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微分方程概观

Overview of differential equations | Chapter 1

Taking the quote from Steven Strogatz,
引用史蒂芬·斯特罗加兹的一句话:
“Since Newton, mankind has come to realize that the law of physics
“自牛顿时代 人类就开始意识到物理定律
are always expressed in the language of differential equations.”
总是可以用微分方程的语言表达”
Of course, this language is spoken
当然这种语言的使用
well beyond the boundaries of physics as well,
远远超出了物理学的范畴
and being able to speak it and read it
学会运用微分方程
adds a new color to how you view the world around you.
能使让你从一个新的角度认识世界
In the next few videos,
在接下来的视频中
I want to give a sort of tour of this topic.
我会介绍一下微分方程这个主题
The aim is to give a big picture view of what this piece of math is all about,
目的是给出这部分数学知识的总览
while at the same time, being happy
同时 我也乐于
to dig into the details of specific examples
在碰到某些具体例子时 对其细节
as they come along.
进行深入分析
I’ll be assuming you know the basics of calculus,
我就假设你们都了解微积分的基础知识
like what derivatives and integrals are,
比如导数和积分
and in later videos, we’ll need some basic linear algebra,
在后面的视频中 我们还要用到一些基本的线性代数
but not much beyond that.
但也不会比这难多少
Differential equations arise
微分方程常在
whenever it’s easier to describe change than absolute amounts.
变化量比绝对数量更易描述的场景下使用
It’s easier to say why population sizes, for example, grow or shrink,
例如 描述为什么种群数量增加或减少
than it is to describe why the have the particular values
比描述为什么在某一时刻有某一特定值
they do at some point of time.
更加容易
It may be easier to describe why your love for someone is changing,
描述为什么你对某人的爱在变化
than why it happens to be where it is now.
比描述你们为什么会变成现在这样更容易
In physics, more specifically, Newtonian mechanics,
在物理学 更准确的说 在牛顿经典力学中
motion is often described in terms of force,
运动通常用力来描述
and force determines acceleration, which is a statement about change.
而力决定了代表状态变化的加速度
These equations come in two different flavors:
微分方程有两种不同类型:
Ordinary differential equations, or ODEs,
常微分方程 即ODE
involving functions with a single input, often thought of as time,
这类方程只有一个自变量 通常为时间
and partial differential equations, or PDEs,
以及偏微分方程 即PDE
dealing with functions that have multiple inputs.
它有多个自变量
Partial differential equations are something
我们将在下一个视频中
we’ll be looking at more closely in the next video.
重点介绍偏微分方程
You often think of them as involving a whole
你可以把偏微分方程想成
continuum of values changing with time,
一组随着时间连续变化的值
like the temperature at every point of a solid body,
比如固体物质在每一点的温度
or the velocity of a fluid at every point in space.
或者是流体在空间中每一点的速度
Ordinary differential equations, our focus for now,
现在 我们把重点放在常微分方程上
involve only a finite collection of values changing with time,
它只涉及一个随时间变化的有限值集合
and it doesn’t have to be time, per se,
事实上它的自变量不一定是时间
your one independent variable could be something else,
也可能是其他变量
but things changing with time
但描述事物随着时间改变
are the prototypical and most common examples of differential equations.
是微分方程最典型和最常见的例子
Physics offers a nice playground for us here,
物理学为我们学习微分方程提供了一个很好的背景
with simple examples to start with,
从一些简单的例子开始
and no shortage of intricacy and nuance as we delve deeper.
而随着研究的深入 也不乏复杂的问题和细微的差别
As a nice warm up,
作为热身
consider the trajectory of something you throw in the air.
先想象一下抛向空中的物体划过的轨迹
The force of gravity near the surface of earth,
地球表面的重力
causes things to accelerate downward at 9.8 m/s per second.
使物体每秒向下增加约9.8m/s的速度
Now unpack what that’s really saying,
现在分析一下刚刚说的场景
it means if you look at that object free from other forces,
这意味着 如果在物体不受其他外力作用下观察它
and record its velocity at every second,
并且记录它在每一秒的速度
these velocity vectors will accrue an additional downward component
这些速度矢量将以每秒9.8米的速度
of 9.8 m/s every second.
产生额外的向下分量
We call this constant 9.8 “g” for gravity.
我们用g=9.8表示重力加速度
This is enough to give us an example of a differential equation,
虽然这很简单 但足以作为
albeit a relatively simple one.
微分方程的一个例子了
Focus on the y-coordinate, as a function of time.
把y坐标看作时间的函数
Its derivative gives the vertical component of velocity,
它的导数就是速度的垂直分量
whose derivative in turn gives the vertical component of acceleration.
而速度的导数又是加速度的垂直分量
For compactness, let’s write that first derivative as y dot,
为描述方便 我们把一阶导记为y’
and that second derivative as y-double-dot.
把二阶导记为y”
Our equation says that y-double-dot is equal to -g, a simple constant.
那我们的方程就是y”=-g 即一个简单的常数
This is one where we can solve by integrating,
这个式子可以通过积分来解决
which is essentially working the question backwards.
