Differential equations, studying the unsolvable

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<div id="description" class="style-scope ytd-video-secondary-info-renderer">An overview of what ODEs are all about Home page: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2F3blue1brown.com%2F&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://3blue1brown.com/</a> Brought to you by you: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=http%3A%2F%2F3b1b.co%2Fde1thanks&v=p_di4Zn4wz4" target="blank" rel="nofollow">http://3b1b.co/de1thanks</a> Need to brush up on calculus? <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/watch?v=WUvTyaaNkzM">https://youtu.be/WUvTyaaNkzM</a> Error correction: At <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/watch?v=p_di4Zn4wz4&t=387s">6:27</a>, the upper equation should have g/L instead of L/g. Steven Strogatz NYT article on the math of love: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fopinionator.blogs.nytimes.com%2F2009%2F05%2F26%2Fguest-column-loves-me-loves-me-not-do-the-math%2F&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://opinionator.blogs.nytimes.com...</a> Interactive visualization of the example from this video, by Ilya Perederiy: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fwww.expunctis.com%2F2019%2F04%2F04%2Fvtvt-another-demo.html&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://www.expunctis.com/2019/04/04/...</a> If you're looking for books on this topic, I'd recommend the one by Vladimir Arnold, "Ordinary Differential Equations" Also, more Strogatz fun, you may enjoy his text "Nonlinear Dynamics And Chaos" Curious about why it's called a "phase space"? You might enjoy this article: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fwww.physics.umd.edu%2Fcourses%2FPhys404%2FAnlage_Spring10%2FThe%2520TangledTaleofPhaseSpace.pdf&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://www.physics.umd.edu/courses/P...</a> From a response on /r/3blue1brown, here are some interactives based on examples shown in the video: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fobservablehq.com%2F%40tophtucker%2Ftales-from-the-romeo-and-juliet-phase-space&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://observablehq.com/@tophtucker/...</a> <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fobservablehq.com%2F%40mbostock%2Fpredator-and-prey&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://observablehq.com/@mbostock/pr...</a> ------------------ Animations made using manim, a scrappy open source python library. <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fgithub.com%2F3b1b%2Fmanim&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://github.com/3b1b/manim</a> If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind. 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Download the music on Bandcamp: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fvincerubinetti.bandcamp.com%2Falbum%2Fthe-music-of-3blue1brown&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://vincerubinetti.bandcamp.com/a...</a> Stream the music on Spotify: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?event=video_description&redir_token=OfObUv0zRhHZ-lCV5ZjJt66xsXR8MTU3MDkyMTMwOUAxNTcwODM0OTA5&q=https%3A%2F%2Fopen.spotify.com%2Falbum%2F1dVyjwS8FBqXhRunaG5W5u&v=p_di4Zn4wz4" target="blank" rel="nofollow">https://open.spotify.com/album/1dVyjw...</a> If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. 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<div class="cue style-scope ytd-transcript-body-renderer">Quoting Steven Strogatz, “Since Newton, mankind has come to realize that the laws</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of physics are always expressed in the language of differential equations.” Of course, this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">language is spoken well beyond the boundaries of physics as well, and being able to speak</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it and read it adds a new color to how you view the world around you.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In the next few videos, I want to give a sort of tour of this topic. To aim is to give a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">big picture view of what this part of math is all about, while at the same time being</div>

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<div class="cue style-scope ytd-transcript-body-renderer">happy to dig into the details of specific examples as they come along.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I’ll be assuming you know the basics of calculus, like what derivatives and integrals</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are, and in later videos we’ll need some basic linear algebra, but not much beyond</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Differential equations arise whenever it’s easier to describe change than absolute amounts.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It’s easier to say why population sizes grow or shrink than it is to describe why</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the have the particular values they do at some point in time; It may be easier to describe</div>

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<div class="cue style-scope ytd-transcript-body-renderer">why your love for someone is changing than why it happens to be where it is now. In physics,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">more specifically Newtonian mechanics, motion is often described in terms of force. Force</div>

