Ellipsoids and The Bizarre Behaviour of Rotating Bodies
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- čas přidán 23. 07. 2020
- Derek's video: The Bizarre Behavior of Rotating Bodies, Explained
• The Bizarre Behavior o...
Based on this amazing footage: Dancing T-handle in zero-g
• Dancing T-handle in ze...
Terence Tao's original answer, with update.
mathoverflow.net/questions/81...
Support me on Patreon and I can make more video like this!
/ standupmaths
We are sat so close because we filmed this in the “before times” of late 2019.
Huge thanks to Helen Czerski for spinning a book in zero-G for us.
• Parabolic Flight - Sci...
cosmicshambles.com/video/webs...
Cosmic Shambles (who convinced ESA to launch Helen) also have a Patreon:
/ cosmicshambles
Ben Sparks made the ellipsoid animations for me. Check out their GeoGebra files here:
www.geogebra.org/m/kuntcc5a
www.geogebra.org/u/sparksmaths
Rotating 3D book was thanks to Tim Waskett of Stone Baked Games.
www.stonebakedgames.com/
There are plenty more Matt and Hugh videos to learn about moments of inertia and suchlike.
• Matt and Hugh play wit...
Hugh Hunt is the Cambridge University Reader in Engineering Dynamics and Vibration. I know!
www2.eng.cam.ac.uk/~hemh1/
Zero G footage courtesy The Cosmic Shambles Network working in association with the European Space Agency. In flight footage shot by Melanie Cowan. External plane footage courtesy Novespace.
CORRECTIONS
- At 11:00 and 14:23 I say the axes correspond to the axes of rotation but technically they represent the three different directions of spin. Which is why they are labelled with omegas representing the angular velocities in those directions.
- At 05:52 the ellipsoid equation should be “volume” not “area”. First pointed out by Ihsan Khairir. My fault for not paying attention editing the text after copying the previous equations.
- Oh my goodness. At 11:50 we missed an “L” in ellipsoid. First pointed out by Daniel Burger. I’m so embarrassed.
- Let me know if you spot any other mistakes. Or, you know, make a whole video about it.
As always: thanks to Jane Street who support my channel. They're amazing.
www.janestreet.com/
Filming and editing by Trunkman Productions
Additional filming by Melanie Cowan
Audio by Peter Doggart
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/products/5b9f...
Nerdy maths toys: mathsgear.co.uk/ - Zábava
Great video Matt! Henry (minutephysics) raised exactly your concern while I was writing my video. We discussed with Grant (3blue1brown), who pointed out if it is the small masses that are spinning and you bump it, the angular momentum is now drastically different from before so you can't make the centrifugal force argument like you can if the big masses are spinning.
Having said that I had a follow-up video of my own planned to talk about ellipsoids because I really do like thinking of it this way too. Who knows I may yet do a follow-up to your follow-up ;)
FIGHT! FIGHT! FIGHT!
Er... I mean... Isn't it lovely to watch scientific arguments unfolding in real time?
This comment should be pinned! Everyone tell matt!
Spicy crossover
I don't really think your explanation is inherently wrong due to that missing information. Rather, Matt's video complements the missing explanation for why the spin does not occur for the small moment of inertia axis.
In fact, I consider Matt's video an excelent video for explaining the conservation part, but misses the mechanics part of the explanation. While the math part explains the various possibilities of the movement, it misses what causes the movement to take all the different arrangements that are given by line of intersection. It can be argumented that due to the higher number of arragements, the thermodynamics explanation would dictate that the object would stay in the most common arragement which exactly is the messy rotation of the object. However, I would like to understand what is the behaviour of the forces the object in to that state.
You get a follow up. You get a follow up. You get a follow up. Everyone gets a follow up!
15:00: "There's no friction or energy loss" said the engineer to the mathematician
Someone mark the calendar, hahaha!
Why is this a big deal? Can I get the back story?
@@goodstudent6157 Typically mathematicians and physicists like to deal with idealized situations that ignore those things, and much of an engineer's job is *about* those things.
@@goodstudent6157 Once apon a time an engineer, swamped with work, asked her mathematician friend to handle a chicken coop design for her. Being a good friend, he did a series of involved, precise calculations and used them to draw up some plans for the coop. He said: "I've proven with absolute certainty that is is the perfect chicken coop", and she gratefully emailed the designs to the builders and went to bed. Naturally, all of the chickens immediately escaped from the coop.
At first the mathematician was confused, but he quickly spotted the problem when he saw the chickens. "Ah, no, these will never do," he said, "the coop is designed for spherical chickens in a vacuum."
