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Taking a quote from Steven Strogatz,
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Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations.
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Of course, this language is spoken well beyond the boundaries of physics as well,
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and being able to speak it and read it adds a new color to how you view the world around you.
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In the next few videos, I want to give a sort of tour of this topic.
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The aim is to give a big picture view of what this piece of math is all about,
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while at the same time being happy to dig into the details of specific examples as they come along.
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I'll be assuming you know the basics of calculus, like what derivatives and integrals are, and in later videos we'll need some basic linear algebra,
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but not too much beyond that.
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Differential equations arise whenever it's easier to describe change than absolute amounts.
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It's easier to say why population sizes, for example, grow or shrink,
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than it is to describe why they have the particular values they do at some point in time.
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It may be easier to describe why your love for someone
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is changing than why it happens to be where it is now.
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In physics, more specifically Newtonian mechanics, motion is often described in terms of force, and force determines acceleration, which is a statement about change.
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These equations come in two different flavors, ordinary differential equations, or ODEs, equations, involving functions with a single input, often thought of as time,
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and partial differential equations, or PDEs, dealing with functions that have multiple inputs.
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Partial differential equations are something we'll be looking at more closely in the next video.
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You often think of them as involving a whole continuum of values changing with time, like the temperature at every point of a solid body,
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or the velocity of a fluid at every point in space.
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Ordinary differential equations, our focus for now, involve only a finite collection of values changing with time.
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And it doesn't have to be time per se, your one independent variable could be something else,
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but things changing with time are the prototypical and most common example of differential equations.
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Physics offers a nice playground for us here, with simple examples to start with, and no shortage of intricacy and nuance as we delve deeper.
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As a nice warm -up, Consider the trajectory of something you throw in the air.
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The force of gravity near the surface of Earth causes things to accelerate downward at 9 .8 meters per second per second.
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Now, unpack what that's really saying.
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It means if you look at that object free from other forces and record its velocity at every second,
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these velocity vectors will accrue an additional downward component of 9 .8 meters per second every second.
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We call this constant g for gravity.
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This is enough to give us an example of a differential equation, albeit a relatively simple one.
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Focus on the y coordinate as a function of time.
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Its derivative gives the vertical component of velocity, whose derivative in turn gives the vertical component of acceleration.
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For compactness, let's write that first derivative as y dot and that second derivative as y double dot.
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Our equation says that y double dot is equal to negative g, a simple constant.
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This is one where we can solve by integrating, which is essentially working the question backwards.
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First, to find velocity, you ask, what function has negative g as a derivative?
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Well, it's negative g times t, or more specifically negative g t plus the initial velocity.
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Notice, there's many functions with this particular derivative, so you have an extra degree of freedom which is determined by an initial condition.
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Now, what function has this as a derivative?
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Well, it turns out to be negative one -half g times t squared plus that initial velocity times t,
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and again we're free to add an additional constant without changing the derivative
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and that constant is determined by whatever the initial position is.
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And there you go, we just solved a differential equation, figuring out what a function is based on information about its rate of change.
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Things get more interesting when the forces acting on a body depend on where that body is.
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For example, studying the motion of planets, stars, and moons, gravity can no longer be considered a constant.
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Given two bodies, the The pole on one of them is in the direction of the other,
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with a strength inversely proportional to the square of the distance between them.
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As always, the rate of change of position is velocity, but now the rate of change of velocity is some function of position,
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so you have this dance between two mutually interacting variables, reminiscent of the dance between the two moving bodies which they describe.
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This is reflective of the fact that often in differential equations, the puzzles you face involve finding a function whose derivative and
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or higher order derivatives are defined in terms of the function itself.
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In physics, it's most common to work with second order differential equations, which means the highest derivative you find in this expression is a second derivative.
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Higher order differential equations would be ones involving third derivatives, fourth derivatives, and so on.
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Puzzles with more intricate clues.
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The sensation you get when really meditating on one of these equations is one of solving an infinite continuous jigsaw puzzle.
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In a sense, you have to find infinitely many numbers, one for each point in time t.
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But they're constrained by a very specific way that these values intertwine with their own rate of change,
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and the rate of change of that rate of change.
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To get a feel for what studying these can look like, I want you to take some time digging into a deceptively simple example, a pendulum.
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How does this angle theta that it makes with the vertical change as a function of time?