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<div class="cue style-scope ytd-transcript-body-renderer">determines acceleration, which is a statement about change.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">These equations come in two flavors; Ordinary differential equations, or ODEs, involving</div>

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<div class="cue style-scope ytd-transcript-body-renderer">functions with a single input, often thought of as time, and Partial differential equations,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">or PDEs, dealing with functions that have multiple inputs. Partial derivatives are something</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we’ll look at more closely in the next video; you often think of them involving a whole</div>

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<div class="cue style-scope ytd-transcript-body-renderer">continuum of values changing with time, like the temperature of every point in a solid</div>

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<div class="cue style-scope ytd-transcript-body-renderer">body, or the velocity of a fluid at every point in space. Ordinary differential equations,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">our focus for now, involve only a finite collection of values changing with time.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It doesn’t have to be time, per se, your one independent variable could be something</div>

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<div class="cue style-scope ytd-transcript-body-renderer">else, but things changing with time are the prototypical and most common examples of differential</div>

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<div class="cue style-scope ytd-transcript-body-renderer">equations. Physics (simple)</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Physics offers a nice playground for us here, with simple examples to start with, and no</div>

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<div class="cue style-scope ytd-transcript-body-renderer">shortage of intricacy and nuance as we delve deeper.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As a nice warmup, consider the trajectory of something you throw in the air. The force</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of gravity near the surface of the earth causes things to accelerate downward at 9.8 m/s per</div>

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<div class="cue style-scope ytd-transcript-body-renderer">second. Now unpack what that really means: If you look at some object free from other</div>

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<div class="cue style-scope ytd-transcript-body-renderer">forces, and record its velocity every second, these vectors will accrue an additional downward</div>

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<div class="cue style-scope ytd-transcript-body-renderer">component of 9.8 m/s every second. We call this constant 9.8 “g”.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This gives an example of a differential equation, albeit a relatively simple one. Focus on the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">y-coordinate, as a function of time. It’s derivative gives the vertical component of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">velocity, whose derivative in turn gives the vertical component of acceleration. For compactness,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">let’s write this first derivative as y-dot, and the second derivative as y-double-dot.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Our equation is simply y-double-dot = -g. This is one where you can solve by integrating,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which is essentially working backwards. First, what is velocity, what function has -g as</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a derivative? Well, -gt. Or rather, -gt + (the initial velocity). Notice that you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">have this degree of freedom which is determined by an initial condition. Now what function</div>

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<div class="cue style-scope ytd-transcript-body-renderer">has this as a derivative? -(½)g*t² + v_0 * t. Or, rather, add in a constant based on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whatever the initial position is.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Things get more interesting when the forces acting on a body depend on where that body</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is. For example, studying the motion of planets, stars and moons, gravity can no longer be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">considered a constant. Given two bodies, the pull on one is in the direction of the other,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">with a strength inversely proportional to the square of the distance between them.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As always, the rate of change of position is velocity, but now the rate of change of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">velocity is some function of position. The dance between these mutually-interacting variables</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is mirrored in the dance between the mutually-interacting bodies which they describe.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So often in differential equations, the puzzles you face involve finding a function whose</div>

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<div class="cue style-scope ytd-transcript-body-renderer">derivative and/or higher order derivatives are defined in terms of itself.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In physics, it’s most common to work with second order differential equations, which</div>

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<div class="cue style-scope ytd-transcript-body-renderer">means the highest derivative you find in the expression here is a second derivative. Higher</div>

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<div class="cue style-scope ytd-transcript-body-renderer">order differential equations would be ones with third derivatives, fourth derivatives</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and so on; puzzles with more intricate clues.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The sensation here is one of solving an infinite continuous jigsaw puzzle. In a sense you have</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to find infinitely many numbers, one for each point in time, constrained by a very specific</div>

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<div class="cue style-scope ytd-transcript-body-renderer">way that these values intertwine with their own rate of change, and the rate of change</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of that rate of change.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I want you to take some time digging in to a deceptively simple example: A pendulum.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">How does this angle theta that it makes with the vertical change as a function of time.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is often given as an example in introductory physics classes of harmonic motion, meaning</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it oscillates like a sine wave. More specifically, one with a period of 2pi * L/g, where L is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the length of the pendulum, and g is gravity.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">However, these formulas are actually lies. Or, rather, approximations which only work</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in the realm of small angles. If you measured an actual pendulum, you’d find that when</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you pull it out farther, the period is longer than what that high-school physics formulas</div>

