能用牛顿的另一个伟大发明解决
to give a simple equation.
即用微积分列出简单的等式
Plug numbers into that equation and its solved.
将数据代入等式 问题就迎刃而解了
Those numbers are the starting positions and velocities of your gravitating bodies,
代入的数据分别是天体的起始位置和速度
plus a value for time.
以及时间值
The equations will then give you the state of the system at that time,
接着 等式能求出这个系统在所求时间点的状态
no matter how far in the past or future.
无论是多遥远的过去或未来
We call such a simple, exactly-solvable equation
我们将这种简单 可精确求解的等式
an analytic expression.
称为解析式
That just means it can be written out with a finite number of mathematical operations and functions.
意思是可以用有限个数学运算和函数将其表达出来
In the case of two gravitating bodies,
在有两个受力天体的情况下
the solutions to Newton’s laws
牛顿定律的解
are just the equations for the path traveled by the bodies,
只是天体运动路径的方程
be it a parabola of a thrown ball,
可能是一个丢出去的球的抛物线轨迹
a circle or ellipse of a planetary orbit,
一个行星轨道的圆或椭圆
or the hyperbola of an interstellar comet.
或者星际彗星的双曲线
In general, conic sections,
总之 它们都是圆锥曲线
the shapes you get when you slice up a cone.
即切开圆锥体时得到的形状
These solutions were so simple
这些解决方案如此简单
that Johannes Kepler figured out much about the elliptical solution for planetary motion
以至于约翰尼斯·开普勒在牛顿定律提出的70年前
70 years before Newton’s laws were even known.
就得到了很多行星运动的椭圆形解法
And after the Principia was published,
在《自然定律》发表之后
many sought simple, analytic solutions for more complex systems,
下一步 自然是引入具有三个引力体的系统
with systems of three gravitating bodies being the natural next step.
很多人为更复杂的系统找到了简单的分析解
But the additional influence of even a single extra body
但是仅一个天体所带来的额外影响
appeared to make an exact solution impossible.
好像让得到准确解法成为了不可能
The three body problem became the obsession for many great mathematicians,
许多伟大的数学家为三体问题着迷
but over the following three centuries,
但在过去的三个世纪中
solutions have been found for very few specialized cases.
数学家只找到了极少数特殊情况的解
Why?
为什么?
Well, in the late 1800s,
在19世纪末期
mathematicians Ernst Bruns and Henri Poincaré
数学家恩斯特·布伦斯和亨利·庞加莱
convincingly asserted
令人信服地断言
that no general analytic solutions exists.
三体问题不存在通解
The reality of the three-body problem is
三体问题的实质是
that the evolution of almost all starting configurations
几乎所有初始状态的演化
is dominated by chaotic dynamics.
都由混沌动力学决定
Future states are highly dependent on small changes in the initial conditions.
未来状态高度依赖于初始条件的细微变化
Orbits tend towards wild and unpredictable patterns,
轨道趋向于野蛮 不可预测的模式
and almost inevitably one of the bodies is eventually ejected from the system.
几乎不可避免的是 其中一个天体最终会弹出三体系统
But despite the apparent hopelessness,
但是除了这种显而易见的无望
there was much profit in learning to predict the gravitational motion of many bodies.
学会预测许多天体的重力运动有很多好处
For most of the three centuries since Newton,
牛顿之后三个世纪的大部分时间里
predicting the motion of the planets and the moon
预测行星和月球运动
was critical for nautical navigation.
对航海学至关重要
Now it’s essential to space travel.
现在 它们对太空旅行来说必不可少
How do we do it?
我们应该怎样做?
Well, just because the three body problem
因为三体问题
for the most part has no useful analytic solution,
多半没有有用的解析解
approximate solutions can be found.
我们可以寻找近似解
For example, if the bodies are far enough apart,
例如 如果天体之间的距离足够远
then we can approximate a many-body system
那么我们可以把一个多天体系统
as a series of two-body systems.
近似看作一系列二体系统
For example, each planet of our solar system
例如 太阳系中的每个行星
can be thought of as a separate two-body system with the Sun.
都可以与太阳一起 看作一个二体系统
That gives you a series of simple elliptical orbits,
这给我们提供了一系列简单的椭圆轨道