积分本质上是一种逆运算
First, to find velocity, you ask, what function has -g as a derivative?
想求出速度 首先要找到哪个方程的导数是-g
Well, it’s -g*t.
没错 是-g×t
Or more specifically, -g*t plus the initial velocity.
或者更确切一点 -g×t+v_0 (初始速度)
Notice there’s many functions with this particular derivative
注意 有很多方程的导数都是-g
so you have an extra degree of freedom
所以你有一个由初始状态决定的
which is determined by an initial condition.
额外自由度
Now what function has this as a derivative?
现在 哪个方程的导数是-g×t+v_0呢?
Well, it turns out to be
由积分可得 该原函数为
-(½)gt^2 plus that initial velocity times t.
-(1/2)gt^2+v_0×t
And again, we’re free to add an additional constant
同样 在导数不变的情况下
without changing the derivative,
我们可以自由添加一个额外的常数
and that constant is determined by whatever the initial position is.
而这个常数是由初始状态决定的
And there you go, we’ve just solved a differential equation,
你看 我们刚刚解出了一个微分方程
figuring out what a function is,
根据函数变化率的信息
based on the information about its rate of change.
解出原函数
Things get more interesting when the forces acting on a body
当物体的位置决定作用在物体上的力时
depend on where that body is.
事情变得更加有趣起来
For example, studying the motion of planets, stars and moons,
比方说 研究行星 恒星和卫星的运动时
gravity can no longer be considered a constant.
引力将不再是一个常量
Given two bodies, the pull on one of them
假定有两个物体 作用在其中一个物体上的力
is in the direction of the other,
指向另一个物体
with a strength inversely proportional to the square of the distance between them.
力的大小与它们之间距离的平方成反比
As always the rate of change of position is velocity,
跟之前一样 位移的变化率是速度
but now the rate of change of velocity, acceleration
但是现在 速度的变化率 也就是加速度
is some function of position.
是位置的函数
So you have this dance between two mutually interacting variables,
所以你得到了两个相互作用的变量
reminiscent of the dance between the two moving bodies which they describe.
表现为他们所描述的两个运动物体之间的相互作用
This is reflective of the fact that often in differential equations,
这反映了一个事实 在微分方程中
the puzzles you face involve finding a function
你所面对的问题是找到一个函数
whose derivative and or higher-order derivatives
其导数 以及高阶导数
are defined in terms of the function itself.
是由函数本身定义的
In physics, it’s most common to work with second order differential equations,
在物理学中 最常用的是二阶微分方程
which means the highest derivative you find in this expression is a second derivative.
这意味着表达式中的最高阶导数是二阶导数
Higher order differential equations,
更高阶微分方程
would be ones involving third derivatives, fourth derivatives, and so on,
将涉及三阶导数 四阶导数等等
puzzles with more intricate clues.
解的过程也更加复杂
The sensation you get when really meditating on one of these equations,
当你真正思考其中某个方程式时 你会感到
is one of solving an infinite continuous jigsaw puzzles.
是在解决一个无穷且连续的拼图游戏
In a sense you have to find infinitely many numbers,
某种意义上来说 你必须找到无穷多的数值
one for each point in time t.
每个数都对应一个时间t
But they are constrained by a very specific way
但这些数受一个具体条件的约束
that these values intertwine with their own rate of change,
即它们的值与自身的变化率
and the rate of change of that rate of change.
以及变化率的变化率相互交织
To get a feel for what studying these can look like,
为了让你知道研究微分方程是什么感受
I want you to take some time digging into a deceptively simple example:
我希望你花时间研究一个看似简单的例子:
A pendulum.
一个单摆
How did this angle theta that it makes with the vertical
它与竖直方向所形成的夹角θ
change as a function of time.
是如何随时间变化的
This is often given as an example in introductory physics classes of harmonic motion,
这个例子常用在物理简谐运动的入门课中
meaning it oscillates like a sine wave.
θ随着摆动的变化图像就像一个正弦波
More specifically, one with a period of 2π*√L/g,
更确切的说 一个周期是2π×√L/g的简谐运动
where L is the length of the pendulum, and g is strength of the gravity.
其中L是单摆长度 g是重力加速度
However, these formulas are actually lies,
然而这些公式实际上是谎言
or rather, approximations which only work in the realm of small angles.
或者说只有在角度较小的情况下会产生近似结果
If you would to go and measure an actual pendulum,
如果你用一个真正的单摆做实验
what you’d find is that as you pull it out farther,
你会发现 当你把它拉远时
the period is longer
它的摆动周期
than what that high-school physics formulas would suggest.
会比高中物理公式得出的要长
And when you pull it out really far,
当你把它拉得足够远时
this value of theta vs time,
θ值随时间的变化曲线
doesn’t even look like a sine wave anymore.
看起来甚至不再像是正弦波了
To understand what’s really going on,
为了明白发生了什么
first things first, let’s set up the differential equation.
首先 让我们建立微分方程
We’ll measure the position of the pendulum’s wave
我们将单摆摆动所对应的弧长x
as a distance x along this arc,
作为测量时单摆的位置
and if the angle theta we care about is measured in radians,
如果用弧度表示我们所关心的θ
we can write x as L*theta,
则可将x写为L×θ
where L is the length of the pendulum.