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This is often given as an example in introductory physics classes of harmonic motion, meaning it oscillates like a sine wave.
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More specifically, one with a period of 2 pi times the square root of L over G, where where L is the length of the pendulum,
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and G is the strength of gravity.
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However, these formulas are actually lies, or rather, approximations which only work in the realm of small angles.
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If you were to measure an actual pendulum, what you'd find is that as you pull it out farther,
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the period is longer than what the high school physics formulas would suggest.
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And when you pull it out really far, this value of theta plotted versus time doesn't even look like a sine wave anymore.
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To understand what's really going on, first things first, let's set up the differential equation.
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We'll measure the position of the pendulum's weight as a distance x along this arc,
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and And if the angle theta that we care about is measured in radians, we can write x as L times theta,
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where L is the length of the pendulum.
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As usual, gravity pulls down with an acceleration of g, but because the pendulum constrains the motion of this mass,
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we have to look at the component of this acceleration in the direction of motion.
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A little geometry exercise for you is to show that this little angle here is the same as theta.
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So the component of gravity in the direction of motion, opposite this angle, will be negative g times sine of theta.
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Here we're considering theta to be positive when the pendulum is swung to the right, and negative when it's swung to the left.
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And this minus sign in the acceleration indicates that it's always pointed in the opposite direction from displacement.
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So what we have is that the second derivative of x, the acceleration, is negative g times sine of theta.
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As always, it's nice to do a quick gut check that our formula makes physical sense.
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When theta is zero, sine of zero is zero, so there's no acceleration in the direction of movement.
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When theta is 90 degrees, sine of theta is one, so the acceleration is the same as it would be for freefall.
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Alright, that checks out.
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And because x is L times theta, that means the second derivative of theta is negative g over L times sine of theta.
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To be a little more realistic, let's add in a term to account for the air resistance,
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which maybe we model as being proportional to the velocity.
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We'll write this as negative mu times theta dot, where mu is some constant that encapsulates all of the air resistance
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and friction and such that determines how quickly the pendulum loses energy.
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Now this, my friends, is a particularly juicy differential equation.
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It's not easy to solve, but it's not so hard that we can't reasonably get some meaningful understanding out of it.
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At first glance, you might think that the sine function you see here relates to the sine wave pattern for the pendulum.
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Ironically though, what you will eventually find is that the opposite is true.
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The presence of the sine in this equation is precisely why real pendulums don't oscillate with a sine wave pattern.
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If that sounds odd, consider the fact that here the sine function is taking theta as an input,
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but in the approximate solution you might see in a physics class, theta itself is oscillating as the output of a sine function.
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Clearly something fishy is afoot.
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One thing that I like about this example is that even though it's comparatively simple,
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it exposes an important truth about differential equations that you need to grapple with.
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They're really freaking hard to solve.
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In this case, if we remove that dampening term, we can just barely write down an analytic solution.
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But it's hilariously complicated.
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It involves all these functions you've probably never heard of, written in terms of integrals and weird inverse integral problems.
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And when you step back, presumably the reason for finding a solution is to then be able to make computations
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and to build an understanding for whatever dynamics you're studying.
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In this case, those questions have just been punted off to figuring out how to compute
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and more importantly understand these new functions.
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And more often, like if we add back in that dampening term, there is not a known way to write down an exact analytic solution.
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Well, I mean, for any hard problem, you could just define a new function to be the answer of that problem.
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Heck, even name it after yourself if you want.
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But again, that's pointless unless it leads you to being able to make computations and to build understanding.
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So instead, in the study of differential equations, we often do a sort of short circuit and skip the actual solution part,
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since it's unattainable, and go straight to building understanding and making computations from the equations alone.
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Let me walk through what that might look like with the pendulum.
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What do you hold in your head, or what visualization can you get some software to pull up for you to understand the many possible ways
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that a pendulum governed by these laws might evolve depending on its starting conditions?
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You might be tempted to try imagining the graph of theta versus t, and somehow interpreting how this slope,
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the position, and the curvature all interrelate with each other.
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However, what will turn out to be both easier
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and more general is to start by visualizing all possible states in a two -dimensional plane.
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What I mean by the state of the pendulum is that you can describe it with two numbers, the angle and the angular velocity.
