20 July 2018
Grant Sanderson (3Blue1Brown), guest-hosting on MinutePhysics · watch on YouTube ↗ · click any timestamp to jump the video
Machine-generated by an AI from the transcript and the top comments. Not my writing, and it may contain errors.
Twenty-one minutes and essentially no padding. The geometry groundwork runs to about 9:07 and is worth not skipping, because the payoff at 18:48 depends entirely on one fact established back at 8:33. The crux — where the inverse square law and Kepler's second law cancel each other — is at 14:41.
Feynman's own definition, in his voice: elementary means very little needs to be known in advance — not that the steps are few or simple. He then adds the condition that you also need an infinite amount of intelligence. It is a joke, but it is also an honest warning about what follows.
The move that makes the whole proof work. Rather than attacking the shape of the path directly, the argument first asks a stranger question: if you take every velocity vector around the orbit and slide them so their tails meet at one point, what shape do the tips trace?
The velocity vectors swing around, lengthening and shortening at different points of the orbit, and yet their tips land exactly on a circle. Nothing about the setup suggests this should be true, which is what makes it the pivot of the argument.
The crux. Slice the orbit into pieces of equal angle. By Kepler's second law the time to cross a slice grows as the square of the distance; by the inverse square law the change in velocity falls as the square of the distance. Multiply them and the distance drops out entirely — so the change in velocity is the same for every slice, no matter where in the orbit it sits.
Because the force always points at the sun, the direction of each velocity change rotates by a constant angle from one slice to the next. Constant length plus constant turning is the definition of a regular polygon — and as the slices get finer, that polygon becomes the circle.
The opening geometry shows that rotating each line 90 degrees about its midpoint produces lines tangent to the emerging ellipse, not just touching it somewhere. Tangency is the whole point, because a tangent direction is what a velocity vector is — and that is the bridge the proof crosses at the end.
The closing trick: rotate the entire velocity diagram a quarter turn, then rotate each individual velocity line back the other way about its own midpoint. That is exactly the construction from the opening, so the ellipse appears — and it has precisely the tangency property the real orbit must have.
A quietly reassuring detail: he admits he could not easily follow Newton's reasoning at this step, so he built his own route instead. The elegance on display is a workaround for a proof he found impenetrable.
It existed only as an unpublished partial transcript with a scattering of notes, sitting in a colleague's office, and crucially missing the blackboard drawings. Judith Goodstein found it; David Goodstein reconstructed the argument well enough to publish.
It needs neither an elliptical orbit nor an inverse square law. The only requirement is that the force points straight at the sun. Historically the implication ran the other way — the equal-area observation is part of what led to understanding angular momentum.
From the Latin for fireplace, because some of the earliest work on ellipses concerned orbits around the sun — the fireplace of the solar system, sitting at one focus.
The reply Feynman's caveat was always going to attract, and the most-liked joke under the video.
Several viewers note the gap between promising no calculus and then, minutes later, taking a polygon to a circle by adding infinitely many infinitely small sections. Fair, and worth knowing before you start.
Someone spotted the symmetry: a guest host covering a lecture Feynman himself gave as a guest.
Auto-generated captions. Click any line or timestamp to seek; the current line highlights as the video plays.
0:03You may be aware that I’m a huge fan of the YouTube channel 3blue1brown, run by Grant Sanderson.
0:07Grant makes excellent videos about math, and mathy aspects of other topics, so I’m letting him take over my channel for the
0:12day. Grant, take it away. (Grant) A week ago, I put out a tweet showing a peculiar place where an ellipse arises,
0:20but what I didn’t mention is that this arbitrary-seeming construction is highly relevant to a once lost lecture by Richard Feynman on
0:26why planets orbit in ellipses. The construction starts by drawing a circle, and choosing some point within the circle which is not
0:35the center, what I’ll call an “eccentric” point.
0:39Then draw a bunch of lines from this eccentric point to the circumference of the circle.
0:43For each of those lines, rotate it 90 degrees about its midpoint.
0:43Once you do this for all the lines, an ellipse emerges in the middle.
0:49Out of context, this a mildly pleasing curiosity, but there’s a much deeper form of satisfaction on its way once you understand
0:56the full story surrounding this. Front and center in that story is Richard Feynman, whose famous along a number of dimensions.