其中L是摆线的长度
As usual, gravity pulls down with acceleration of g,
通常来说 重力产生向下的加速度g
but because the pendulum constrains the motion of this mass,
但是因为单摆限制了这个物体的运动
we have to look at the component of this acceleration,
我们不得不考虑加速度
in the direction of motion.
在运动方向上的分量
A little geometry exercise for you is to show that this little angle here
运用一点几何知识 你会发现这里的角度
is the same as theta
就等于θ
So the component of gravity in the direction of motion,
所以重力加速度在运动方向上的分量
opposite this angle, will be
在这个角的对面 那就是
-g*sin(theta).
-g×sinθ
Here we’re considering theta to be positive
这里我们认为θ
when the pendulum is swung to the right,
在单摆摆向右边时是正数
and negative when it’s swung to the left,
在单摆摆向左边时是负数
and this minus sign in the acceleration indicates that
这个在加速度前的负号意味着
it’s always pointed in the opposite direction from displacement.
它总是跟位移的方向相反
So what we have is the second derivative of x,
所以我们知道了x的二阶导数
the acceleration, is -g * sin(theta).
即加速度 是-g×sinθ
As always, it’s nice to do a quick gut check
像往常一样 最好快速检查一下
that our formula makes physical sense.
函数在物理上是否有意义
When theta is zero, sin0 is zero,
当θ为0时 sin0等于0
so there is no acceleration in the direction of the movement.
所以在运动方向上没有加速度
When theta is 90 degrees,
当θ为90°时
sin(theta) is one,
sinθ等于1
so the acceleration is the same as what it would be for free fall.
此时的加速度和自由落体的加速度是一样的
Alright, that checks out.
没错 函数没问题
And because x is L*Θ,
并且因为x是L×θ
that means the second derivatives of theta
那就意味着θ的二阶导数
is -g over L times sin(theta).
就是 -g×sinθ/L
To be a little more realistic,
为了更实际一点
let’s add in a term to account for the air resistance,
我们在函数中加一项以表示空气阻力
which maybe we model as being proportional to the velocity.
假设空气阻力与速度成正比
We’ll write this as -μ times theta-dot,
我们将其写为-μ×θ’
where μ is some constant that
这里的μ是一个常数
encapsulates all the air resistances and frictions
代表了所有的空气阻力和摩擦等
and such that determines how quickly the pendulum loses it’s energy.
它决定了单摆损失能量的速度
Now this, my friends, is a particularly juicy differential equation.
朋友们 这个是一个特别有趣的微分方程
It’s not easy to solve,
它很难求解
but is’s not so hard that
但也没有困难到
we can’t reasonably get some meaningful understanding out of it.
我们无法从中得到一些有意义的推论
At first glance you might think of the sine function you see here,
乍一看 你可能会想到这里的正弦函数
relates to the sine wave pattern for the pendulum.
与单摆正弦波形式的摆动有关
Ironically though, what you’ll eventually find is that the opposite is true.
但戏剧性的是 你最终会发现事实正好相反
The presence of the sine in this equation
在这个方程中出现的正弦函数
is precisely why real pendulum
恰好解释了为什么现实中的单摆
don’t oscillate with the sine wave pattern.
不以正弦波的形式摆动
If that sounds odd, consider the fact that here,
如果听起来很奇怪 考虑一下这件事
the sine function takes theta as an input,
这里的正弦函数把θ当做一个输入值
but in the approximate solution you might see it in the physics class,
但根据你在物理课上可能见到的近似解
theta itself is oscillating as the output of the sine function.
θ本身作为正弦函数的输出而变化
Clearly something fishy is afoot.
很明显 某种可疑的事情正在发生
One thing I like about this example is that
我喜欢这个例子的一个原因在于
even though it’s comparatively simple,
即使它比较简单
it exposes an important truth about differential equations
但它仍揭示了微分方程式中一个
that you need to be grapple with:
你需要面对的重要事实
They’re really freaking hard to solve.
它们真的太难求解了
In this case, if we remove that damping term
在这个例子里 假如我们去除了阻尼项
we can just barely write down an analytic solution,
也只能勉强写下一个解析解法
but it’s hilariously complicated,
但是它非常复杂
it involves all these functions you’re probably never heard of,
包括了一些你可能从未听过的函数方程
written in terms of integrals and weird inverse integral problems.
方程里有积分术语和奇怪的逆积分问题
And when you step back,
当你退一步想
presumably the reason for finding a solution
也许 寻找解决方案的原因
is to then be able to make computations,
是为了能够计算
and to build an understanding for whatever dynamics you’re studying.
并理解你所研究的动力学
In this case, those questions have just been punted off
在这种情况下 为了知道如何计算求解
to figuring out how to compute
以及更重要的 理解这些新函数
and more importantly understand these new functions.
我们先将那些问题搁置在一边
And more often, like if we add back in that dampening term,
而常见情况下 如果我们把阻尼项加回去
there is not a known way to write down an exact analytical solution.