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<div class="cue style-scope ytd-transcript-body-renderer">would suggest. And when you pull it really far out, the value of theta vs. time doesn’t</div>

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<div class="cue style-scope ytd-transcript-body-renderer">even look like a sine wave anymore.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">First thing’s first, let’s set up the differential equation. We’ll measure its</div>

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<div class="cue style-scope ytd-transcript-body-renderer">position as a distance x along this arc. If the angle theta we care about is measured</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in radians, we can write x and L*theta, where L is the length of the pendulum.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As usual, gravity pulls down with acceleration g, but because the pendulum constrains the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">motion of this mass, we have to look at the component of this acceleration in the direction</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of motion. A little geometry exercise for you is to show that this little angle here</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is the same as our theta. So the component of gravity in the direction of motion, opposite</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this angle, will be -g*sin(theta).</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Here we’re considering theta to be positive when the pendulum is swung to the right, and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">negative when it’s swung to the left, and this negative sign in the acceleration indicates</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that it’s always pointed in the opposite direction from displacement.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So the second derivative</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of x, the acceleration, is -gsin(theta). Since x is Ltheta, that means the second</div>

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<div class="cue style-scope ytd-transcript-body-renderer">derivative of theta is -(g/L) * sin(theta). To be somewhat more realistic, let’s add</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in a term to account for air resistance, which perhaps we model as being proportional to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the velocity. We write this as -mu * theta-dot, where -mu is some constant determining how</div>

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<div class="cue style-scope ytd-transcript-body-renderer">quickly the pendulum loses energy.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is a particularly juicy differential equation. Not easy to solve, but not so hard</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that we can’t reasonably get some meaningful understanding of it.</div>

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<div class="cue style-scope ytd-transcript-body-renderer active">At first you might think that this sine function relates to the sine wave pattern for the pendulum.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Ironically, though, what you'll eventually find is that the opposite is true. The presence</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of the sine in this equation is precisely why the real pendulum doesn't oscillate with</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the sine wave pattern.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If that sounds odd, consider the fact that here, the sine function takes theta as an</div>