1:06To scientists, he’s a giant of 20th century physics, winner of the Nobel prize for his foundational insights for Quantum Electro Dynamics
1:11among many other things. To the public, he’s a refreshing contradiction to stereotypes about physicists: A safe-cracking, bongo-playing, mildly-philanderous non-conformist whose heavily
1:20brooklyn-accented voice you’ve probably heard either relaying some bit of no-nonsense pragmatic wisdom about the only sensible way to view the world,
1:29or else some wry joke told through a crooked smile.
1:33But to physics students, he was an exceptionally skillful teacher, both for his charisma and his uncanny ability to make complicated topics
1:38feel natural and approachable. Many of the lectures he gave as a CalTech freshman course are immortalized in the now famous “Feynman
1:47lectures”, whose three volumes are available for free online.
1:51But not all of the lectures he gave made it into this collection.
1:55One in particular, a guest lecture lecture given in March 13, 1964 entitled “The motion of planets around the sun”, survived only
2:00as an unpublished partial transcript with a smattering of notes buried in the office of one of Feynman’s colleagues until it was
2:08eventually dug up by Caltech archivist Judith Goodstein.
2:12Despite the absence of some crucial blackboard drawings to follow what Feynman was saying, her husband David eventually reconstructed the argument of
2:16the lecture, which the two of them published in a book titled “Feynman’s lost lecture”, conveying both the lecture itself and the
2:24surrounding story in a really beautiful way.
2:27Here, I’d like to give a more animated and slightly simplified retelling of the argument Feynman presented.
2:35The lecture itself is about why planets and other astronomical objects orbit the sun in ellipses.
2:40It ultimately has to do with the inverse square law, the fact that the gravitational force pulling an object towards the sun
2:43is inversely proportional to the square of the distance between the orbiting object and the sun.
2:47But why? How exactly does that give rise to an ellipse, of all shapes?
2:57Of course, the gravitational attraction between planets, moons comets and such means that no one orbit is a perfect ellipse, but to
3:03a very good approximation this is the shape of an orbit.
3:07You can solve this analytically, setting up the appropriate differential equation and seeing the formula for an ellipse pop out, but Feynman’s
3:16goal was not to rely on any heavy mathematical machinery.
3:16In fact, let’s take a listen to him articulate his own goal.
3:22I am going to give what I will call an elementary demonstration.
3:28But elementary does not mean easy to understand.
3:28Elementary means that very little is required to know ahead of time in order to understand it, except to have an infinite
3:40amount of intelligence. There may be a large number of steps that hard to follow, but to each does not require already
3:47knowing the calculus or Fourier transforms.
3:56Yeah, that’s all, infinite intelligence. I think you’re up to that, don’t you?
4:00I’ve done what I can to simplify things down further, but that’s not to say a good deal of focus won’t be
4:05required. First thing’s first, we need some definition of an ellipse, otherwise there’s no hope of proving that they’re the shape of
4:11orbits. Some of you may be familiar with the classic way to construct an ellipse using two thumbtacks and a piece of
4:15string. Use the thumb tacks to fix the ends of a small length of string in place, then pull the string taut
4:20with a pencil, and trace out a curve while keeping the string taut.
4:28It’s similar to how you might use a single pushpin to construct a circle, where the fixed length of string guarantees that
4:33every point you trace is a constant distance from the thumbtack.
4:38But in this case, with two thumbtacks, what property are you guaranteeing about each point you trace?
4:44Well, at every point, the sum of the distances from that point to each of the thumbtacks will be the full length
4:50of the string, right? So the defining property of this curve is that when you draw lines from any point on the
5:01curve to these two special thumbtack locations, the sum of the lengths of those lines is a constant, namely the length of
5:07the string. Each of these points is called a “focus” of your ellipse, collectively called “foci”.
5:11Fun fact, the word focus comes from the latin for “fireplace”, since one of the first places ellipses were studied was for
5:17orbits around the sun, a sort of fireplace of the solar system, sitting at one of the foci of a planet’s orbit.
5:26Making up a bit of terminology, let’s call this constant sum of the distances from any point on the ellipse to the
5:30two foci the “focal sum” of the ellipse.