like those predicted by Kepler.
例如开普勒预测的那些
But those orbits eventually shift
但是那些轨道
due to the interactions between the planets.
最终会因行星之间的相互作用而发生变化
Another useful approximation
另一种有用的近似情况
is when one of the three bodies has a very low mass
是当三体中一个天体 与其他两个天体相比
compared to the other two.
质量非常小时
We can ignore the minuscule gravitational influence of the smaller body
我们可以忽略小天体对另两个天体极小的引力作用
and assume that it moves within the completely solvable
并假设它在大天体同伴
two-body orbits of its larger companions.
完全可解的二体轨道内运动
We call this the reduced three-body problem.
我们称之“简化的三体问题”
It works very well for tiny things like artificial satellites around the Earth.
这个模型非常适合人造地球卫星之类的小天体
It can also be used to approximate the orbits of the moon relative to the Earth and Sun,
也可以联系地球和太阳 近似计算月球轨道
or the Earth relative to the Sun and Jupiter.
或者联系太阳和木星 近似计算地球轨道
These approximate solutions are useful,
这些近似解有用
but ultimately fail to predict perfectly.
但最终也不能完美地进行预测
Even the smallest planetary bodies have some mass,
即使最小的行星体也具有一定质量
and the solar system as a whole has many massive constituents.
同时 整个太阳系中也有很多大质量天体
The Sun, Jupiter and Saturn alone
仅太阳 木星和土星
are automatically a three-body system with no analytic solution,
在不考虑地球时
before we even add in the Earth.
就是一个天然的无解析解的三体系统
But the absence of an analytic solution doesn’t mean the absence of any solution.
但是 不存在解析解不等于不存在任何解
To get an accurate prediction for most three-body systems,
为了得到一个适用于大多数三体系统的精确预测
you need to break the motion of the system into many pieces,
你需要把这个系统的运动分解成很多部分
and solve them one at a time.
然后一点一点地解决
A sufficiently small section of any gravitational trajectory
任何引力轨迹的某一足够小部分
can be approximated with an exact, analytical solution.
都可以用一个准确的解析解近似表示
Perhaps a straight line
可能是一条直线
or a segment of two-body path
或是以大质量天体为整个系统中心
around the center of mass of the entire system,
形成的二体系统路径的一部分
assuming everything else stays fixed.
假设其他条件保持不变
If you break up the problem into tiny enough paths segments
如果你把这个问题分解为足够小的路径段
or time-steps,
或时间段
then the small motions of all bodies in the system
然后系统内所有天体的微小运动
can be updated step by step.
就能被一步一步地更新了
This method of solving differential equations one step at a time
这种一步一步解决微分方程的方法
is called numerical integration.
叫作数值积分
And when applied to the motion of many bodies,
当这种方法应用于多天体运动时
it’s an N-body simulation.
就是一个N体模拟
With modern computers,
采用现代计算机
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,
行星在遥远的未来的运动
or solve for millions of objects
或求解数百万个天体的轨迹
to simulate the formation and evolution of entire galaxies.
以模拟整个星系的形成和演化
But these numerical solutions
但是这些数值解
didn’t begin with the invention artificial computers.
并不是人造计算机的发明之后出现的
Before that, these calculations had to be done by hand,
在此之前 这些计算必须手动完成
in fact by many hands.
实际上是很多人手动完成
The limitations of approximate solutions,
近似解的局限性
the laboriousness of pre-computer numerical integration,
计算机出现以前 数值积分工作的繁琐性
and also the legendary status of the three-body problem
以及三体问题的传奇地位
inspired generations of physicists and mathematicians
鼓舞着几代物理学家和数学家
to continue to seek exact, analytic solutions.
继续寻求精确的解析解
And some succeeded,
一些人成功了
albeit in very specialized cases.
尽管是针对某些非常特殊的情况
The first was Leonhard Euler,
首先是莱昂哈德·欧拉
who found a family of solutions for three bodies
他为绕着共同质心运动的三个天体
orbiting around a mutual center of mass,
找到了一系列解决方案
where all bodies remain in a straight line,
这种情况下 所有天体始终共线
essentially in permanent eclipse.
本质上是永久的日蚀