没有已知的方法能够写出一个准确的解析解
Well I mean for any hard problem,
甚至对于任何一个困难的问题
you could just define a new function to be the answer to that problem,
你都可以定义一个新函数作为它的解
heck, even name it after yourself if you want.
如果你想 你甚至可以用自己的名字来命名
But again, that’s pointless
但它毫无意义
unless it leads you to being able to make computes
除非它能够让你计算出具体解
and to build understanding.
并能帮助理解这个方程
So instead, in the study of differential equations,
所以 在学习微分方程式时
we often do a sort of short-circuit,
我们经常走一些捷径
and skip the actual solution part since it’s unattainable
跳过计算求解部分 而直接理解方程并计算
and go straight to build understanding and make computations from the equations alone.
因为我们无法解得具体的答案
Let me walk through what that might look like with the pendulum.
让我们通过单摆问题研究一下它可能的样子
What do you hold in your head,
当你在理解这些
or what visualization can you get some software to pull up for you,
不同初始状态的单摆在一些规则的限制下
to understand the many possible ways
产生的运动时
that a pendulum governed by these laws
脑海中会想到什么
might evolve depending on its starting conditions?
或者会用软件来绘制怎样的可视图形?
You might be tempted to try imagining the graph of theta vs t,
你可能试着去想象θ和t的图像
and somehow interpreting how the slope, the position and the curvature
并且尝试去解释为什么斜率 位置和曲率之间
all interrelated with each other.
存在相关性
However, what will turn out to be both easier and more general
然而 更加简单并且常规的方法
is to start by visualizing all possible states
是在二维平面上把所有可能的状态
in the two-dimensional plane.
可视化表现出来
What means by the state of the pendulum
你可以用两个数值来描述
is that you can describe it with two numbers:
单摆的状态
the angle, and the angular velocity.
即角度和角速度
You can freely change either one of these two values
你可以在不影响另一个值的情况下
without necessarily changing the other,
随意改动其中一个值
but the acceleration is purely a function of these two values.
但是加速度是只关于这两个数值的函数
So each point of this two-dimensional plane
所以这个二维平面上的每个点
fully describes the pendulum at any given moment.
都能代表单摆在任何一个时刻的状态
You might think of these as
你也许会把这些看作是
all possible initial conditions of that pendulum.
单摆的所有可能的初始状态
If you know the initial angle and the initial angular velocity,
如果你知道初始的角度和角速度的话
that’s enough to predict
那就足以预知
how the system will evolve as time moves forward.
随着时间推移系统将如何变化
If you haven’t worked with them before,
如果你以前没有接触过这种方法
these sorts of diagrams can take a little getting used to.
可能需要一点时间来适应
What you’re looking at now, this invert spiral,
现在你看到的这个反螺线
it’s a fairly typical trajectory for our pendulum.
是一个相当典型的单摆运动轨迹
So take a moment to think carefully about what is being represented.
所以请花点时间仔细想想这代表了什么
Notice how at the start, as theta decreases,
注意到从一开始 随着θ角度的减小
theta-dot, the y-coordinate gets more negative, which make sense
θ’即y轴上的值 随着θ的变化减小的更快 这很正常
because the pendulum moves faster in the leftward direction
因为单摆在接近最低点时
as it approaches the bottom.
会向左作加速运动
Keep in mind, even though the velocity vector on this pendulum is pointing to the left,
记住 虽然单摆的速度矢量方向向左
the value of that velocity is always being represented
但是速度的值 在我们的二维空间中
by the vertical component of our space.
由速度的垂直分量表示
It’s important to remind yourself that
记得提醒自己
this state space is an abstract thing,
这种状态空间是抽象的
and is distinct from the physical space
和单摆实际存在和运动的
where the pendulum itself lives and moves.
物理空间并不同
Since we’re modeling this as losing some of its energy to air resistance,
因为我们建立的模型是单摆受空气阻力影响损失能量
this trajectory spirals inward,
它的运动轨迹向内部旋转
meaning the peak velocity and peak displacement
意味着速度和位移的峰值
each go down a bit with each swing.
都会在每次摆动后减少一点
Our point is, in a sense, attracted to the origin
某种意义上说 我们的点被原点处
where theta and theta-dot both equal 0.
即θ和θ’的值都等于0的点所吸引
With this space, we can visualize a differential equation as a vector field.
在状态空间中 我们可以把微分方程可视化为矢量场
Here, let me show you what I mean.
这里我需要解释一下
The pendulum state is a vector, (theta, theta-dot),
单摆的状态是一个矢量(θ, θ’)
Maybe you think of that as an arrow from the origin,
你可能会认为它是一个从原点出发的箭头
or maybe you think of that as a point.
或者是一个点
what matters is that it has two coordinates, each a function of time.
重点是 它的两个坐标都是时间的函数
Taking the derivative of that vector gives you its rate of change,
而这个矢量的导数 就是它在图表中运动时
in direction and speed that it will tend to move in this diagram.
在方向和速度上的变化率
That derivative is a new vector
那个导数是一个新的矢量
(theta-dot, theta-double-dot),
坐标是(θ’, θ”)
which we visualize as being attached to the relevant point in this space.