5:37We’ll get to orbital mechanics in a moment, but first let’s turn back to that construction I showed at the beginning, which
5:42will come up again later in the story.
5:46Remember, we take all these lines from an eccentric point of a circle to its circumference, and rotate each 90 degrees about
5:51its center, why on earth should an ellipse pop up?
5:59You could just take my word for it, but I think you’ll be much more satisfied in the end if we take
6:04the time now for a brief sidestep into geometry proof land.
6:10First off, there are really only two special points in this diagram, the eccentric point from which all the lines emerge, and
6:14the center of the circle, so you might guess that each of these is a focus of the ellipse.
6:19Given the defining property of an ellipse, you know you’re going to want to look at the sum of the distances from
6:25these two points to...something. Also, if you’re doing a geometry problem involving a circle, you’ll very likely want to draw a radius
6:33of that circle, and at some point use the fact that this radius has a constant length no matter where you draw
6:38it. I mean, that’s what defines a circle, so you’ll probably need to incorporate that somewhere.
6:47With those two thoughts in the back of our mind, let’s limit our attention to just one of these lines, touching some
6:50point P on the circle. Remember what happens in our construction: You rotate this line from the eccentric point 90 degrees about
6:58its center, and the geometry enthusiasts in the room might fancifully call this a “perpendicular bisector” of the original line.
7:08Take a moment to think about the sum of the distances from our two proposed focus points to any point Q along
7:12this perpendicular bisector. The key insight here is that you can find two similar triangles to conclude that the distance from the
7:29eccentric point to Q is the same as the distance from Q to P.
7:33So, that means adding the distances to each focus is the same as adding the distances from the center to Q, then
7:40Q to P. Now there are two key things I want you to notice: First, at the point where this perpendicular bisector
7:46intersects the radius, that sum is clearly the radius of the circle.
7:54Since that radius is a constant no matter where we draw it, the focal sum at that intersection point stays constant, which
7:58by definition means it traces out an ellipse, specifically an ellipse whose focal sum equals the radius of this circle.
8:10Isn’t that neat? Second, because the sum of these two lengths at every other point on this perpendicular bisector is larger than
8:16the radius, meaning the sum of the distances to the foci from those points are bigger than the ellipse’s focal sum, all
8:27other points of this line must lie outside the ellipse.
8:33What this means, and this will be important, is that this perpendicular bisector, the line we got after our special 90 degree
8:38rotation, is tangent to the ellipse.
8:43So the reason all the lines we drew earlier make an ellipse appear is because we’re drawing a bunch of that ellipse’s
8:50tangent lines. The reason this will be important, as you’ll say later, is that this tangency direction will correspond to the velocity
8:58of an orbiting object. Okay, geometry proofiness done, onto some actual physics and orbital mechanics!
9:07The first fact to use is Kepler’s (very beautiful) second law, which says that as an object orbits around the sun, the
9:12area it sweeps out during a given amount of time, like 1 day, will be a constant, no matter where you are
9:18in the orbit. For example, think of a comet whose orbit is very skewed.
9:22Close to the sun, it’s getting whipped around very quickly, so it covers a larger arc length during a given time interval.
9:32Farther away, it’ll move slower, so covers a shorter arc length during that same time.
9:39And this trade off between radius and arc length balances in just such a way that the swept area is the same.
9:44A quick way to see why this is true is to leverage conservation of angular momentum.
9:49For a tiny time step, delta t, the area swept out is essentially a triangle.
9:56In principle you should think of this as a small sliver for a tiny time step, but I’ll draw it thicker so
10:00we can better see all the parts.
10:06The area is ½ base times height, right?
10:10The base is the distance to the sun, and the height will be this little length here, which you can think of
10:17as the component of the object’s velocity perpendicular to the line to the sun, which I’ll call v_perp, multiplied by the small
10:22duration of time. So the area is ½ R * (v_perp) * (delta t).
10:26Conservation of angular momentum with respect to a given origin point, like the sun, tells us that this radius time the component
10:36of velocity perpendicular to it will remain constant, so long as all forces acting on the object are directed towards that origin.