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You can freely change either one of those two values without necessarily changing the other, but the acceleration is purely a function of those two values.
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So each point of this two -dimensional plane fully describes the pendulum at any given moment.
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You might think of these as all possible initial conditions of that pendulum.
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If you know the initial angle and the initial angular velocity, that's enough to predict how the system will evolve as time moves forward.
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If you haven't worked with them before, these sorts of diagrams can take a little getting used to.
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What you're looking at now, this inward spiral, is a fairly typical trajectory for our pendulum,
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so take a moment to think carefully about what is being represented.
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Notice how at the start, as theta decreases, theta dot, the y coordinate, gets more negative.
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Which makes sense, because the pendulum moves faster in the leftward direction as it approaches as the bottom.
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Keep in mind, even though the velocity vector on this pendulum is pointed to the left,
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the value of that velocity is always being represented by the vertical component of our space.
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It's important to remind yourself that this state space is an abstract thing,
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and is distinct from the physical space where the pendulum itself lives and moves.
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Since we're modeling this as losing some of its energy to air resistance,
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this trajectory spirals inward, meaning the peak velocity and peak displacement each go down a bit with each swing.
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Our point is, in a sense, attracted to the origin, where theta and theta dot both equal zero.
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With a space, we can visualize a differential equation as a vector field.
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Here, let me show you what I mean.
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The pendulum state is a vector, theta, theta dot.
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Maybe you think of that as an arrow from the origin, or maybe you think of it as a point.
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What matters is that it has two coordinates, each a function of time.
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Taking the derivative of that vector gives you its rate of change, the direction and speed that it will tend to move in this diagram.
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That derivative is a new vector, theta dot, theta double dot, which we visualize as being attached to the relevant point in space.
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Now take a moment to interpret what this is saying.
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The first component for this rate of change vector is theta dot, which is also a coordinate in our space.
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So the higher up we are in the diagram, the more the point tends to move to the right, and the lower we are,
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the more it tends to move to the left.
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The vertical component is theta double dot, which our differential equation lets us rewrite entirely in terms of theta and theta dot itself.
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In other words, the first derivative of our state vector is some function of that vector itself,
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with most of the intricacy tied up in that second coordinate.
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Doing the same at all points of this space will show how that state tends to change from any position.
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As is typical with vector fields, we artificially scale down the vectors when we draw them to prevent clutter, but use color to loosely indicate magnitude.
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Notice, we've effectively broken up a single second order equation into a system of two first order equations.
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You might even give theta dot a different name to emphasize that we're really thinking of two separate values,
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intertwined via this mutual effect they have on one another's rate of change.
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This is a common trick in the study of differential equations.
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Instead of thinking about higher order changes of a single value, we often prefer to think of the first derivative of vector values.
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In this form, we have a wonderful visual way to think about what solving the equation means.
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As our system evolves from some initial state, our point in this space will move along some trajectory in such a way that at every moment,
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the velocity of that point matches the vector from this field.
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And again, keep in mind, this velocity is not the same thing as the physical velocity of the pendulum.
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It's a more abstract rate of change, encoding the rates of change for both theta and theta -dot.
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You might find it fun to pause for a moment
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and think through what exactly some of these trajectory lines say about the possible ways the pendulum evolves from different starting conditions.
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For example, in regions where theta dot is quite high,
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the vectors guide the point to travel to the right quite a ways before settling down into an inward spiral.
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This corresponds to a pendulum with a high enough initial velocity
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that it fully rotates around several times before settling into a decaying back and forth.
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Earth.
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Having a little more fun, when I tweak this air resistance term,
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mu, say increasing it, you can immediately see how this will result in trajectories that spiral inward faster,
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which is to say the pendulum slows down faster.
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That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context,
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not knowing that they described a pendulum.
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It's not obvious just looking at them that increasing this value of mu means
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that the system as a whole tends towards some attracting state faster.
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So getting some software to draw these vector fields for you
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can be a great way to build an intuition for how they behave.
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What's wonderful is that any system of ordinary differential equations can be described by a vector field like this,
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so it's a very general way to get a feel for them.
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Usually though, they have many more dimensions.
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For example, consider the famous three -body problem,
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which is to predict how three masses in three -dimensional space evolve if they act on each other with gravity,
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and if you know their initial positions and velocities.