我们可以把它放在空间中对应的点上可视化出来
Now take a moment to interpret what this is saying,
在这里花一秒钟想一下这意味着什么
The first component for this rate-of-change vector is theta-dot,
这个变化率矢量的第一个分量是θ’
which is also a coordinate in our space,
也是空间中的一个坐标
so the higher up we are in the diagram,
所以这个点在图象中的位置越高
the more the point tends to move to the right,
它越倾向于向右移动
and the lower we are,
而点的位置越低
the more it tends to move to the left.
它越倾向于向左运动
The vertical component is theta-double-dot,
垂直方向的矢量分量是θ”
which are differential equation,
它是一个微分方程
let’s us rewrite entirely in terms of theta and theta-dot itself.
我们可以只用θ和θ’重新书写这个方程
In other words, the first derivative of our state vector
换句话说 状态空间中矢量的一阶导数
is some function of that vector itself,
是矢量本身的函数
with most of the intricacy tied up in that second coordinate.
它与坐标的第二项有复杂的关系
Doing the same at all points of the space
对空间中的每一点做同样的操作
will show how that state tends to change from any position.
我们就能看出任意一点状态的变化趋势
As it’s typical with vector fields,
在这个典型的向量场中
we artificially scaling down the vectors when we draw them to prevent clutter,
为防止看起来杂乱 我们会刻意把所绘向量缩小
but using color to loosely indicate magnitude.
并用不同的颜色表示其大小
Notice, we’ve effectively broken up a single second order equation,
注意 我们实际上把一个二阶微分方程分解成了
into a system of two first order equations.
两个一阶微分方程的组合
You might even give theta-dot a different name
你甚至可以给θ’换个名字
to emphasize that we’re thinking of two separate values,
来强调这是两个不同的值
intertwined via this mutual effect they have
它们通过彼此变化率的相互作用
on one and other’s rate of change.
纠缠在一起
This is a common trick in the study of differential equations,
这是学习微分方程的常用技巧
instead of thinking about higher order changes of a single value,
与其研究某一个值的高阶导数
we often prefer to think of the first derivative of vector values.
我们通常更喜欢研究矢量值的一阶导数
In this form, we have a wonderful visual way
在这种形式下 我们通过一种完美的可视化方法
to think about what solving this equation means.
来思考这个方程的解意味着什么
As our system evolves from some initial state,
由于我们的系统从某个初始状态开始发展
our point in this space will move along some trajectory
空间中的点会沿着轨道运动
in such way that at every moment, the velocity of that point
该点每一刻的速度
matches the vector from this field.
都与场内向量吻合
And again, keep in mind,
再强调一遍
this velocity is not the same thing
这里的速度跟
as the physical velocity of the pendulum.
单摆的物理速度不是一个概念
It’s a more abstract rate of change,
它是代表了θ和θ’变化率的
encoding the rates of change for both theta and theta-dot.
更加抽象的变化率
You might find it’s fun to pause for a moment
你可以先暂停一下视频
and think through what exactly some of these trajectory lines say
研究一下这些代表着不同初始状态下
about the possible ways the pendulum evolves from different starting conditions.
单摆的运行轨迹 究竟意味着什么
For example, in regions where theta-dot is quite high,
例如 当θ’数值比较大时
the vectors guide the point to travel to the right quite a ways
对应的点在进入稳定内螺旋运动前 沿着所处位置的向量
before settling down into an inward spiral.
向右运动了一段不小的距离
This corresponds to a pendulum with a high enough initial velocity,
这正对应了速度足够大时单摆的运动情况
that it fully rotates around several times
在它左右摆动之前
before settling into a decaying back and forth.
会完整地旋转几圈
Having a little more fun,
更有趣的是
when I tweak this air resistance term μ
当我们把空气阻力项μ
say increasing it.
增大一些
you can immediately see how this will result in trajectories
你马上就能发现它对轨迹的影响
that spiral inward faster,
更多的迹线进入内螺旋了
which is to say the pendulum slows down faster.
这也意味着单摆减速更快了
That’s obvious why I call it the air resistant term,
这就是为什么我称它为空气阻力项
but imagine that you saw the equations out of context,
但想象你在脱离情境时看到这个公式
not knowing that they describe a pendulum,
你不知道它描述了一个单摆运动
it’s not obvious just looking at them
只看公式的话 并不容易发现
that increasing this value of μ
增加μ的值
means the system has a whole
会导致系统整体更快地
tends towards some attracting state faster.
趋向于某个稳定状态
So getting some software to draw these vector fields for you,
所以 用一些软件画出矢量场
can be a great way to build an intuition for how they behave.
是一种让物体运动更直观的好方法
What’s wonderful is that
奇妙的是
any system of ordinary differential equations
任何常微分方程组
can be described by a vector field like this,
都能用这样一个矢量场来描述
so it’s a very general way to get a feel for them.
所以这是理解它们的一种常见方法
Usually, though, they have many more dimensions.