@@goodstudent6157 It is a trope that mathmeticians are picky about being exact. A physicist or engineer will have a mathematical model to explain and predict something in the real world, there are often margins of acceptable error and many things that are considered "close enough" (e.g. a lens doesnt have to be a PERFECT porabola to be useful as a lens, you can even use spherical lenses, but they have a few issues). When proving a mathmatical therom is a bit closer to philosophy and imagining theroetical constructs imo, all the rules have to be followed to the letter. An engineer may use Pi = 3.14 in a schematic or design for something, but mathmaticians use Pi in its pure form in their mathmatical constructs.
hope i didnt make this to word-vomity
Note that this only works with Elen Czerski's book.
It almost works with Matt's books. Not quite but close enough.
it's just one diagonal that doesn't quite work, isn't it?
Matt's 1st book requires following the path of 4D hyper-ellipsoids
Typical! A Parker Book.
At least Matt's book gave it a go and, in the end, that's what matters.
I understood that reference
Book authors: Vectors are bold
Normal people: Vectors have an arrow over them
Societal deviants: Vectors have bars over them
This absolute madlad: Vectors have a tilde under them
Personally I like an underline :)
I was taught that unit vectors had a “hat” over them.
Well, to be fair, bars and arrows are totally sensible for handwritten notation, since you can't exactly write unambiguous bold face on the chalkboard or a notebook.
... just like you'll find plenty of print representations of boolean expressions where inversion is done with ' marks instead of overbars. I guess it's easier to type in casual discussion, but if you're going to do it in LaTeX ... idk.
@@PSUQDPICHQIEIWC That's why I said book authors use bold, because they have the means.
@@112048112048 ... you did say that, didn't you. How did I not... today is brainfart day, it seems. You win this time!
Have a nice day despite my obliviousness and subsequent silliness.
A quick historical note: When Richard Feynman went to Cornell University at the end of the World War II, he was having a bit of trouble rekindling his love for physics after having spent years on what might be considered the ultimate applied physics project. One day in the cafeteria he saw someone toss a plate through the air and noticed that the wobble of the plate was not synchronized with its spin and determined to work out why. He was delighted to work out the answer to this ultimate "no application" research question, and this newfound enthusiasm for basic physics spilled over into enthusiasm for quantum electrodynamics.
@zztop3000 The plate wasn't the ultimate applied physics project, the Manhattan project was.
that escalated quckly quantum elecrodynamicwhatthefuck lol
I bet he'd struggle to get funding for that research, however.
@zztop3000 he said the Manhatten Project was ultimate and it was for a lot of people
I would just be inspired to get the Beef Wellington.
The veritasium had a crazy spinning animation to explain it intuitively but just seeing two ellipses hugging, now that's intuitive.
it honestly is, tho. Like, it's not intuitive how you get to them, but once you have them, they are very easy to understand.
@@TheAntibozo why? It is intuitive. It might be hard to figure out on one's own, but it is easy to follow - and once you see the two ellipsoids it just doesn't leave any room for doubt.
With other explanations, there's always some step where you don't quite understand why something has to be the case - here, as long as you understand the conversation of momentum and of energy, it is easy to see why every step in the reasoning is true.
(it is intuitive in the sense that it is a satisfying and, once you've seen it, easy to visualize answer. If you want a stricter definition of intuitive, I don't think you'll get that in maths or physics)
@@Iwasneverevenhere The issue I have with this way of looking at it is that this does not explain why the effect happens. It clearly explains why it cannot happen for the two extreme cases. It explains why it can happen in the intermediate case. But it does not explain why it actually does happen. Just because there is a line of possible solutions to travel along, there is no obvious reason why it also would actually travel along it.
In the explanation of the other video it is the other way around. You get a reason for why it happens in the intermediate case. Mostly because of centrifugal force. However it did not explain why it does not also happen in the two extremcases as well.
In the end I have still no clue why the things happen exactly the way they do. Whats missing for me is the interaction between this two exolanations.
Yes I agree the first time I learned about this on my own that made it much more clear how everything relates and why some things happen with spins but not others.
@@DerIntergalaktische Shut up and calculate.
The answer is actually quite simple. The simulator we live in doesn't properly use quaternions.
yeah it's very hard to get a book unstuck from gimbal lock. so patch the issue we just reset the orientation.
Thought they fixed that bug ages ago. Used to play around with that when I as a teenager was in the "getting home from school"-speedrunning community, but could never figure out any use for it.
Have they removed Herobrine yet?
Seems to me that it does follow quaternions.
that would be my fault, i started losing control back in late 2013. sorry.
It's slighly inconvenient to have two ellipsoids. It's even better to express everything, not in angular velocities, but angular momentum. If you do this (let's call it h, as in your video), the first ellipsoid becomes
h1^2/(2A) + h2^2/(2B) + h3^/(2C) = energy
but the second one is then just a sphere - h1^2+h2^2+h3^2=h^2.
This way, the shape of the body is ENTIRELY contained in the first ellipsoids (once you know the moments of inertia, the shape of the ellipsoid is fixed), and initial conditions just change the relative size of the ellipsoid with respect to the sphere - just grow a sphere and check for intersections. It is also easier to see what happens for symmetric bodies - if two moments of inertia are the same.