10:47Well, specifically it says this quantity times the mass of the object stays constant, but the mass of an orbiting object won’t
10:51be changing. So! Our expression for the area swept out depends only on the amount of time that has passed, delta t.
11:04Historically, this went the other way around, and Kepler’s second law is one of the empirical facts that led to an understanding
11:10of angular momentum. I should emphasize, this law does not assume that the orbit is an ellipse.
11:13Heck, it doesn’t even assume the inverse square law, the only thing needed for this to hold is that the only force
11:24acting on the orbiting object is directed straight towards the sun.
11:29This is a fact Feynman spent much more time showing, recounting an argument by Newton in his Principia, but it kind of
11:33distracts from our main target, so I figure assuming conservation of angular momentum is good enough for our purposes here, albeit at
11:38some loss of elementarity. At this point we don’t know the shape of an orbit; for all we know it’s some wonky
11:54non-elliptical egg shape. The inverse square law will help pin down that shape precisely, but the strategy is a little indirect.
11:59Before showing the shape of the path traced by the orbiting object, we’ll show the shape traced out by the velocity vectors.
12:10Here, let me show you what I mean by that.
12:14As the object orbits, its velocity will be changing, always tangent to the curve of the orbit, longer at points where the
12:20object moves quickly, and shorter at points where it moves more slowly.
12:25What we’ll show is that if you take all these velocity vectors, and collect them together so that their tails all sit
12:29a single point, their tips actually trace out a perfect circle.
12:35This is a pretty awesome fact, if you ask me.
12:41The velocity spins around and gets faster and slower at various angles, but evidently the laws of physics cook things up just
12:46right so that these trace out a perfect circle.
12:52The astute among you might have a little internal lightbulb starting to turn on at the sight of this circle with an
12:56off-center point. Now, why on earth should this be true?
12:59Feynman describes being unable to easily follow Newton at this point, so instead he comes up with his own elegant line of
13:05reasoning to explain where this circle comes from.
13:09He starts by looking at the orbit, and slicing it up into little pieces which all cover the same angle with respect
13:15to the sun. Alright, now think about how the amount of time it takes the orbiting object to traverse one of these
13:25equal-angle slices changes as it gets farther away.
13:30Well, by Kepler’s 2nd law, it’s proportional to the area swept out, right?
13:30And because these slices have the same angle, as you get farther away from the sun, not only does the radius increase,
13:41but the component of arc length perpendicular to that radial line goes up in proportion to that radius.
13:47So the area of one of these slices, and hence the time it takes the object to traverse it, is proportional to
13:54the distance away from the sun squared.
13:59In principle, we’ll ultimately be considering very small slices, so there won’t be ambiguity in what I mean by the radius from
14:05the planet to the sun on a given slice, and the relevant bits of arc length will be effectively straight.
14:07Alright, now think about how the inverse square law comes into play.
14:13At any given point, the force the sun imparts on the object is proportional to 1/(the radius)^2, but what does that really
14:19mean? What force is is the acceleration on the object, the amount that it’s velocity changes per unit time, multiplied by that
14:32object’s mass. This is enough to give us a super useful bit of information about how the velocity of our orbiting object
14:35changes from one slice to the next.
14:41The change in velocity is acceleration times change in time, right?
14:41Which means its proportional to the change in time over the radius squared.
14:56But since the time it takes to traverse one slice is proportional to the radius squared, these terms cancel, so the change
15:02in velocity as the object traverses a given slice is actually some constant that doesn’t depend on the slice at all.
15:11In other words, if you look the velocity at the start of the slice, and at the end of the slice, then
15:18directly compare them by joining their tails, looking at the difference between the two, the vector joining their tips, this difference has
15:24the same length no matter which slice of the orbit you were looking at.
15:29Also! Since the force vector is always pointing towards the sun, as we go from the start of one slice to the
15:44next, that force vector is turning by a constant angle.
15:51In geometry lingo, you might say that all the “external angles” of this polygon that has formed will be equal.
15:58I know this is a little tricky, but hang in there!
16:05Remember that all you need to follow along is infinite intelligence.
16:09Take a moment to make sure it’s clear what’s happening without velocity diagram: The change from one vector to the next, the
16:15little difference vector joining one tip to the next, will always have the same length, which was a consequence of the perfect
16:21cancelation that happens when mixing Kepler’s second law with the inverse square law.
