Balls dropped on a double well curve (~ x^4 - x^2)
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- čas přidán 14. 05. 2023
- 1, 2, and 1000 balls dropped on the function y = x^4 - 3x^2.
In the 1000-ball part balls are initially mutually separated by less than a millionth of the plot width.
Thanks to those suggesting this system! If you announce yourself in the comments, I'll give you credit of course.
Music by @gpcbass. The cool song was also used in our last video, well worth a second listening. It's called Polar_Mist.
Visuals in Python & FFmpeg. - Věda a technologie
I felt so happy for the first one when it got to the right side
I actually told the ball when it got close, "No way mate, you're stuck on that side for good"
If you were to drop a ball from each pixel on the left side, and then color that pixel based on the number of bounces it takes to make it to the right side. I wonder how the pattern would turn out. Would it be fractal like?
Probably
There must be periodic orbits that stay on one side, and as you move away from those there must be orbits that move to the other side, so that pretty much guarantees that there must be some fractal structure surrounding those.
@@landsgevaerThat doesn't necessarily mean it's fractal. There could be sharp boundaries between regions.
@@DavidGuild Not sure what you mean by "sharp boundary". Mandelbrot has a sharp boundary, but it is infinitely convoluted.
I can't draw or post links here unfortunately. Hard to explain in text. It is comparable to the fractal that you get when looking at multiple mirroring balls: images of balls in balls in balls etc. corresponding with light rays bouncing from one to another. Here you have domains that switch from the left concavity to the right, in which you have regions that switch to the left again, in which.. etc. If you record the bounces taking place on the Left or Right, you get sequences like LLRRRLRRLLLLRLRRRL.. similar to how that occurs in the logistic map.
Sorry that I cannot explain it better.
I suspect it would be really random. Butterfly effect.
The best part about this channel is how the animations seamlessly links deterministic dynamics with statistical mechanics.
Would love to see a wave packet falling into this double well!
Same! The time evolution of a wave function in a double potential well can have really interesting behavior.
I came here to say this.
I concur
"When fleas are placed in a jar, they try to jump out but after the lid is placed on, the fleas stop attempting the escape, as they learn the boundaries of the new environment they’re in. Even when the lid is removed, the fleas never jump out. Their thinking has created the ‘lid’ as the boundary, so has conditioned them to limit their jumping."
The satisfaction of watching the first ball finally make it over the middle to the other side was unreal
Wow the green blob really sticked together until the end
3:10 is it just me or do these middle balls sync up with the music? Either way, nice simulation!
Came here to say exactly this. It seems like the music was timed around this point
Just a lucky coincidence. The music was specifically made for another of my visuals.
@@animations_ag Which one? Would love to watch it.
@@CountDooku420 czcams.com/video/q66kwizsUvk/video.html
@@animations_ag when I click the read more nothing happens 😂
This really came alive with the many balls. The two wells lend an interest to the video that you don't get with just different shapes of well. Next, how about a sinusoid y = sin x? So that the balls don't disperse too far too soon, make the wells high enough that jumping from one well to another is rare but still possible.
Make that y= sin x * |x|
Excellent choice of music! Really mesmerizing result
I think looking at cosh x could be really interesting! And also, maybe a video comparing x^2, x^4, and cosh x, and seeing how fast they go chatoic would be really interesting!
Just to clarify, the ball strikes the surface with perfect elasticity and its overall energy (KE+PE) is conserved?
That's right -- otherwise the balls would reach lower maximum heights with each bounce, and eventually be unable to ever cross between the wells again.
ball strikes.
still loving the content, keep it up!!
❤ the moment when separation starts....love it
3:09 this was timed perfectly with the music
Nice… a beautiful demonstration of the progression of entropy…
Would be cool to have a marker that tracks the location of the center of mass. Maybe even draw the path.
Awesome! Would be super cool to add a slight attraction or repulsion between them, or color them based on their speed
That was the cricket glove leg on the slip from the mop while the other leg was a iron needle
A great demonstration of simulated annealing.
Gorgeous visual and good music. What more can we ask ?
Underrated video
The music makes me yearn to see a metropolitan magazine writer profess her love to an Australian rustic in a NY subway station
Fascinating
Man, this song absolutely *slaps*. Total banger right here.
1000 balls never ceases to amaze! 😮
The music in this video is such a banger
Relaxing!
I love all the music in each video, I am returning just for the music and I guess the balls too.
i like how the music bops even harder at 2x speed.
2:40 interesting that the points diverge almost immediately after many smaller collisions.
omg hii peterscrapsss
iam a fan
1:09 I LOVE HOW SMOOTHLY IT BOUNCED
This makes me think of quantum wave functions 🌊⚛🤩. Fascinating
its cool to see sliding arise with a system where you just bounce balls back
Do you know what kind of shape induces the fastest divergence? Of course something like a stochastic process would be an overkill. I just would like to have a function that maps curves into a scalar meaning how “chaotic” they are
Wow that's the perfect illustration of thermodynamics and liquid physics, would be even more interesting if you add colour change depending on the speed of the ball
What is the offset in the initial condition of many balls?
I always love the falling ball vids! I was wondering how you generate the colour spectrum for them? Are you using RGB triples? If so, how do you obtain the appropriate values? If not, how do you create the spectrum?