What makes this way particularly useful is that you can actually color the ellipsoid according to the distance of each point to the center - you now put all cases on the same picture, and you can probably even make a colored 3D model that you can flip around.
Question: Constant angular momentum of the released rigid body means both the magnitude and direction are separately constant, right? So the initial rotation about the intermediate axis which evolves in time to a more "complicated" motion still maintains the same angular momentum vector as the initial motion? Clearly the decomposition of the total angular momentum into momenta about the three principal axes is not unique. Why does the initial starting rotation about the intermediate axis promote shifting the angular momentum between the momenta about the three principal axes to an extent that the other two initial starting points do not?
@@jamesstewart2543 Indeed, both direction and magnitude of the angular momentum are conserved. The main relationship you need is Γ = Jω, where Γ is the angular momentum vector, ω is the angular velocity vector (pointing along the axis of rotation) and J is the inertia tensor... in general that means that the angular momentum does *not* point in the direction of the rotation axis (because angular velocity components are multiplied by bigger numbers in directions of larger inertia). Because the body actually rotates about ω, not Γ, the main axes of the body move relative to the angular momentum. But that means that to keep the same angular momentum, the *axis* must move. This is the "not unique" decomposition - the decomposition *is* unique for *one* orientation, but if orientation changes, then the decomposition must change, too. And the orientation *must* change because the body is not rotating around a fixed axis unless it is a principal axis. And among these principal axes, the intermediate axis has the property that it's not a stable equilibrium - if the axes are not *exactly* aligned, the slight deviations will amplify away from this axis - kind of like balancing a pencil on its tip. This instability follows from equations of motion - see Euler's equations for rigid body dynamics. It's not that easy to show, but the differences of moments of inertia in the equations are the core reason.
In short: angular momentum is a fixed vector. Angular velocity (the axis of rotation) is spinning around the angular momentum. The body (and its principal axes) are spinning around the angular velocity. It's a double-nested orbit, which is why this motion is so complicated.
Whoa Matt, I was totally unprepared to find another ellipsoid. I had to stop the video, compose myself and then return to video after some meditation and yoga.
What? Another ellipsoid?? NO! It can't be! We only just defeated the first one, and now there's more! I should have heeded the old wizard's warnings!
I'm mesmerized by the gaggle of water-drinking birds in the background.. IT'S DRINKING THE WATER
It was in this position that veritasium resigned the explanation
Just a nice sip of their water.
So It is as of move √2 that we have a completely new game!
So what's the idea here
@@non-inertialobserver946 so am i, but with respect to you.
(Nice name)
"Terrance edited an old answer because of this...", it's things like this that make me celebrate the internet and appreciate the world collaborating together to improve ourselves!
And this is how science makes progress.
@@joshuarosen6242 there IS NO CANON.
@@R1ckr011 Yes there is. My favourite canon is Canon 1 à 2 cancrizans from Bach's Musikalisches Opfer. I'm also rather fond of Sumer is Ycumen in.
I really liked the ellipsoid intersection idea. But I don’t see that it really helps in understanding the striking behavior of the wing nut, in that it stays in (near) intermediate axis rotation for a while and then flips very sudden. I’d like to see a part 2, showing how the point of rotation travels along the intersection of the ellipsoids.
Here is why the wing nut flips suddenly. When the objects energy state is at each end of the ellipsoid along each axis it is spinning in the opposite direction. The stable paths stay at one end.
But whenever the path on the ellipsoid crosses the yz plane it means a flip to the x spin. It's an oscillation on each axis. Fast tight spin -> slower -> slow wobble -> flip -> and then back up to a fast tight opposite spin.
The two possible circle paths in the diagram are which way it rotates on the other two axis for the flip - but both require spin flips. It's kind of like juggling a gyroscope, in order to smoothly deal with the wobbly bits and pass spin energy from x to y to z, the right-hand-rule requires an opposite spin when the energy gets back to x.
Bruce Leenstra I’m a dum dum but that makes intuitive sense to me and was what I assumed. I would love to see the intersecting ellipsoids of the original spinning handle.
I believe that the "striking behavior" of the "wingnut animation" at least partially is due to it being a "3D illustration". And that it has (in my oppinion) been made to look a bit to "perfect".
It would have been more "illustrative" if they had made the "transitional stage" "longer" (i.e. more revolutions) so that one could see that this is an "unstable state" and that it is "undulating" between them.
As it was made it looked "perfectly stable" in it's two "extreme states" and then at regular intervals it seemingly "spontaneously" just "suddenly flips" over to the "other extreme" and "stabilises" in that "state"... It all looks "too clean" and (in my opinion) we miss the "instability" that is the actual "driver/cause" of this phenomenon.
At least that's what it looked like to me,
Best regards.
@@onlyeyeno
...The Wingnut is a recorded video. It's an actual behavior of an actual physical system, not a constructed illustration.