然而通常情况下 它们有更多维度
For example, consider the famous three-body problem,
举个例子 想一想著名的三体问题
which is to predict how three masses
它研究的是 如果三个物体
in 3-dimensional space evolve if they act on each with gravity,
已知初始位置和速度 且仅受互相的引力作用
and if you know their initial positions and velocities.
它们将在三维空间做怎样的运动
Each mass has three coordinates describing its position,
每个物体都有三个坐标描述它们的位置
and three more describing its momentum,
以及三个坐标描述它们的动量
so the system has 18 degrees of freedom in total,
所以系统中一共有18个自由度
and hence an 18-dimensional space of possible states.
因此可能的状态构成了18维空间
It’s a bizarre thought, isn’t it?
简直不可思议 不是吗?
A single point meandering through an 18-dimensional space
一个单一的点在一个无法具象化的
that we can not visualize,
18维空间中游荡
obediently taking steps through time
它基于前一时刻所在向量
based on whatever vector it happens to be sitting on from moment to moment,
一步一步地随着时间移动
completely encoding the positions and momentums
完全决定了三个物体在
of the three masses we see in ordinary, physical, 3-d space.
原始物理三维空间中的位置和动量
In practice, by the way, you can reduce the number of dimension here
顺便一提 在实际应用中 你可以利用建立起的
by taking advantage of the symmetries of your set up,
对称性来减少维度的数量
but the point that more degrees of freedom results
但重点是更高的自由度仍然会
in higher-dimensional state basis remains the same.
导致更高维的状态空间
In math, we often call a space like this a “phase space”.
在数学中 我们通常把这个空间称为“相空间”
You’ll hear me use that term broadly
你会经常听见我用这个术语
for spaces encoding all kinds of states of changing systems.
来指代描述变化的系统所有状态的空间
But you should know that in the context of physics,
但你应该知道 在物理学背景下
especially Hamiltonian mechanics,
尤其是哈密顿力学中
the term is often reserved for a more special case,
这个术语通常用于描述一个更特殊的例子
namely a space whose axes represent position and momentum.
即一个以位置和动量为坐标的空间
So a physicist would agree that the 18-dimensional space
所以物理学家可能认为 这个描述三体问题的
describing the 3-body problem is a phrase space.
18维空间 是一个相空间
But they might ask that we make a couple of modifications to our pendulum set up
但是 他们也许会要求我们调整一下单摆设定
for to properly deserve the term.
以便恰当地配置这个术语
For those viewers who just watched the block collision videos,
对于那些看过我们滑块碰撞视频的观众
the planes we worked with there
那个视频中我们所用的平面
would happily be called phase spaces by math folk,
数学家很乐意称其为相空间
a physicist might prefer other terminology.
而一个物理学家可能更倾向于其他术语
Just know that the specific meaning may depend on your context.
只要知道 一个词在不同的语境中会有不同的意义
It may seem like a simple idea,
这可能看上去很简单
depending on how well indoctrinated you are to modern ways of thinking about math,
只要你对现代数学的思维方式足够熟悉
but it’s worth keeping in mind that
但值得牢记的是
it took humanity quite a while to really embrace thinking of dynamics
人类花了相当多的时间 才找到这种
specially like this,
用相空间描述动力学的方法
especially when the dimensionals get very large.
尤其是在规模非常大的时候
In his book Chaos,
James Gleick的书《混沌》中
the author James Gleick describes phase space as
作者将相空间描述为
“one of the most powerful inventions of modern science.”
“现代科学最伟大的发明之一”
One reason it’s powerful is
其中一个原因是
that you can ask questions not just about a single initial condition,
你能从中得到的不只是一个单一的初始条件
but about a whole spectrum of initial states.
而是整个系统初始状态的集合
The collection of all possible trajectories
所有可能的轨迹构成的集合
is reminiscent of a moving fluid,
就像是流动的流体
so we call it phase flow.
所以我们称之为相流
To take one example of
举一个例子来说明
why phase flow is a fruitful idea,
为什么相流是一个富有成效的概念
consider the question of stability,
细想一下稳定性的问题
the origin of our space
空间的原点
corresponds to the pendulum standing still,
对应于静止状态的单摆
and so does this point over here,
而这个点也是
represently when the pendulum is perfectly balanced upright.
代表单摆在正上方也能保持平衡
These are the so-called fixed points of our system,
这些就是所谓的系统不动点
and one natural question to ask is
一个很自然的问题产生了
whether or not they are stable.
这些不动点是否稳定
That is, will tiny nudges to the system
当系统中出现一个微小的扰动
result in a state that tends back towards that fixed point
它是否会趋向于回到这个不动点
or away from it.
还是远离它
Physical intuition for the pendulum makes the answer here kind of obvious,
利用对单摆的物理直觉 答案很明显
but how would you think about stability just looking at the equations
但是如果换成无法用直觉判断的情景
say if they arose in some completely different less intuitive context?
你要如何只通过公式就看出稳定性呢?
We’ll go over how to compute the answer to questions like this in following videos,
我们将会在接下来的视频中谈到如何计算这样的问题
and the intuition for the relevant computations are guided heavily
这些计算背后的直觉思路很大一部分在于
by the thought of looking at small regions in space around the fixed point
取不动点附近的一小块区域
and asking whether the flow tends to contract or expand.