They may use hsb colour values instead of RGB, and changing the hue incrementally
@@haggisllama2630 Do you mean HSV?
I had always heard that all of the variance in the universe was caused by a tiny discrepancy in the initial Big Bang singularity. This reminds me of that dynamic.
NOW THESE ARE SOME SERIOUS BALLS
Very cool system! One part of the song reminds me of Dear Prudence.
I noticed that!
I find it curious that:
The 1000 balls lost synchronization after sliding in the half pipe and kept synchronization when bouncing against the walls with an orthogonal angle.
And eventually an even distribution shows up, with 1 clump of balls traveling together but seemingly losing mass.
Making it seem that eventually it will end up with all the balls scattered and approximately an even amount apart.
Is this because of the second law of thermodynamics?
It’s a related phenomenon in that statistically the balls will always separate, just as all energy is statistically converted to unusable energy over time. But this system is chaotic, which means the paths of the balls have a sensitive dependence on initial conditions. The balls are released at slightly different points, so very slightly different that it looks to be the same to the naked eye.
The green clump represented an outlier clump that had more tendency toward similar behavior over a longer period of time. Eventually, though, the entire system would look completely random.
Well I’m subscribed now
the edge is unreal
We can't predict what a ball will do in a double parabola thing, but we can apparently predict the climate for the next 100 years (a complex and chaotic system).
The prophecy is true!
And that kids, is why you always have to brush your teeth
YES!
The dance of chaos
3:09 great timing with the music
A catenary curve would be a good candidate for a similar animation
when your balls dropped on a double well
this is what happens when confetti is brought to life
Yo sick dude my balls dropped at Olive Garden💀
I was stoked when that first ball got over the hump.
These are the 2 iron balls from yesterday which are short thats why the ball didnt travel into the hands
I really just watched a 4+ minute video on balls dropping.
Imagine a boomerang roller coaster based on this. That hill in the middle would produce some WICKED airtime!
I like how at the top after a bit of the 1000 balls it seems like tuere is an invisible barrier
You start to see it at 3:33
I bet I'm not the only person who cheered out loud at 0:33
I wonder how this will look if the balls are made so that they can bounce with each other too. That would be total chaos.
running in reverse time would be very spectacular.
How do you calculate the collisions? Id love to see an explanation video or tutorial or whatever else on how you achieve these simulations as they are super neat
its all done by counting a derivative in x point and mirroring an angle
How about intensity of the rebound? They seem to always bounce up to a fixed height..
@@radioforthebirds That's just the natural way they would bounce if there was no loss of momentum or kinetic energy due to friction or distortion. Since those other realistic factors haven't been simulated here, they always bounce up to the height from which they were originally dropped.
It's a good thing that friction isn't taken into consideration here, because then the video would be boring. Eventually, all the balls would wind up on the bottom, and they'd quickly lose any opportunity to get over the hill in the middle.
@@PhilBagels OTOH the rotational momentum transferred to the balls would give an interesting additional aspect. You could still do it energetically ideal.
@@HotelPapa100:
Yeah, zero torque makes this less interesting.
I want this graphed 3D for each frame with the third dimension being time :D
then we can rotate the camera angle
What about computing a location probability map? The probability of the ball being in any point of the surface?
Me: If he gets to that other well... Everyone is dead.
Ball: *Immediately goes in the other well*
Me: ...Well S#!t
1:10 i like how the balls just slide on it instead of bouncing
The bacterica inside of a tooth be like:
Imagine for April fool's next year it's 1000 balls in a square
w
well,
Oh, my heart broke a little when the 2 balls were separated :(
I'd love to see this but with each of the 1000 balls having a very slightly different mass and starting from the same position/same velocity
I can hear snippets of Dear Prudence buried in the music
Love the amongus legs!
Help me understand. These balls seem to have infinite bounce, are we presuming zero friction ? What other presumptions ?
Can you do it with the ball losing energy
This is assuming the collisions are completely elastic.
Why did the green balls stay together for so long?
3:10 nice timing
The plot thickens
really want to see balls dropped on a catenary curve y=cosh(x)
The 1000 balls by the end is acting like a fluid
3:13 the music was a bit late, if it were earlier then it would perfectly sync with the balls
It feels like the music is synced to the balls
This would be great at a Dentist office
🦷🥶Tooth Sensitivity Pain Visualized 😂🤣
Cavities forming in real time
At 1:42 or so we have officially achieved chaotic motion
tbh i want to see a website with 1000 balls in a random curve that occasionally changes
the more you watch the more they look like patrick star's pants
Cavities otw to my nerve:
2:51 goes so hard
water but each particle is almost weightless and super bouncy
nah whose molar did they do this in
Balls
This is the perfect illustration of why all the air never randomly bunches up in one end of a room for a moment. Just imagine if the particles interacted with each other as well as the surface.
So I found the scientific name for a tooth shape. :D
whoa my double well curve tastes like colors 2:47
Hey, i tried to put (~ x^4 - x^2) into a graphing calculator but an error said something needs to be on both sides of ~.
love your vids btw
Sorry, the ~ symbol just means that the polynomial is "proportional to"... The true polynomial used here is x^2 - 3x^2. Thus, just remove ~.
It’s interesting how there’s a height limit so to speak
Where does the ball get it's energy from?