You have to create a special device for optimal optical support of our capabilities.
Instead of a book, we take a cuboid of an appropriate material, and place 8 colored LED (8 different or at least 4 different colors) in each corner. Then we dim the light while spinning, filming with high frequency and doing a slower playback.
Ideally, we see tracing lines to follow each corner.
Since we need power to feed the LEDs, we might use epoxy for the book-shaped figur to hold batteries, LED and cables.
Realizing that the conservation laws are satisfied was where these two ellipsoids intersected was a mind blow moment. Great video!
For those looking deeper into the physics, I'd just like to point out that at 7:48 that equation for angular momentum is only true on the so called principle axes of the object. In general you have to multiply the angular velocity vector with the moment of inertia matrix. However, if we switch our basis to the principle axes (by diagonalizing the inertia matrix), that's where the expression for angular momentum they give at 12:51 comes from.
Thanks - I was a little bothered by that also!!
That bothered me too…
PS it is “principal” axes… I often have to stop to think about that…
It's less of "why does the intermediary case flip" and more of "why are the extreme cases so stable" - and that seems tied to the shape, mass distribution, etc etc as you described.
Im a chemist by training but these videos, the practical application of maths really brings me joy. I still can't get my head round more abstract maths.
I use this effect to flip the remote control when I don't pick it up in the right orientation. I have tried to extend my usage of this effect with my phone... I am not doing it anymore with the new one ^^
Great video Matt! I wrote a paper inspired by the dancing T-handle in space (aapt.scitation.org/doi/abs/10.1119/1.5093302 or arxiv.org/abs/1807.03867), but I got side tracked by geometric phase! Two things that I would like to comment on: 1) I never liked the words stability and instability since the motion is set once the initial conditions are started. Also 2) there is so much more here! I write down full analytic formulas for the motion of an asymmetric top, but one neat aspect is that they are NOT periodic in general. After a cycle, the object is in a different orientation... This new orientation relative to the beginning is the geometric phase which is a world of its own!
I just read the response from Derek Muller. In my opinion the objection does indeed *refute* the attempted explanation in terms of some 'centrifugal force'. About the 2019 update by Terence Tao:.that may save the explanation mathematically, but at a total loss of intuitive accessibility. This implies that the 'centrifugal force' suggestion never had any intuitive accessibility, but that it was an illusion all along.
In mathematical physics there is the distinction between being able to *account* for something and being able to *explain* something. The conservation principles have strong calculational power, but in general the form of the calculation will obscure the physics taking place, rather than clarify it.
The direct, visceral, expression is Newton's second law, F=ma.
The work-energy theorem is mathematically equivalent to F=ma, but it is a more abstract way of expressing it. Angular momentum is a more abstract concept than momentum (Angular momentum is an integral; the linear momenta of constituent parts integrated around the axis of rotation.) The inner product of the angular momentum vector with itself is yet another abstraction level. The abstraction level tends to make calculation easier, but it obstructs visceral understanding.
The two ellipsoids representation does a good job of accounting for the fact that two axes are stable and the intermediate axis is not. But it gives no clue how the transition occures from one end to the other.
Presumably: in the idealized case the process will be cyclic, with a consistent time period from cycle to cycle. (Some people appear to suggest that maybe each transition is a random flop from one semi-stable state to another, which is of course wrong). The ellipsoids representation does not give a clue how to understand the form of the cycle, it only suggests there will be a cyclic process.
Later edit:
This comment has likes trickling in (currently at 16). I decided to expand.
In another comment to this video a user 'physnick' pointed out his own paper:
arxiv.org/abs/1807.03867
This is a very interesting paper.
About intuitive accessibility:
On my own website there is an article where I demonstrate that it is possible to explain gyroscopic precession in a way that is both intuitive and correct. The explanation is for the most symmetrical case, and capitalizes on that symmetry.
www.cleonis.nl/physics/phys256/gyroscope_physics.php
For the assymmetric top the challenge is to find an approach that opens the case to intuitive understanding. Is it possible? My best guess: when the book is spinning around its shortest axis the case is effectively the same as the well known case of 'Feynman's wobbling plate'. Feynman's wobbling plate is the same dynamics as nutation. My best guess is that if intuitive understanding is possible it will involve some expansion of the case of Feynman's wobbling plate.
Damn. Great comment.
Parker debunks Veratasium? That’s got to be interesting.
*grabs popcorn*
It is more like Parker demystifies Veritasium. It is still veryawesome.
This felt like the educational youtube equivalent of throwing down the gauntlet.
@@Treviisolion Yeah I found this video challenging too.
Doesn't sound that hard...
11:49 You got to love Parker-Spelling, makes everything look intuitive and not right at the same time.