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<div class="cue style-scope ytd-transcript-body-renderer">input, but the approximate solution has the value theta itself oscillating as a sine wave.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Clearly something fishy is afoot.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">One thing I like about this example is that even though it’s comparatively simple, it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">exposes an important truth about differential equations that you need to be grapple with:</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">They’re really freaking hard to solve.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In this case, if we remove the damping term, we can just barely write down an analytic</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">solution, but it’s hilariously complicated, involving all these functions you’re probably</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">never heard of written in terms of integrals and weird inverse integral problems.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Presumably, the reason for finding a solution is to then be able to make computations, and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to build an understanding for whatever dynamics your studying. In a case like this, those</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">questions have just been punted off to figuring out how to compute and understand these new</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">functions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And more often, like if we add back this dampening term, there is not a known way to write down</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">an exact solution analytically. Well, for any hard problem you could just define a new</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">function to be the answer to that problem. Heck, even name it after yourself if you want.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">But again, that’s pointless unless it leads you to being able to compute and understand</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the answer.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So instead, in studying differential equations, we often do a sort of short-circuit and skip</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the actual solution part, and go straight to building understanding and making computations</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">from the equations alone. Let me walk through what that might look like with the Pendulum.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Phase space What do you hold in your head, or what visualization</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">could you get some software to pull up for you, to understand the many possible ways</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a pendulum governed by these laws might evolve depending on its starting conditions?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">You might be tempted to try imagining the graph of theta(t), and somehow interpreting</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">how its position, slope, and curvature all inter-relate. However, what will turn out</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to be both easier and more general is to start by visualizing all possible states of the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">system in a 2d plane.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The state of the pendulum can be fully described by two numbers, the angle, and the angular</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">velocity. You can freely change these two values without necessarily changing the other,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">but the acceleration is purely a function of these two values. So each point of this</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">2d plane fully describes the pendulum at a given moment. You might think of these as</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">all possible initial conditions of the pendulum. If you know this initial angle and angular</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">velocity, that’s enough to predict how the system will evolve as time moves forward.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">If you haven’t worked with them, these sorts of diagrams can take a little getting used</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to. What you’re looking at now, this inward spiral, is a fairly typical trajectory for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">our pendulum, so take a moment to think carefully about what’s being represented. Notice how</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">at the start, as theta decreases, theta-dot gets more negative, which makes sense because</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the pendulum moves faster in the leftward direction as it approaches the bottom. Keep</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in mind, even though the velocity vector on this pendulum is pointed to the left, the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">value of that velocity is being represented by the vertical component of our space. It’s</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">important to remind yourself that this state space is abstract, and distinct from the physical</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">space where the pendulum lives and moves.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Since we’re modeling it as losing some energy to air resistance, this trajectory spirals</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">inward, meaning the peak velocity and displacement each go down by a bit with each swing. Our</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">point is, in a sense, attracted to the origin where theta and theta-dot both equal 0.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">With this space, we can visualize a differential equation as a vector field. Here, let me show</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The pendulum state is this vector, [theta, theta-dot]. Maybe you think of it as an arrow,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">maybe as a point; what matters is that it has two coordinates, each a function of time.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Taking the derivative of that vector gives you its rate of change; the direction and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">speed that it will tend to move in this diagram. That derivative is a new vector, [theta-dot,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">theta-double-dot], which we visualize as being attached to the relevant point in this space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Take a moment to interpret what this is saying.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The first component for this rate-of-change vector is theta-dot, so the higher up we are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on the digram, the more the point tends to move to the right, and the lower we are, the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">more it tends to move to the left. The vertical component is theta-double-dot, which our differential</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">equation lets us rewrite entirely in terms of theta and theta-dot. In other words, the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">first derivative of our state vector is some function of that vector itself.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Doing the same at all points of this space will show how the state tends to change from</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">any position, artificially scaling down the vectors when we draw them to prevent clutter,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">but using color to loosely indicate magnitude.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Notice that we’ve effectively broken up a single second order equation into a system</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of two first order equations. You might even give theta-dot a different name to emphasize</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that we’re thinking of two separate values, intertwined via this mutual effect they have</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on one and other’s rate of change. This is a common trick in the study of differential</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">equations, instead of thinking about higher order changes of a single value, we often</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">prefer to think of the first derivative of vector values.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In this form, we have a nice visual way to think about what solving our equation means:</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">As our system evolves from some initial state, our point in this space will move along some</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">trajectory in such a way that at every moment, the velocity of that point matches the vector</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">from this vector field. Keep in mind, this velocity is not the same thing as the physical</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">velocity of our pendulum. It’s a more abstract rate of change encoding the changes in both</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">theta and theta-dot.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">You might find it fun to pause for a moment and think through what exactly some of these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">trajectory lines say about possible ways the pendulum evolves for different starting conditions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">For example, in regions where theta-dot is quite high, the vectors guide the point to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">travel to the right quite a ways before settling down into an inward spiral. This corresponds</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to a pendulum with a high initial velocity, fully rotating around several times before</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">settling down into a decaying back and forth.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Having a little more fun, when I tweak this air resistance term mu, say increasing it,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you can immediately see how this will result in trajectories that spiral inward faster,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which is to say the pendulum slows down faster. Imagine you saw the equations out of context,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">not knowing they described a pendulum; it’s not obvious just-looking at them that increasing</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the value of mu means the system tends towards some attracting state faster, so getting some</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">software to draw these vector fields for you can be a great way to gain an intuition for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">how they behave.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">What’s wonderful is that any system of ordinary differential equations can be described by</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a vector field like this, so it’s a very general way to get a feel for them.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Usually, though, they have many more dimensions. For example, consider the famous three-body</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">problem, which is to predict how three masses in 3d space will evolve if they act on each</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">other with gravity, and you know their initial positions and velocities.