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三体问题的解释

Newton’s three-body problem explained - Fabio Pacucci

“理论指引实验 还是实验指引理论?” 刘慈欣《三体》
In 2009, two researchers ran a simple experiment.
2009年 两位研究员做了一个简单的实验
They took everything we know about our solar system
他们应用人类关于太阳系的一切认知
and calculated where every planet would be
计算了50亿年以后
up to 5 billion years in the future.
每颗行星的位置
To do so they ran over 2,000 numerical simulations
为此 他们在保持初始条件不变的情况下
with the same exact initial conditions
进行了2000多次数值模拟
except for one difference: the distance between Mercury and the Sun,
除了一个条件的不同:水星到太阳的距离
modified by less than a millimeter from one simulation to the next.
每次模拟改变的数值小于1毫米
Shockingly, in about 1 percent of their simulations,
令人震惊的是 在约1%的模拟中
Mercury’s orbit changed so drastically
水星的轨道出现翻天覆地的变化
that it could plunge into the Sun or collide with Venus.
以至于它可能坠入太阳或撞上金星
Worse yet, in one simulation
更糟糕的是 在一次模拟中
it destabilized the entire inner solar system.
它打破了整个内太阳系的稳定
This was no error;
这并非出错
the astonishing variety in results reveals the truth
模拟结果的惊人变化揭示了一个事实
that our solar system may be much less stable than it seems.
太阳系可能远没有它看上去那么稳定
Astrophysicists refer to this astonishing property of gravitational systems
天体物理学家把引力系统这种惊人的特性
as the n-body problem.
称为“多体问题”
While we have equations that can completely predict
尽管我们可以用方程完全预测
the motions of two gravitating masses,
两个引力天体的运动
our analytical tools fall short
但在面对更多天体的系统时
when faced with more populated systems.
我们的分析工具无能为力
It’s actually impossible to write down
事实上 根本不可能写出
all the terms of a general formula
一个包含所有变量的通用方程式
that can exactly describe the motion of three or more gravitating objects.
来确切描述三个或多个引力天体的运动
Why? The issue lies in how many unknown
为什么呢?问题在于一个多体系统中
variables an n-body system contains.
包含多少未知变量
Thanks to Isaac Newton, we can write a set of equations
幸亏有艾萨克·牛顿 我们可以写下一套方程
to describe the gravitational force acting between bodies.
来描述两个天体间的万有引力作用
However, when trying to find a general solution
然而 当试图为这些方程中的未知变量
for the unknown variables in these equations,
找到一个通解时
we’re faced with a mathematical constraint:
我们遇到一个数学上的约束
for each unknown, there must be at least one equation
每个未知变量必须至少有一个
that independently describes it.
单独描述它的方程
Initially, a two-body system appears to have more unknown variables
起初 二体系统中关于位置和速度的未知变量
for position and velocity than equations of motion.
似乎多于它的运动方程
However, there’s a trick:
然而 有个小技巧
consider the relative position and velocity of the two bodies
就是考虑两个天体关于系统重心的
with respect to the center of gravity of the system.
相对位置和相对速度
This reduces the number of unknowns
这可以减少未知变量的数量
and leaves us with a solvable system.
并得到一个可解的系统
With three or more orbiting objects in the picture,
当有三个或更多绕行轨道的天体时
everything gets messier.
一切更复杂了
Even with the same mathematical trick of considering relative motions,
即使同样使用考虑相对运动的数学技巧
we’re left with more unknowns than equations describing them.
未知变量的个数依然多于描述它们的方程
There are simply too many variables for this system of equations
这组方程中有太多变量
to be untangled into a general solution.
以至于无法得到一个通解
But what does it actually look like for objects in our universe
但是宇宙中的天体到底是如何
to move according to analytically unsolvable equations of motion?
遵循这些无解的运动方程运行的呢?
A system of three stars — like Alpha Centauri —
像半人马座α星系这样三颗恒星组成的系统
could come crashing into one another or, more likely,
可能会彼此相撞 或者更可能
some might get flung out of orbit
在维持长时间的表面稳定后
after a long time of apparent stability.
某颗恒星会被甩出轨道
Other than a few highly improbable stable configurations,
除了少数极不可能的稳定配置外
almost every possible case is unpredictable
长期看 几乎每一种可能情形
on long timescales.
都是不可预测的
Each has an astronomically large range of potential outcomes,
位置和速度的最微小差异会导致
dependent on the tiniest of differences in position and velocity.
每个天体的潜在结果出现巨大变化
This behaviour is known as “chaotic” by physicists,
这种行为被物理学家称为“混沌”
and is an important characteristic of n-body systems.
它也是多体系统的一个重要特征
Such a system is still deterministic,
这个系统依然有确定性
meaning there’s nothing random about it.
意味着它不具备随机性
If multiple systems start from the exact same conditions,
如果多个系统的初始条件完全相同
they’ll always reach the same result.
它们总会得到相同结果
But give one a little shove at the start,
但如果在开头给它一个微小扰动
and all bets are off.
一切都不成立了
That’s clearly relevant for human space missions,
这与人类的太空使命确实息息相关
when complicated orbits need to be calculated with great precision.
因为需要非常精确的计算才能得到复杂轨道
Thankfully, continuous advancements in computer simulations
喜人的是 计算机模拟技术的不断进步
offer a number of ways to avoid catastrophe.
提供了无数方法避免灾难
By approximating the solutions with increasingly powerful processors,
利用日渐强大的处理器估算解决办法
we can more confidently predict the motion of n-body systems
我们可以更自信地预测
on long time-scales.
多体系统的长期运行轨迹
And if one body in a group of three is so light
如果三体系统中的某一个天体质量很小
it exerts no significant force on the other two,
对另两个天体的作用力微乎其微
the system behaves, with very good approximation, as a two-body system.
这个系统的运行就非常近似二体系统
This approach is known as “the restricted three-body problem”.
这种情形被称为“限制性三体问题”
It proves extremely useful in describing, for example,
它在描述天体运动上很有用 比如
an asteroid in the Earth-Sun gravitational field,
地球-太阳引力场中的一颗小行星
or a small planet in the field of a black hole and a star.
或者黑洞-恒星引力场中的一颗小行星
As for our solar system, you’ll be happy to hear
关于太阳系 喜闻乐见的是
that we can have reasonable confidence in its stability
我们有充足的自信认为
for at least the next several hundred million years.
它能保持接下来至少几亿年的稳定
Though if another star,
尽管如此
launched from across the galaxy, is on its way to us,
如果有一颗恒星正穿越银河系 朝我们飞来
all bets are off.
一切都不成立了

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从牛顿开始,人们对三体问题的探究及应用

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视频来源

https://www.youtube.com/watch?v=D89ngRr4uZg

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