I did something similar a while back. The initial problem is given a set of billiard balls, by choosing some setup where the billiard balls are not moving and then you hit an initial ball with some speed v, what is the maximum speed of a ball that gets redirected in a direction perpendicular to the velocity of the first ball? There is no friction, air resistance, or rotational momentum, the only things that matter are conservation of energy and momentum. Of course the balls all have the same mass and the collisions are perfectly elastic. Just to reiterate you decide what the setup looks like, you are just looking to maximize the velocity of one of the billiard balls in a way that it is also moving perpendicular to the original velocity vector.
The idea here is to figure the possible scenarios before and after a single collision. Initially the ball has velocity v, and momentum conservation means that the velocities after the collision must sum to v. Since energy is conserved, the sum of the squares of the final velocities' magnitudes must sum to the original magnitude of v. If you were to draw an arrow representing v, then the possibilities for the final velocities v1 and v2 must be such that the tail of v1 starts at the tail of v, the tip of v1 lies on a circle drawn with the arrow v as its diameter, and v2 is drawn from the tip of v1 to the tip of v2.
Note that this just represents the possibilities resulting from one collision. If there are n billiard balls in addition to the initial ball with speed v, then assuming we focus on creating n collisions in our attempt to transfer the velocity perpendicularly, and supposing each collision occurs at some angle θ, then the angles θ1, θ2, ..., θn must sum to π/2 or 90 degrees. It should also be noted that the velocity of the second ball after a collision has magnitude |v|cos(θ).
Thus in the n ball case we are looking to maximize cos(θ1)*cos(θ2)*...*cos(θn) under the constraint that the sum of the angles is π/2. We can simplify this by assuming all the angles are the same, and as such θ=π/(2n) and with n billiard balls the solution becomes (cos(π/(2n)))^n. For n=2 we see that we can obtain half the original velocity, n=3 gives 3/4, and as n approaches infinity the solution approaches 1. This means even if you're looking to have the final ball point in the opposite direction of the original, or even do a couple full rotations of collisions, given enough billiard balls you can end up with a ball that gets arbitrarily close to the original velocity while pointing in any direction that is desired.
What I like about this problem is it takes a problem in a physical realm and converts it into a phase space and along the way has a bit of fun with calculus and limits. I don't know if anyone will read this or really understand it, but I thought it'd be fun to post it nonetheless.
I 'code' this type of stuff into hardware accelerators as part of my job - it is always great to see other examples of people who have discovered these abstract representations of the real world
I'm way too tired to really understand this, but I'm responding in the hope that I get a notification when I'm more awake. It seems really cool and interesting!
Wow!! Great job guys, my brain actually was on the same page by the end of the video!! Totally makes sense now.
Beautiful way for explaining what is going on, love it. Great video!
can you explain how a boomerang works i am Australian and make them myself and i have absolutely no clue, now that is a bizarre rotating body.
Don’t get Hugh started…
Boomerangs are shaped vaguely like an airfoil, so when moving through the air it generates lift. Since throwing it gives it a spin, one side ends up moving faster relative to the air than the other side, and thus generates more lift and makes the boomerang tilt. The combined lift from the entire boomerang is greater than zero, so it also starts turning while tilted. In a perfect world boomerangs would actually fly in a spiral, because the speed of turning is proportional to the tilt angle, which changes at a constant speed.
On a side note, this is exactly why you can throw a boomerang wrong, because too much spin will make the combined lift smaller and make it turn too slow and tilt too fast; on the other hand, too little spin makes it tilt less, which makes it turn too slowly.
I have honestly no idea why boomerangs have to be V-shaped, I personally have seen and used Y- and X-shaped ones
@@pocarski not quite, being a boomerang maker myself there is no aerofoil of any kind necessary, and it will still work. it just needs to be aerodynamic, you can try it yourself and just smooth a boomerang edges totally equal and you can see it will still work even when thrown upside down.
@@pocarski There are 2 torques and a force acting on a thrown boomerang, and you left one of the torques out.
The lift is intuitive. The differential rotation that causes the boomerang to roll, which you described.
There's also an L x {omega} torque that causes the boomerang to precess, which causes yaw.
Dr Hunt can definitely explain how boomerangs work, and will do so at the least provocation. He also regularly demonstrates in Great Court, where Matt does the introduction to this video.
The most interesting thing to me is the possibility of "reversed" spinning.
In all but the most precise cases, there will be two very strange loops, one "forward" and one "backward". But they won't exchange. The "flipping" behavior depends on which part of the loop they're in, the "top" half of the loop, or the "bottom" half of the loop. This matches the periodical, predictable, procession of the wing nut in space.
In the perfect set-up, there will be the complicated structure (non-smooth algebraic variety) that can be described as two vertices and 4 paths between the two of them. The object's state will process along one of these paths until it reaches a vertex (singular point on the variety). Once there, it'll be chaotic which path it takes. Every time it reaches one of the vertices, it will go down another path (probably has to be one of the other three). It should be possible for the state of rotation to process along one path, go in a full ellipse, and process backwards along the first path.