Joseph-Louis Lagrange
约瑟夫·路易斯·拉格朗日
found solutions in which the three bodies form an equilateral triangle.
找到了三个天体形成等边三角形的情况的解
In fact, for any two bodies orbiting each other,
实际上 对于任意两个互相绕行的天体
the Euler and Lagrange’s solutions
欧拉和拉格朗日的解
define 5 additional orbits for a third body
为第三个天体定义了5个
that can be described with simple equations.
可用简单方程描述的附加轨道
These are the only perfectly analytical solutions to the three body problem that exist.
这是现存对三体问题仅有的完美解析解
Place a low-mass object on any of these 5 orbits
在这5个轨道中的任何一个上放置一个低质量天体
and it will stay there indefinitely,
它都将永远停留在该轨道上
tracking the Earth’s orbit around the Sun.
沿着地球绕太阳运行的轨道
We now call these the Lagrange points,
我们现在称这些轨道为“拉格朗日点”
and they’re useful places to park our spacecraft.
它们是我们停放航天器的好地方
There was a bit of a gap after Euler and Lagrange,
在欧拉和拉格朗日之后 研究进展很少
because to discover new specialized three-body solutions,
因为要发现新的三体问题特解
we had to search the vast space of possible orbits
我们必须使用计算机搜索广阔的太空
using computers.
寻找可能的轨道
The key was to find three-body systems that had periodic motion.
关键是找到进行周期运动的三体系统
They evolve, sometimes in complex ways,
它们有时以复杂的方式
back to their starting configuration.
演变回初始构形
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
找到了一组解
involving two masses bouncing back and forth
包含两个天体在第三天体轨道的中心
in the center of a third body’s orbit.
来回穿梭
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.
构成的稳定8字形轨道
The numerical discovery of the figure-8 solution
Alain Chenciner和Richard Montgomery
was proved mathematically
从数学角度证明了
by Alain Chenciner and Richard Montgomery,
8字形特解的数值发现
and insights gained from that proof
从该证明中得到的经验
led to a boom in the discovery of new periodic three body orbits.
也导致了多个新周期性三体轨道的发现
Some of these periodic solutions are incredibly complex,
其中一些周期性特解非常复杂
but Montgomery came up with a fascinating way
但是Montgomery想出了一种有趣的方式
to depict them in the absence of simple equations.
在不用简单方程的情况下描绘它们
It’s called the shape-sphere,
这种方法叫做shape-sphere(球状体)
and it works like this.
就像这样
Imagine the bodies in 3-body system
想象一下 三体系统中的天体
are the vertices of a triangle,
是三角形的三个顶点
whose center is the center of mass of the system.
三角形中心是三体系统的质心
The evolution of the system can be expressed
系统的演变可以通过
through the changing shape of that triangle.
该三角形的形状变化来表示
We throw away certain information,
我们忽略一些信息
the size of the triangle and its orientation,
例如三角形的大小及方向
keeping only information about the relative lengths of the edges,
只保留边的相对长度
or equivalently the angles between the edges.
或边之间夹角的信息
Now we map that information on the surface of a sphere.
现在 我们将这些信息映射到一个球体的表面上
We only need the 2-D surface,
我们只需要2D平面
because if we know 2 internal angles of the triangle
因为如果知道三角形的2个内角角度
we also know the 3rd.
也就能知道第三个了
So, the equator of the sphere represents both angles being zero.
球体的赤道表示两个三角形内角均为零
That’s a fully collapsed triangle.
就是一个完全平放的三角形
The 3-bodies are in a straight line,
3个天体处在一条直线上
as in Euler’s solutions.
正如欧拉特解一样
The poles are equilateral triangles.
天体位于极点上时是等边三角形
So, Lagrange’s solutions.
也是拉格朗日特解的一种
All other orbits move on this sphere
随着轨道定义的三角形的演变
as the triangle defined by the orbits evolves.
所有其他轨道都在该球面上移动
It turns out that the periodic motion on the shapesphere
人们发现 与天体本身的运动相比
appears much simpler and easier to analyze
球体上的周期性运动
than the motion of the bodies themselves.
似乎更容易分析
Now hundreds of stable 3-body orbits are known,
现在 数百个稳定的三体轨道已被发现
although it should be noted
尽管需要注意