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Each mass has three coordinates describing its position, and three more describing its momentum.
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So the system has 18 degrees of freedom in total, and hence an 18 -dimensional space of possible states.
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It's a bizarre thought, isn't it?
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A single point meandering through an 18 -dimensional space that we cannot visualize,
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obediently taking steps through time based on whatever vector it happens to be sitting on from moment to moment,
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completely encoding the positions and the momenta of the three masses we see in ordinary, physical, 3D space.
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In practice, by the way, you can reduce the number of dimensions here by taking advantage of the symmetries of your setup,
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but the point that more degrees of freedom results in higher dimensional state spaces remains the same.
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In math, we often call a space like this a phase space.
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You'll hear me use that term broadly for spaces encoding all kinds of states of changing systems,
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but you should know that in the context of physics, especially Hamiltonian mechanics, the term is often reserved for a more special case,
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namely a space whose axes represent position and momentum.
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So a physicist would agree that the 18 -dimensional space describing the three -body problem is a phase space,
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but they might ask that we make a couple of modifications to our pendulum setup for it to properly deserve the term.
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For those of you who just watched the block collision video, the planes we worked with there would happily be called phase spaces by math folk,
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though a physicist might prefer other terminology.
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Just know that the specific meaning may depend on your context.
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It may seem like a simple idea, depending on how well indoctrinated you are to modern ways of thinking about math,
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but it's worth keeping in mind that it took humanity quite a while to really embrace thinking of dynamics spatially like this,
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especially when the dimensions get very large.
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In his book
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Chaos, the author James Glick describes phase space as, One of the most powerful inventions of modern science.
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One reason it's powerful is
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that you can ask questions not just about a single initial condition but about a whole spectrum of initial states.
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The collection of all possible trajectories is reminiscent of a moving fluid, so we call it phase flow.
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To take one example of why phase flow is a fruitful idea, consider the question of stability.
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The origin of our space corresponds to the pendulum standing still, and so does this point over here, representing when the pendulum is perfectly balanced upright.
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These are the so -called fixed points of our system, and one natural question to ask is whether or not they're stable.
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That is, will tiny nudges to the system result in a state that tends back towards that fixed point, or away from it?
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Physical intuition for the pendulum makes the answer here kind of obvious, but how would you think about stability just looking at the equations,
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say if they arose in some completely different, less intuitive context?
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We'll go over how to compute the answers to questions like this in following videos, and the intuition for the relevant computations are guided heavily by
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the thought of looking at small regions in space around a fixed point
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and asking whether the flow tends to contract or expand.
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And speaking of attraction and stability, let's take a brief sidestep to talk about love.
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The Strogatz quote
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that I mentioned earlier comes from a whimsical column in the New York Times on the mathematics of modeling affection,
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an example well worth pilfering to illustrate that we're not just talking about physics here.
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Imagine you've been flirting with someone, But there's been some frustrating inconsistency to how mutual your affection seems.
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And perhaps during a moment when you turn your attention towards physics to keep your mind off the romantic turmoil,
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mulling over the broken -up pendulum equations, you suddenly understand the on -again -off -again dynamics of your flirtation.
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You've noticed that your own affection tends to increase when your companion seems interested in you, but decrease when they seem colder.
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That is, the rate of change for your love is proportional to their feelings for you.
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But this sweetheart of yours is precisely the opposite, strangely attracted to you when you seem uninterested, but turned off once you seem too keen.
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The phase space for these equations looks very similar to the center part of your pendulum diagram.
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The two of you will go back and forth between affection and repulsion in an endless cycle.
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A metaphor of pendulum swings in your feelings would not just be apt, but mathematically verified.
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In fact, if your partner's feelings were further slowed when they feel themselves too in love, let's say out of a fear of being made vulnerable,
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we'd have a term matching the friction in the pendulum, and you too would be destined to an inward spiral towards mutual ambivalence.
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I hear wedding bells already.
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The point is that two very different seeming laws of dynamics, one from physics involving a single variable, and another from, uh, chemistry with two variables,
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actually have a very similar structure, easier to recognize when you're looking at the phase diagram.
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Most notably, even though the equations are different, for example there's no sine function in the romance equations, the phase space exposes an underlying similarity nevertheless.
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In other words, you're not just studying a pendulum right now.
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Now, the tactics you develop to study one case have a tendency to transfer to many others.