This "reversed" procession would be super awesome to video. Great video, it would be very cool to see more.
I wonder if you could analyze that system like a superposition of the limits of those 'imperfect' systems. You could think of a "real" system for which you know it's very close to the 'perfect' system, but you don't know if the loop orientation is 'forward-backward' or 'up-down.'
You could perhaps analyze this as some probability of being forward-backward, and some probability of being up-down. In this limit, you can see that as you approach the 'perfect' system from 'imperfect' ones, the probability of proceeding directly across the vertex is always 0.
(Perhaps until you get so close to the 'perfect' system that the error allows you to "jump" across from moving towards the vertex in 'forward-backward' to moving towards the same vertex in 'forward-backward' from the other side, *then* hopping from 'forward-backward' to 'up-down.' You should then have gone to the side of the vertex opposite where you started, and moving away. This requires that the system has certain properties similar to quantum-mechanical ones, I think.)
No, it is actually quite simple. Matt and Hugh misinterpreted their own diagram. The extreme cases have a circle at each end, representing symmetric 'paths' in opposite directions.
The intermediate case is one of two possible circular paths _which pass through the areas of opposite direction._ The 'flip' happens as the path crosses the plane that separated the circles in the other diagram. *Not* because it switched paths. So the 'perfect' spin has a single circle embedded in that plane.
A point on the path represents the axis of rotation and magnitude of angular velocity. The point moving along the path represents a change in that axis. Such a change *must* follow that path, and cannot jump (without substantial and precise perturbance). You are right that the switches are not a switch in paths. The direction of this precession is arbitrary (if somewhat 'inertial'), so each opposite side will not in general precess oppositely. From the way the precession looks in videos, the flip happens very quickly through that plane along the path. This was, from my understanding, what was presented in the video.
It's clear from the graphs that there is one extreme case where the intersection of the ellipsoids is a single circle, so far as to be unphysical, I think. Perhaps some special case of rotationally symmetric objects rotated perpendicular to their symmetrical axis would be an example. This would indicate such rotation would precess, at least if perturbed.
Our discussion of 'perfect' and 'imperfect' is unrelated to a 'perfectly stable' rotational axis. We're discussing the particular edge case which is shown with the intersection being two ellipses intersecting at two points. This is the limit of cases with 2 disjoint closed paths.
When I talked about 'jumping,' I was referring to something related to this special case of 'perfectly' intersecting ellipses where the path that is taken *doesn't* correspond to the limit of the paths along the closed curves, at least not exactly. I was trying to construct a scenario where you might continuously move 'across' the intersection point, which is inconsistent with the limits of those paths.
That was great. The visualizations really helped me get it. Thanks to all involved.
Great stuff Matt! Asking questions and analyzing problems like that - that's the stuff! Love it :)
And to all Stand-up Maths Patrons - thanks a lot for making it possible. Much appreciated!
Halfway through: this is intuitive?
End of video: oh, that is intuitive
SAME!!! 😂
The flock of drinking birds vibing in the background
That's a fantastic visualization, explains it exceptionally well! Thanks Matt
Great high-quality video Matt! Thanks and greetings from the Netherlands
"I'd like to thank my Patrons so you don't have to", nice reference to the video from Veritassium.
"... remembers your password so you don't have to"
Thankyouverymuch.
@@standupmaths hey, it would be fun to see how much of this is true in the fourth spatial dimention!
I thought Nostalgia Critic have copyrighted "I do X, so you don't have to"
@@standupmaths
What about air turbulence?
When you're spinning around the "wobbly axis" the propeller effect will be at its greatest - which means that any deviation from a "perfect" axial spin will result in the most turbulence.
My life in a nutshell(3:27): "I'm too occasional a mathematician"
That was beautiful to watch, and a joy to understand. Thank you!
This is the definitive video on this subject. As a mechanical engineering student specializing in mechanics, I learned this as the Poinsot construction, wuth the intersection path being the polhode: en.wikipedia.org/wiki/Poinsot%27s_ellipsoid
Modern computer graphics shows this much clearer than a clunky static diagram. Well done!
8:34 Hugh says "we're in 3 dimensions" and you can see Matt try his hardest to stop himself going "Actually..."
In Terrence's defense, he's a renowned Mathematician, and not a renowned Physicist, also it is a particularly deceptively tricky question.
aaand I'm transported to my nonlinear dynamics class when we went over fixed points and stability analysis.
This is unusually intuitive honestly. Derek's video was great, but I appreciate the effort you've put in to correcting its shortcomings.
I can not believe how FABULOUS this video was! What a wonderful explanation! And the way you both supported each other and did not awkwardly get in each other's way during the presentation was charming! Well done.
Me nodding along as if I understand: sees I, J, K “Ooooh quaternions, I’ve heard of that!”
Engineer and Matt: “nah those just disappear when we square them”
Me: oh sure right yep
Quarternions are just imaginary so of couse they disappear when you square them. I deal with all of my imaginary issues the same way.