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Each mass has three coordinates describing its position and three more describing its</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">momentum, so the system has 18 degrees of freedom, and hence an 18-dimensional space</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of possible states. It’s a bizarre thought, isn’t it? A single point meandering through</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and 18-dimensional space we cannot visualize, obediently taking steps through time based</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on whatever vector it happens to be sitting on from moment to moment, completely encoding</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the positions and momenta of 3 masses in ordinary, physical, 3d space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">(In practice, by the way, you can reduce this number of dimension by taking advantage of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the symmetries in your setup, but the point of more degrees of freedom resulting in a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">higher-dimensional state space remains the same).</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In math, we often call a space like this a “phase space”. You’ll hear me use the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">term broadly for spaces encoding all kinds of states for changing systems, but you should</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">know that in the context of physics, especially Hamiltonian mechanics, the term is often reserved</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for a special case. Namely, a space whose axes represent position and momentum.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So a physicist would agree that the 18-dimension space describing the 3-body problem is a phase</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">space, but they might ask that we make a couple of modifications to our pendulum set up for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">it to properly deserve the term. For those of you who watched the block collision videos,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the planes we worked with there would happily be called phase spaces by math folk, though</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a physicist might prefer other terminology. Just know that the specific meaning may depend</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on your context.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">It may seem like a simple idea, depending on how well indoctrinated you are to modern</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ways of thinking about math, but it’s worth keeping in mind that it took humanity quite</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a while to really embrace thinking of dynamics spatially like this, especially when the dimensions</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">get very large. In his book Chaos, James Gleick describes phase space as “one of the most</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">powerful inventions of modern science.”</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">One reason it’s powerful is that you can ask questions not just about a single initial</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">state, but a whole spectrum of initial states. The collection of all possible trajectories</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is reminiscent of a moving fluid, so we call it phase flow.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">To take one example of why phase flow is a fruitful formulation, the origin of our space</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">corresponds to the pendulum standing still; and so does this point over here, representing</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">when the pendulum is balanced upright. These are called fixed points of the system, and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">one natural question to ask is whether they are stable. That is, will tiny nudges to the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">system result in a state that tends back towards the stable point or away from it. Physical</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">intuition for the pendulum makes the answer here obvious, but how would you think about</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">stability just by looking at the equations, say if they arose from some completely different</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and less intuitive context?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">We’ll go over how to compute the answer to a question like this in following videos,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and the intuition for the relevant computations are guided heavily by the thought of looking</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">at a small region in this space around the fixed point and asking about whether the flow</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">contracts or expands its points.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Speaking of attraction and stability, let’s take a brief sidestep to talk about love.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The Strogatz quote I referenced earlier comes from a whimsical column in the New York Times</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on mathematical models of love, an example well worth pilfering to illustrate that we’re</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">not just talking about physics.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Imagine you’ve been flirting with someone, but there’s been some frustrating inconsistency</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to how mutual the affections seem. And perhaps during a moment when you turn your attention</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">towards physics to keep your mind off this romantic turmoil, mulling over your broken</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">up pendulum equations, you suddenly understand the on-again-off-again dynamics of your flirtation.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">You’ve noticed that your own affections tend to increase when your companion seems</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">interested in you, but decrease when they seem colder. That is, the rate of change for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">your love is proportional to their feelings for you.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">But this sweetheart of yours is precisely the opposite: Strangely attracted to you when</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you seem uninterested, but turned off once you seem too keen.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The phase space for these equations looks very similar to the center part of your pendulum</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">diagram. The two of you will go back and forth between affection and repulsion in an endless</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">cycle. A metaphor of pendulum swings in your feelings would not just be apt, but mathematically</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">verified. In fact, if your partner’s feelings were further slowed when they feel themselves</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">too in love, let’s say out of a fear of being made vulnerable, we’d have a term</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">matching the friction of your pendulum, and you two would be destined to an inward spiral</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">towards mutual ambivalence. I hear wedding bells already.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The point is that two very different-seeming laws of dynamics, one from physics initially</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">involving a single variable, and another from...er...chemistry with two variables, actually have a very similar</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">structure, easier to recognize when looking at their phase spaces. Most notably, even</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">though the equations are different, for example there’s no sine in your companion’s equation,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the phase space exposes an underlying similarity nevertheless.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In other words, you’re not just studying a pendulum right now, the tactics you develop</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to study one case have a tendency to transfer to many others.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Okay, so phase diagrams are a nice way to build understanding, but what about actually</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">computing the answer to our equation? Well, one way to do this is to essentially simulate</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what the world will do, but using finite time steps instead of the infinitesimals and limits</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">defining calculus.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The basic idea is that if you’re at some point on this phase diagram, take a step based</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on whatever vector your sitting on for some small time step, delta-t. Specifically, take</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a step of delta-T times that vector. Remember, in drawing this vector field, the magnitude</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of each vector has been artificially scaled down to prevent clutter. Do this repeatedly,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and your final location will be an approximation of theta(t), where t is the sum of all your</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">time steps.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">If you think about what’s being shown right now, and what that would imply for the pendulum’s</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">movement, you’d probably agree it’s grossly inaccurate. But that’s just because the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">timestep delta-t of 0.5 is way too big. If we turn it down, say to 0.01, you can get</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a much more accurate approximation, it just takes many more repeated steps is all. In</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this case, computing theta(10) requires a thousand little steps. Luckily, we live in</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a world with computers, so repeating a simple task 1,000 times is as simple as articulating</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that task with a programming language.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In fact, let’s write a little python program that computes theta(t) for us. It will make</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">use of the differential equation, which returns the second derivative of theta as a function</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of theta and theta-dot. You start by defining two variables, theta and theta-dot, in terms</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of some initial values. In this case I’ll choose pi / 3, which is 60-degrees, and 0</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for the angular velocity.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Next, write a loop which corresponds to many little time steps between 0 and 10, each of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">size delta-t, which I’m setting to be 0.01 here. In each step of the loop, increase theta</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">by theta-dot times delta-t, and increase theta-dot by theta-double-dot times delta-t, where theta-double-dot</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">can be computed based on the differential equation. After all these little steps, simple</div>