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Okay, so phase diagrams are a nice way to build understanding.
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But what about actually computing the answer to our equation?
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Well, one way to do this is to essentially simulate what the universe would do,
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but using finite time steps instead of the infinitesimals and limits defining calculus.
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The basic idea is that if you're at some point in this phase diagram,
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take a step based on the vector that you're sitting on for a small time step, delta t.
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Specifically, take a step equal to delta t times that vector.
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As a reminder, in drawing these vector fields, the magnitude for each vector has been artificially scaled down to prevent clutter.
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When you do this repeatedly, your final location will be an approximation of theta t, where t is the sum of all of those time steps.
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If you think about what's being shown right now, though, and what that would imply for the pendulum's movement,
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you'd probably agree that this is grossly inaccurate.
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But that's only because the time step delta t of 0 .5 is way too big.
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If we turned it down, say to 0 .01, you can get a much more accurate approximation.
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It just takes more repeated steps is all.
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In this case, computing theta of 10 requires 1 ,000 little steps.
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Luckily, we live in a world with computers,
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so repeating a simple task a thousand times is as simple as articulating that task with a programming language.
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In fact, let's finish things off by writing a little
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Python program that computes theta for us.
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What it has to do is make use of the differential equation,
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which returns the second derivative of theta as a function of theta and theta dot.
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You start off by defining two variables, theta and theta dot, each in terms of some initial conditions.
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In this case I'll have theta start at pi thirds, which is 60 degrees, and theta dot start at zero.
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Next, write a loop that corresponds to taking many little time steps between zero and your time t, each of size delta t,
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which I'm setting here to be 0 .01.
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In each step of this loop, increase theta by theta dot times delta t, and increase theta dot by theta double dot times delta t,
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where theta double dot can be computed based on the differential equation.
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After all these little time steps, simply return the value of theta.
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This is called solving a differential equation numerically.
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Numerical methods can get way more sophisticated and intricate than this to better balance the tradeoff between accuracy and deficiency,
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but this loop gives the basic idea.
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So even though it sucks that we can't always find exact solutions,
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there are still meaningful ways to study differential equations in the face of this inability.
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In the following videos we will look at several methods for finding exact solutions when it's possible,
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but one theme I'd like to focus on is how these
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exact solutions can also help us to study the more general, unsolvable cases. But it gets worse.
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Just as there's a limit to how far exact analytic solutions can get us, one of the great fields to have emerged in the last century,
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chaos theory, has exposed
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that there are further limits on how well we can use these systems for prediction with or without solutions.
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Specifically, we know that for some systems, small variations to the initial conditions, say the kind due to necessarily imperfect measurements,
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result in wildly different trajectories.
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We've even built some good understanding for why this happens.
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The three -body problem, for example, is known to have seeds of chaos within it.
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So, looking back at the quote from earlier, it seems almost cruel of the universe to fill its language with riddles that we either can't solve,
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or where we know that any solution would be useless for long -term prediction anyway.
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It is cruel, but then again, it should also be reassuring.
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It gives some hope that the complexity we see in the world around us can be studied somewhere in this math,
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and that it's not hidden away in the mismatch between model and reality.
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Vocabulário e dicas de fala para esta lição
Esta aula de conversação usa o vídeo “Differential equations, a tourist's guide”. As palavras que mais se repetem: theta, equation, pendulum, space, differential. Este vídeo tem 287 frases e 4581 palavras para praticar shadowing. A fala dura 26:50. O falante fala em ritmo natural, cerca de 171 palavras por minuto, próximo de uma conversa do dia a dia. Apenas 78% das palavras estão entre as 3.000 mais comuns do inglês, por isso o vocabulário é exigente.