Those are not i, j, k from quaternions. Here i is [1, 0, 0], quaternion i is [0, 1, 0, 0]. We are using boring old 3d vectors, not imaginary cool 4d ones. Altough i wonder how this problem would look like in quaternion space.
@@igornoga5362 Yes, I have absolutely no idea where this dude got quarternions from.
Igor Noga, Zaheer Coovadia: Fun fact, y'all: physicists use i, j, k for 3d coordinates as a holdover from when physicists used quaternions (when the physicist WR Hamilton invented quaternions, vectors weren't yet a thing, so they made do with quaternions)
@@Ramzuiv ooo think i saw a vid on that
I just watched Derek's video yesterday for the first time. Stop this, get out of my head! xD
The Algorithm....
Excellent. You succeeded in keeping it intuitive while giving a more accurate answer.
This was a very helpful and visually satisfying explanation. Thank you to all involved.
I was waiting for the "Matt and Hugh play with a thing and then do some working out" title card and it never came...
You don't call them flipping that book around "playing"? WILL YOU NEVER BE SATISFIED?
Same honestly
@@U014B Note the quotation marks
The intuition feels off here too... I want to say that the reason it can be unstable is because the intersection of the ellipsoids creates a trajectory that doesn’t enclose the intermediate axis, whereas in the other two cases it does. This would leave rotation free to fluctuate between positive and negative values of angular momentum along the intermediate axis. It doesn’t require the two trajectories where the ellipsoids intersect to connect or be close like your animation shows - in fact most values of omega 1, 2, 3 will not exist on the particular configuration with two intersecting paths. I’m not sure if I’m misreading the intuition you are trying to convey.
To put it another way (which I have yet to prove to myself), it seems that all trajectories formed by the intersection of the ellipsoids form a closed loop around either the axis of largest or the axis of smallest moment of inertia.
@@frederickcoburn All rotations about the largest and smallest moment form the closed loops. the intermediate axis can and does cross multiple axes in the intersection.
frollard Yes, I understand what you are saying and agree. The exact point that lies on the intermediate axis (the pole) does lie on the intersection of two possible trajectories. What I mean to say is that almost all points nearby but not on that pole won’t lie on the trajectories that intersect, but rather on the kind of separated trajectories that are shown in the animation leading up to the intersecting trajectories. In real life, we see the flipping effect every time we use initial conditions very close to that pole, not just with carefully chosen components of omega. That’s why I think this explanation is incorrect in attributing the flipping to the intersecting trajectories.
@@frederickcoburn Fair. My thought is that there's a stitching-on-a-baseball pattern at play where the shapes quickly become asymptotic in one axis as soon as it deviates away from the polar region, but it involves all three axes, not just the convenient intersections shown as you mention.
I thought the video's point was that only when the two ellipsoids coincide so that the two "poles" kiss can the lamina escape rotating about a single axis. If the explanation seems unintutive because that case is easy to achieve experimentally despite being infinitesimally slim in theory, I think the answer lies in the geometry of a rectangular lamina. Maybe you could prove mathematically that such a lamina will produce this condition if you rotate it about one of the three axes.
Wow. Loved the explanation! Very intuitive.
Thanks, Matt and Hugh, for the very elegant explanation
I'd like to point out an error at 5:53 where you described the volume of an ellipsoid but the equation shown in the video says area
I haven't watched the rest of the video yet so I am not sure if Matt makes a correction later.
You are totally right! That was my mistake. I’ve added it to the corrections. Thanks!
Classic Matt.
@@standupmaths just wait for my video Matt
@@veritasium Oh hello! Nice surprise to see you here!
@@veritasium veritasium is excited now
I like this explanation a lot!
However, it doesn't address why the spinning handle seems to flip between two semi-stable positions. I assume the main difference between the spinning book and the spinning handle is the their symmetry and asymmetry, respectively. It would have been interesting to see the intersecting ellipsoid graphs of the asymmetric spinning handle.
But it does. The spinning handle must have a mass giving it two ellipsoids that overlap exactly like the middle case such that it can flip between two positions seemingly at random. I know that throwing knives are specifically designed such that the handle and blade are equal in mass to facilitate the throwing so I can easily imagine other things being designed in similar ways such as the spinning handle in the video.
Beautiful ! A nice moment! Thanks!
Once I saw the book with the bubble, I immediately remembered the video with Helen Czerski! What a nice throwback!! :D
There's another notation for vectors with tilde underneath. Dots, hats, arrows, bold letters, why cant we stick with one lol
dots for marking vectors? Newton would be turning in his grave :D
@@ibonitog Dots on the undersides if I'm not mistaken, so it wouldn't get confused with the time-derivative notation
don't forget bra-ket for QM
there are truly too many notations for vectors
@@francescosorce5189 ah that makes more sense :D thanks for clearing that up, I've never seen such notation.