16:27Now, because those constant-length change vectors rotate by a constant angle each time, it means they form regular polygon.
16:33As we consider finer and finer slices of the original orbit, based on smaller and smaller angles for those slices, the relevant
16:39regular polygon defining the tips of the vectors in our velocity diagram will approach a circle.
16:50Isn’t that really neat? Hopefully, at this point you’re looking at this circle with a special eccentric point, and your just itching
16:57to see it give rise to an ellipse the way we saw earlier.
16:57But, it’s a little weird, right?
17:02We’re looking at a diagram in velocity space, how exactly will this give us the shape of the orbit?
17:07What follows is tricky, but very clever.
17:14Step back and consider what we know: We don’t know the specific shape of the orbit, only the shape the velocity vectors
17:18trace. But more specifically than that, we know that once the planet has turned an angle theta off the horizontal with respect
17:24to the sun, this corresponds to walking theta degrees around our circle in the velocity diagram, since the acceleration vectors rotate just
17:35as much as the radius vector.
17:40This tells us the tangency direction for each point on the orbit; whichever vector from our velocity diagram touches that point theta
17:45degrees around, that’s the velocity vector of our orbiting object, and hence the tangency direction of the curve.
17:55In fact, let me just start drawing all those velocity vectors as lines, since all we’ll need to use is the information
18:01they cary about the slope of the orbit curve; the specific magnitude of each velocity will not be as important.
18:08Notice, it's not that the angle of the velocity vector at this point is an angle theta off the vertical.
18:16No no no. The angle I’m referencing in the velocity diagram is with respect to the circle’s center, which is almost certainly
18:21a little different from where the velocity vectors are rooted.
18:26So the question is, what special curve satisfies the property that the tangency direction for a point theta radians off the horizontal
18:31is given by this vector from a special eccentric point of a circle to a point theta degrees around the circle from
18:45the vertical? Well, here’s the trick.
18:48First, rotate this whole circle setup 90 degrees.
18:56Then take each of those individual velocity directions and rotate them 90 degrees back the other way, so that they’re oriented as
19:01they were before, it’s just that each is rooted in a different spot.
19:05Aha! We’ve spotted our ellipse! But we still have just a little thinking ahead of us to really understand how this emergent
19:11ellipse is related to the astronomical orbit.
19:16Importantly, I didn’t just rotate these lines about any point, I rotated each about its center, which means we can leverage the
19:26geometric proof we saw several minutes ago.
19:26And this is the moment where you kind of have to furrow your brow and think back “wait, what was going on
19:31in that proof again?” One of the key points was that when you have two lines, one from the center of the
19:37circle, one from the eccentric point, both to a common spot on the circle’s circumference, the perpendicular bisector to the eccentric line
19:48will be tangent to the ellipse.
19:48What’s more, the point of tangency is where it intersects with the radial line from the center.
20:03What that means is that the point of our little ellipse which is theta degrees off the horizontal, with respect to the
20:08circle’s center, has a tangent slope perpendicular to this eccentric line.
20:13And because of the whole 90 degree rotation, this means it’s parallel to the velocity vector we need it to be.
20:20So this little emergent curve inside the velocity diagram has exactly the tangency property we need our orbit to have!
20:29And hence, the shape of the orbit must be an ellipse.
20:29QED. Alright, pat yourself on the back, because there’s no small amount of cleverness required to follow this.
20:41First there was this peculiar way of constructing an ellipse, requiring some geometry savviness to prove.
20:49Then there’s the pretty clever step of even thinking to ask the question about what shape the velocity vectors trace out when
20:53you move all their tails to the same spot.
20:58And showing that this is a circle requires mixing together the inverse square law with Kepler’s second law in another sly move.
21:02But the cleverness doesn’t end there!
21:02Showing how this velocity diagram with vectors rooted at a point off the circle’s center implies an elliptical orbit brings in this
21:11neat 90 degree rotation trick. I just love this.
21:16Watching Feynman do physics, even elementary physics, is like watching Bobby Fischer play chess.
21:22(Henry): Thanks again to Grant, and you should definitely go check out his videos on 3blue1brown