Vocabulário principal deste vídeo
15 palavras do vídeo que vale a pena aprender, com pronúncia e significado:
| Palavra | Pronúncia | Significado |
|---|---|---|
| equation substantivo | /ɪˈkweɪ.ʒən/ | equação |
| derivative substantivo | /dɪˈɹɪvətɪv/ | derivado |
| vector substantivo | /ˈvɛktɚ/ | vector, vetor |
| velocity substantivo | /vəˈlɑsəti/ | velocidade |
| physics substantivo | /ˈfɪz.ɪks/ | física |
| acceleration substantivo | /əkˌsɛl.əˈɹeɪ.ʃən/ | aceleração |
| solve verbo | /sɒlv/ | resolver, solucionar |
| component substantivo | /kəmˈpoʊ.nənt/ | componente |
| trajectory substantivo | /tɹəˈd͡ʒɛktəɹi/ | trajetória, trajectória |
| involve verbo | /ɪnˈvɑlv/ | envolver |
| gravity substantivo | /ˈɡɹævɪti/ | gravidade |
| dimensional adjetivo | /daɪˈmɛn.ʃə.nəl/ | dimensional |
| diagram substantivo | /ˈdaɪ.ə.ɡɹæm/ | diagrama |
| angle substantivo | /ˈæŋ.ɡəl/ | ângulo |
| mathematics substantivo | /mæθ(.ə)ˈmæt.ɪks/ | matemática |
Phrasal verbs que você vai ouvir
| Palavra | Pronúncia | Significado |
|---|---|---|
| figure out verbo | descobrir, deduzir | |
| write down verbo | anotar | |
| check out verbo | verificar, sacar só | |
| go back verbo | voltar | |
| go down verbo | descer | |
| move forward verbo | adiantar | |
| set up verbo | /ˌsɛt ˈʌp/ | preparar |
| slow down verbo | desacelerar |
Gramática neste vídeo
As estruturas que o falante mais usa, com as palavras exatas do vídeo:
| Estrutura | No vídeo |
|---|---|
| Voz passiva be + particípio passado — o foco está no que acontece, não em quem faz | are always expressed · is spoken · is often described |
| Present perfect have/has + particípio passado — uma ação passada que ainda importa agora | has come · you've probably never heard · have just been punted |
| Orações relativas who / which + oração — informação extra sobre uma pessoa ou coisa | acceleration, which is · integrating, which is · freedom which is |
Pronúncia para ficar de olho
O falante usa 40 contrações e formas reduzidas, como you're, we'll, we're. Diga-as na forma curta, do jeito que você ouve.
- Os sons de “th”: mathematics /mæθ(.ə)ˈmæt.ɪks/, nevertheless /ˌnɛvɚðəˈlɛs/, theta /ˈθiːtə/
- Os sons de “sh” e “zh”: equation /ɪˈkweɪ.ʒən/, differential [dɪfəˈɹənʃəɫ], acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, affection /əˈfɛk.ʃən/
- Palavras longas — acerte a sílaba tônica: differential [dɪfəˈɹənʃəɫ], derivative /dɪˈɹɪvətɪv/, velocity /vəˈlɑsəti/, acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, trajectory /tɹəˈd͡ʒɛktəɹi/
Sons difíceis para falantes de português:
- Consoante final — sem acrescentar um “i” depois: component /kəmˈpoʊ.nənt/, coordinate /koʊˈɔɹ.dəˌneɪt/, loop /luːp/, correspond /ˌkoɹəˈspɑnd/, indicate /ˈɪndɪˌkeɪt/
- /l/ final — a língua toca o céu da boca, não vira “u”: differential [dɪfəˈɹənʃəɫ], dimensional /daɪˈmɛn.ʃə.nəl/, angle /ˈæŋ.ɡəl/, vertical /ˈvɝ.tɪ.kəl/, spiral /ˈspaɪɹəl/
- /h/ — um sopro suave, diferente do “r”: hence /ˈhɛns/, humanity /hjuˈmænɪti/, happily /ˈhæp.ə.li/, heck /hɛk/
Como praticar com este vídeo
- Ouça o vídeo inteiro uma vez sem falar e anote as palavras que você não conhece.
- Comece na velocidade 0,75×, faça shadowing frase por frase e volte à velocidade normal quando ficar fácil.
- Grave a sua voz e compare com o original, prestando atenção a palavras como equation, derivative, vector.
O que é a Técnica de Shadowing?
Shadowing é uma técnica de aprendizado de idiomas com base científica, originalmente desenvolvida para o treinamento de intérpretes profissionais. O método é simples, mas poderoso: você ouve áudio em inglês nativo e repete imediatamente em voz alta — como uma sombra seguindo o falante com 1-2 segundos de atraso. Pesquisas mostram melhora significativa na precisão da pronúncia, entonação, ritmo, sons conectados, compreensão auditiva e fluência na fala.