@@ibonitog I've seen it like once or twice, but it's definitely not mainstream
6:03 When you said "Ellipsoids!" they sounded like some sort of medication ..... Looked like it, too! 8-o
Best matheneering video in quite a while
I love this explanation! I first noticed this phenomena in grade school by spinning my books! Never saw a satisfying explanation until now! Wonderful!
I reeeaaally want to read a book about bubbles for some reason now... anyone got any recommendations?
I also want to
I am waiting for Derek's comment.......
It's there! 😄 Matt pinned it
Thank you. The intersection of the energy and momentum ellipsoids explained it quite well. Especially the one that detailed the intermediate axis case.
Maybe it's because I am a mechanical engineer and also occasional mathematician, but I really do enjoy seeing Dr. Hunt in your videos.
frequent uploads go brrrrrrrrrrrrrrrrrr
Simp
Nice seeing you here!
Yeye
beyblade beyblade let it rip
Wow! That is totally fascinating. Thanks for the great explanation.
Damn now THAT is intuitive. Excellent video! One of the shortest 25 minutes on youtube
Why are you using Helen Czerski's book ... OK that's why - long live the queen of bubbles
Now I just want to buy the Bubbles book.
1
Hi papa
This was my ib extended essay project and this video would have definitely saved me a lot of time. Best explanation I have ever heard!
excellent work at clearing up what was murky before.
ngl that crash course on how to calculate ellipsoid area is something i absolutely needed, thanks matt!
This video is so good ! So well explained , the ideas flow through before even finishing the video.
Meanwhile veritasium's was lots of rambling and rambling and honestly, painted the basic phenomenon into an even more mysterious way.
25 minutes of delight. I could follow every word, and what is best, I think I will be able to recall the theory years from now. If only my lessons at school had been like this video...
Now that makes great sense and very well explained.
Wish there was a super-like button available for 1 video per day on youtube. You made my day.
I'm excited to watch this! I had the exact same problem with Derek's video, and no one gave an answer
This is great. Differing opinions fleshes out each persons arguments and helps creates better explanations
Just saying I enjoy the addition of the scans to the videos, makes things a lot easier to look at than the old Go Pro pointing at the paper at a wonky angle for static things. Keep up the great work Matt, you’ve been keeping my brain sharp for the past few months
Thanks! I’m trying new things with the scan vs GoPro.
Thanks to you and your patreons for providing me with knowledge i probably should have been given at school.
The math is not intuitive as most people don't think about 3d unit vectors...but the resultant animations and intersections make it clearly obvious and intuitive. I love when someone can boil a simple thing down to its constituent complex math, and then boil that math down to a simple diagram. Awesome work!
What's really interesting is the singular case where the ellipsoids intersects at two intersecting curves. I assume that the flipping T-handle (or whatever that was) was close to that: the intersection would consist of two smooth curves, each having two almost-sharp bends, and the wrench was flipping when the state was going through those bends. Since those were rapid turns in the angular velocity space, the observed behavior of the wrench was visibly "jumpy". But what would happen if the bends were truly sharp and the curves intersected exactly? Would the behavior at the intersections become indeterministic? Or is there a deeper physical law that would force the system to move continuously through an intersection?
Thanks for doing the heavy lifting, walking me through the exotic concepts! I feel clever, picking up what you lay down. Thank you!
It''s really neat the way you integrate the drawings in the video!
Well, THIS IS PHYSICS at the highest explanatory level! Congratulations!
Great explanation!!!
This guy is good. I love where he was heading with the 3 axis of rotation and the different angular momentum and kinetic energy of each.
This was your best video, not sure you will ever be able to top this one.
Awesome video, thanks a lot!
Intuitive, economical, but not economical with the facts. Educational CZcams needs more of this.
Great video. This resolves the centrifugal force issue really nicely. What it doesn't do, that the Veritasium video does, is show why rotations about the intermediate appear to flip between meta-stable states.
This is great stuff! thanks!! I learned all about those ellipsoids and stable/unstable rotation around those axes about 20 yrs ago, but I never understood it intuitively. Now I will dust off my old books and look at them again
This is an intuitive way to explain something connected to, but not quite the thing that confuses people in the first place
Thanks, that was marvellous. I remember learning it at Bristol University in my physics course but could never quite intuitively feel the answer.
11:46 That's one hell of a drinking party in the background.
I always like the working out but seeing the translation of kinetic energy and angular momentum into ellipsoid spaces was by far the most interesting working out I’ve seen you two do so far!!! Fascinating!!!
Very good explanation :)
Amazing video Matt! :D
Absolutely mind blowing proof, amazing video
Wow, this is a genius explanation! Math is amazing.
that was one of the nicest things I learned in the first year of my physics studies. the reasoning was the same, but your explanation is way more intuitive :)
a most excellent video, thanks
11:48 really went all out on the effects there and I love it. Very fancy.