观察流动趋向于紧缩还是扩张
And speaking of attraction and stability,
说到吸引和稳定性
let’s take a brief sidestep to talk about love.
不妨顺便岔开话题谈一谈爱情
The Strogatz quote that I mentioned earlier comes from a whimsical column
我之前提到的斯特罗加兹在纽约时报
in the New York Times on mathematics of modeling affection,
发表了一个关于爱情数学模型的古怪专栏
an example well worth pilfering to illustrate that
这是一个很好的例子
we’re not just talking about physics here.
说明了我们不是只能在这儿讨论物理
Imagine you’ve been flirting with someone,
想象一下 你和某人正在暧昧阶段
but there’s been some frustrating inconsistency to how mutual the affections seems,
但是你们之间的感情却令人沮丧的多变
and perhaps during a moment when you turn your attention towards physics
也许过段时间 当你希望摆脱这种浪漫的困扰
to keep your mind off this romantic turmoil.
而把注意力转向物理时
Mulling over the broken up pendulums,
你思考着那些破碎的单摆公式
you suddenly understand the on again off again dynamics of your flirtation.
突然你就明白了你们之间跌宕起伏的的关系是怎么回事了
You’ve noticed that your own affections tend to increase
你注意到 当对方对你有兴趣时
when your companion seems interested in you,
你的爱意会增加
but decrease when they seem colder.
但如果对方变得冷淡 你的爱意就会减少
That is, the rate of change for your love
即 你好感的变化速率
is proportional to their feelings for you.
正比于 对方对你的好感
But the sweetheart of yours is precise the opposite:
但是你的心肝宝贝正好与你相反
Strangely attracted to you when you seem uninterested,
你表现的越冷淡 她越喜欢
but turned off once you seem too keen.
但当你变得热情 她却变得冷淡
The phase space for these equations
这些公式构成的相空间
looks very similar to the center part of your pendulum diagram.
看起来跟你那单摆图的中心部分很相似
The two of you will go back and forth
你们两个会陷入一个无止境的循环
between affection and repulsion in an endless cycle.
在喜欢和厌恶之间不断徘徊
A metaphor of pendulum swings in your feelings would not just be apt,
用单摆的摆动来比喻你们的感情不仅很恰当
but mathematically verified.
而且已经被数学证明了
In fact, if your partner’s feelings were further slowed
实际上 如果你的对象发现自己爱的太深
when they feel themselves too in love,
就会变得更谨慎
let’s say out of fear of being made vulnerable,
可能是出于害怕自己变得脆弱
we’d have a term matching the friction in the pendulum,
这一项就对应着单摆的摩擦阻力
and you two would be destined to an inward spiral towards mutual ambivalence.
而且你们俩注定会被卷入这个情感漩涡
I hear wedding bells already.
我仿佛已经听到婚礼的钟声了
The point is that two very different seeming laws of dynamics,
关键是这两件看上去截然不同的事
one from physics involving a single variable,
一件是由单变量引起的物理现象
and another from chemistry, with two variables,
而另一件是由两个变量引起的化学反应
actually have a very similar structure,
实际上有着非常相似的结构
easier to recognize when you are looking at the phase diagram.
当你看他们的相图就能很容易地看出来
Most notably, even though the equations are different,
最重要的是 即使公式不同
for example there’s no sin function in the romance equations,
比如说爱情方程中并没有正弦函数
the phase space exposes an underlying similarity nevertheless.
而相空间能够揭示背后类似的本质
In other words, you’re not just studying a pendulum right now,
换句话说 你现在不仅仅在研究一个单摆
the tactics you developed to study one case
你还能学到一种从一件事
have a tendency to transfer to many others.
推广到其他很多事情的能力
Okay, so phase diagrams are a nice way to build understanding,
相空间确实是一个帮助理解的好办法
but what about actually computing the answer to our equation?
但是要怎么计算出我们方程的答案
Well, one way to do this is to essentially simulate what the universe would do,
一种方法是从本质上模拟宇宙的演化
but using finite time steps instead of the infinitesimals and limits defining calculus.
但要用有限的时间步长来定义微积分 而不是无穷小和极限
The basic idea is that if you’re at some point in this phase diagram,
基本思路是 假设你在这个相图中的某一点
take a step based on the vector that you’re sitting on
基于你所处位置的向量方向
for some small time step, delta-t,
迈出一小步 时长为Δt
specifically take a step equals to delta-t times that vector.
准确地说 步长等于Δt乘以这个向量
As a reminder, in drawing this vector field,
注意在画这个向量场时
the magnitude of each vector has been artificially scaled down to prevent clutter.
为避免凌乱 每个向量的长度已经刻意按比例缩小了
When you do this repeatedly,
当你重复这个步骤时
your final location will be an approximation of theta(t),
你最终所处的位置大概在θ(t)
where t is the sum of all of those time steps.
这里的t就是走过的所有时间之和
If you think about what’s being shown right now, though,
但如果你仔细思考一下现在显示的图像
and what that would imply for the pendulum’s movement,
以及实际的单摆运动
you’d probably agree it’s grossly inaccurate.