that besides the Euler and Lagrange solutions,
除了欧拉特解和拉格朗日特解
none of these are likely to occur in nature.
这里的其他任何轨道 都不可能出现在自然界中
So their practical use may be limited.
所以它们的实际用途可能是有限的
Very recently, a new approach to solving the three-body problem has appeared,
最近 出现了一种解决三体问题的新方法
which transforms the chaotic nature of three-body interactions
该方法将三体相互作用的混乱性质
into a useful tool,
转变为一种有用的工具
rather than a liability.
而不是一种妨碍
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
混沌运动的关键在于
is that the state of the system seems to get randomly shuffled over time.
系统的状态似乎会随时间的流逝而随机改变
The motion is actually perfectly deterministic,
在某一瞬间与下个瞬间之间
defined between one instant and the next,
运动实际上完全是确定性的
but can be thought of as approximately random over long intervals.
但可以认为它在很长的间隔内近似随机
Such a pseudo-random system will, over time,
随着时间的推移 这种伪随机系统
explore all possible configurations consistent with some basic properties
可能出现某些基本属性一致的所有构型
like the energy and angular momentum of the system.
这些基本属性包括系统的能量和角动量
The system explores what we call a phase space,
该系统探索所谓的相空间
a space of possible arrangements of position and velocity.
即位置和速度可能的排列空间
Well, for a pseudo-random system,
对于一个伪随机系统来说
statistical mechanics lets us calculate the probability of the system
统计力学使我们能够计算该系统
being in any part of that phase space at any one time.
在任何时间处于相空间任何部分的概率
How is this useful?
这有什么用?
Well, actually,
实际上
almost all three-body systems eject one of the bodies,
几乎所有三体系统都会弹出其中一个天体
leaving a nice, stable two-body system, a binary pair.
留下一个美妙且稳定的二体系统 即一对天体
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.
可能发生这种弹出的相空间区域
And by doing so,
通过这样
they could map the range of likely orbital properties
他们可以映射弹出后
for the two objects left behind after the ejection.
两个遗留天体可能轨道的属性范围
This looks to be incredibly useful for understanding
这似乎对理解
the evolution of dense regions of the universe,
宇宙密集区域的演化非常有用
where three-body systems of stars or black holes
在这种区域中 三体恒星系统或三体黑洞系统
may form and then disintegrate very frequently.
可能非常频繁地形成 然后瓦解
One last thing about the three-body problem.
关于三体问题的最后一点
Henri Poincare
昂利·庞加莱认为
thought the general case could not be solved.
无法求出一般的三体问题的解析解
In fact he was wrong.
其实他错了
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
芬兰数学家Karl Sundman
found a solution to the general three-body problem.
找到了一般三体问题的一个解析解
It was a converging infinite series
这是一个收敛的无穷级数
that added together an endless chain of terms
是将无穷项数字相加
to solve the orbital calculation.
来解决这种轨道计算问题
The convergence of the series,
这个级数的收敛
meant that successive terms diminished to effectively nothing.
意味着逐项递减至0
So in principle the equation could be written out on paper.
所以原则上这个式子是可以写在纸上的
However, the convergence of Sundman’s series is so slow
然而 Sundman的级数收敛得太慢了
that it would take something like 10^8 million terms
以致于需要大约10^14项的收敛
to converge for a typical calculation in celestial mechanics.
才能进行天体力学的典型计算
That is a lot of sheets of paper.
这需要很多张纸
So there you have it.
所以你就明白了
The three-body problem is perfectly solved uselessly,
三体问题的完美解析解用处不大
or for seemingly useless and bizarre orbits.
或者说是针对看似无用和离奇的轨道的
And it can be approximately solved
对于所有有用和实用的目的
for all useful and practical purposes
它都可以近似地解决
with enough precision to work just fine.
并且精度足够
Good to know,
好消息
next time you’re in a chaotic orbit,
下次进入一个混乱的轨道时
trying to astronavigate around two other gravitating denizens of space time.
你可以尝试围绕另外两个天体运行