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<div class="cue style-scope ytd-transcript-body-renderer">return the value of theta.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is called solving the differential equation numerically. Numerical methods can get way</div>

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<div class="cue style-scope ytd-transcript-body-renderer">more sophisticated and intricate to better balance the tradeoff between accuracy and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">efficiency, but this loop gives the basic idea.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So even though it sucks that we can’t always find exact solutions, there are still meaningful</div>

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<div class="cue style-scope ytd-transcript-body-renderer">ways to study differential equations in the face of this inability.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In the following videos, we will look at several methods for finding exact solutions when it’s</div>

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<div class="cue style-scope ytd-transcript-body-renderer">possible. But one theme I’d like to focus is on is how these exact solutions can also</div>

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<div class="cue style-scope ytd-transcript-body-renderer">help us study the more general unsolvable cases.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But it gets worse. Just as there is a limit to how far exact analytic solutions can get</div>

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<div class="cue style-scope ytd-transcript-body-renderer">us, one of the great fields to have emerged in the last century, chaos theory, has exposed</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that there are further limits on how well we can use these systems for prediction, with</div>

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<div class="cue style-scope ytd-transcript-body-renderer">or without exact solutions. Specifically, we know that for some systems, small variations</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to the initial conditions, say the kind due to necessarily imperfect measurements, result</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in wildly different trajectories. We’ve even built some good understanding for why</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this happens. The three body problem, for example, is known to have seeds of chaos within</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So looking back at that quote from earlier, it seems almost cruel of the universe to fill</div>

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<div class="cue style-scope ytd-transcript-body-renderer">its language with riddles that we either can’t solve, or where we know that any solution</div>

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<div class="cue style-scope ytd-transcript-body-renderer">would be useless for long-term prediction anyway. It is cruel, but then again, that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">should be reassuring. It gives some hope that the complexity we see in the world can be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">studied somewhere in the math, and that it’s not hidden away in some mismatch between model</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and reality.</div>

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Differential equations, studying the unsolvable

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