你可能会觉得这个图太不准确了
But that’s just because the timestep delta-t of 0.5 is way too big.
但这只是因为给时间步长Δt取值0.5实在是太大了
If we turn it down, say to 0.01,
如果我们把它缩小到0.01
you can get a much more accurate approximation,
你会得到一个精确得多的近似图
it just takes many more repeated steps is all.
只需要更多次步骤
In this case, computing theta(10)
对于这个例子 计算θ(10)
requires one thousand little steps.
需要重复1000步
Luckily, we live in a world with computers,
幸运的是 我们生活在一个电脑无处不在的世界
so repeating a simple task 1,000 times
所以重复一个简单的任务1000次
is as simple as articulating that task with a programming language.
只需要用编程语言阐明这个任务就行
In fact, let’s finish things off by write a little python program
那么让我们写一个计算θ(t)的python程序
that computes theta(t) for us.
来结束这个视频
What it has to do is make use of the differential equation,
它要做的就是利用微分方程
which returns the second derivative of theta as a function of theta and theta-dot.
输出由θ和θ’构成的一个方程来表示θ的二阶导数
You start off by defining two variables, theta and theta-dot,
首先需要定义两个变量θ和θ’
each in terms of some initial conditions.
并分别给它们赋初值
In this case I’ll have theta start at pi/3, which is 60-degrees,
这里我把θ的初始值赋为π/3 即60°
and theta-dot start at 0.
把θ’的初始值赋为0
Next, write a loop
接着写一个循环
which corresponds to many little time steps between 0 and your time t,
把0到t的时间分成许多小步
each of size delta-t,
每个间隔为Δt
which I’m setting here to be 0.01.
这里我设置为0.01
In each step of the loop,
在循环的每一步中
increase theta by theta-dot times delta-t,
都在θ上加θ’*Δt
and increase theta-dot by theta-double-dot times delta-t,
以及在θ’上加θ”*Δt
where theta-double-dot can be computed based on the differential equation.
而θ”可以根据微分方程计算得出
After all these little steps, simple return the value of theta.
在整个循环结束后 只需要返回θ的值
This is called solving the differential equation numerically.
这就叫做微分方程的数值解法
Numerical methods can get way more sophisticated and intricate than this
数值解法有时候可能非常复杂繁琐
to better balance the tradeoff between accuracy and efficiency,
但能在准确度和效率之间达到更好的平衡
but this loop gives the basic idea.
这个循环给出了最基本的思路
So even though it sucks that we can’t always find exact solutions,
所以尽管我们总是不能获得精确解 这让人感觉很糟糕
there are still meaningful ways to study differential equations in the face of this inability.
面对这种无奈 学习微分方程仍然是很有意义的
In the following videos,
在接下来的视频中
we will look at several methods for finding exact solutions when it’s possible.
我们会探讨几种可能的获得精确解的方法
But one theme you might like to focus on is
但是你可能会注意到的一个问题是
how these exact solutions can also help us study the more general unsolvable cases.
我们该如何利用这些精确解去研究更普遍但尚未解决的问题
But it gets worse.
但是不容乐观
Just as there is a limit to how far exact analytic solutions can get us,
正如这些准确的分析解法能帮助我们是有限的
one of the great fields to have emerged in the last century, chaos theory,
上个世纪出现的一个伟大理论之一 混沌理论
has exposed that there are further limits
揭示了我们利用这些系统进行预测时
on how well we can use these systems for prediction,
还有更多的限制
with or without solutions.
不管解是否存在
Specifically, we know that for some systems,
特别地 我们知道对于某些系统而言
small variations to the initial conditions,
初始条件略有不同
say the kind due to necessarily imperfect measurements,
比如无法避免的测量误差
results in wildly different trajectories.
就会导致截然不同的运行轨迹
We’ve even built some good understanding for why this happens.
我们甚至建立了一些模型帮助理解这种现象发生的原因
The three body problem, for example, is known to have seeds of chaos within it.
例如三体问题就是混沌运动的著名例子
So looking back at that quote from earlier,
所以 回顾最开始的引言
it seems almost cruel of the universe to fill its language with riddles that we either can’t solve,
这看起来太残酷了 宇宙中充满了谜语 要么我们解不出来
or where we know that any solution would be useless for long-term prediction anyway.
要么让我们知道任何解从长远来看都是无意义的
It is cruel,
这很残酷
but then again, it should also be reassuring.
但它也会让人鼓起勇气
It gives some hope that the complexity we see in the world around us
它给了我们希望 也许我们所看到的世界的错综复杂
could be studied somewhere in the math,
也可以通过数学进行研究
and that it’s not hidden away in some mismatch between model and reality.
而且它也不会在理论和现实之间的误差之中隐匿

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视频概述

深入浅出地描绘了微分方程的世界,介绍了一种分析问题的数学工具

听录译者

收集自网络

翻译译者

钉子酷咸鱼

审核员

审核员1024

视频来源

https://www.youtube.com/watch?v=p_di4Zn4wz4

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