A few weeks ago, I invited Matt to come to Fermilab
几周前 我邀请了Matt来费米国立加速器实验室
to make an awesome crossover video on the subject of neutrinos.
制作关于中微子的超棒跨界视频
He accepted and the rest,
他接受了邀请 而接下来的
as they say, is history.
正如人们所说 就是历史性的时刻了
There were some great questions in the comments
评论区有一些非常棒的问题
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.
现在开始吧
Sanskar Jain asks
Sanskar Jain提问
what it means for a neutrino to go with a particular lepton,
中微子与特定的轻子一起运动意味着什么
meaning electron, muon or tau.
也即电子 μ子或τ子
It turns out that over short distances
原来 在短距离内
and before neutrinos have a chance to oscillate,
且在中微子有机会振荡之前
they remember how they were made.
它们会记住自己是如何产生的
Neutrinos made in nuclear reactors are made with electrons,
核反应堆中产生的中微子带有电子
and if they interact again, they make only electrons.
如果再次反应 只会放出电子
In particle beams, neutrinos are made with muons,
在粒子束中 中微子由μ子构成的
and can subsequently only make muons.
随后也只能构成μ子
In fact, this observation in 1962
实际上 1962年的这一发现
led to the discovery that there were different kinds of neutrinos and,
导致人们发现了各种不同的中微子
subsequently, to the 1988 Nobel Prize in physics.
并随后获得了1988年诺贝尔物理学奖
Gede Ge asks
Gede Ge提问
why we use Argon in our neutrino detectors,
为什么我们在中微子探测器中使用氩气
and that’s a great question.
这是个好问题
The answer is that we don’t always.
答案是并不总是这样
Neutrino detectors have been made of water, metal, dry cleaning fluid,
中微子探测器的构成材料有 水 金属 干洗液
even baby oil doped with a chemical called scintillator.
甚至是掺有被称为“闪烁体”的化学物质的婴儿油
We use Argon because it ionizes very easily.
我们使用氩气是因为它很容易电离
That means when a neutrino DOES interact in the Argon,
这意味着 当中微子粒子确实在氩气中发生反应时
we can see the path of the particles made in the interaction.
我们可以看到反应中产生的粒子的路径
From that, we can reconstruct the collision
由此 我们可以重建碰撞过程
and learn more about neutrinos.
了解有关中微子的更多信息
Nexus void asks
Nexus void提问
how we’ll learn if neutrinos prefer to interact with matter
我们如何知道中微子是更喜欢跟物质反应
or antimatter.
还是跟反物质反应?
Actually, what we’ll do is a little different.
实际上 我们的做法稍有不同
We want to see if matter or antimatter neutrinos
我们想看看物质和反物质中微子
change their identity at different rates.
是否以不同的速率改变
We do that by performing the experiment with a beam of neutrinos
我们的做法是 对中微子束进行实验
and then repeating it with a beam of antineutrinos.
然后对反中微子束重复该实验
If they’re different, we may be on to something.
如果速度不同 那么我们就可能有一些发现了
And, if you want to learn more about that,
如果你想了解更多相关信息
I recommend my video on Leptogenesis on the Fermilab YouTube channel.
建议你观看我的油管频道Fermilab上有关“轻子生成”的视频
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.
我们会是的
Although we hadn’t met before we started filming,
虽然录影前我们没见过面
we are kindred spirits,
但是我们志趣相投
interested both in cutting edge science
都对前沿科学感兴趣
and making videos that share the excitement with everyone.
都通过制作视频来和大家分享
I’m a huge fan of PBS Space Time,
我的PBS太空的忠实粉丝
and if you like them, you’ll like ours as well.
如果你喜欢他们 你也会喜欢我们的
In fact, I’d like to invite you to subscribe to the Fermilab YouTube channel.
实际上 我想邀请你订阅油管频道Fermilab
Our videos cover some of the most interesting topics in all of physics.
我们的视频涉及物理学中最有趣的话题中的一部分
And that’s saying something, because physics…
这能说明一些问题 因为物理学……
and Space Time of course…is everything.
当然还有太空……就是一切

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视频概述

简单介绍了三体问题相关的物理知识,以及三体问题的研究简史。

听录译者

收集自网络

翻译译者

Icyyyy🌸

审核员

审核员MS

视频来源

https://www.youtube.com/watch?v=et7XvBenEo8

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