Exact Solution of the Nonlinear Pendulum [No Approximations, engis gtfo]
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- เผยแพร่เมื่อ 10 พ.ย. 2020
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Elliptic Integral Series Expansion: • Complete Elliptic Inte...
Physics Playlist: • Newtonian mechanics: R...
Today we solve the equation of motion of a free undamped pendulum EXACTLY without small angle approximations. We reduce the order of the nonlinear second order differential equation and proceed to integrate like madlads. We finally arrive at an expression for the period time in terms of the Complete Elliptic Integral of the 1st Kind! :D Enjoy :3 Video sponsored by Brilliant btw! :)
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"I'm not gonna explain the double angle formula" a minute later "we can change the order because addition is commutative"
tWo Is NoT zErO bEcaUsE iT's ThE sUcCeSsOr Of OnE
xDDD
@@PapaFlammy69 we love you
such a good comment lol, keeled over immediately
tbh that the sign of greatness right there.
Mathematicians: "we have an exact solution to this differential equation!"
Also Mathematicians: *invents a new function to deal with an integral they can't simplify*
elliptical integrals go brrr
It's kinda a theme. Same with having sin(x). We don't have a human-calculatable exact expression for that. :( We're so used to f(x)= , but have to do with sin(x)= .
And call them "special functions"
@@JoonasD6 even real number exponentiation is defined as a^b = exp(b log(a)), which is an infinite series. (And all exponentiation like this uses some sort of limit.)
We have an exact solution to the diameter of a circle!
*Invents the character pi*
The easiest method to solve the equation completely :
Let us define a special function £(t) as a solution of this equation. Hence £(t) is the solution of the equation. QED
Initial conditions are always zero. I would expect a mathematician to know that.
It's basically what he did though. That K(k) IS a special function.
You have to prove a solution exists before you can do that
@@edmundsmaths3980 Since the function f(x,y,z)=(-g/l)sin(y) satisfies a Lipschitz condition in every subdomain of R^3 and it is continuous hence a corollary of existence and uniqueness theorem guarantees that there exists a unique solution of the given equation for every initial condition, and which is defined for all x in R ( I am taking x for independent variable and y for dependent variable)
@@davidherrera4837 You are right, and even if it is proved that a given differential equation does not a solution which can be expressed implicitly or explicitly as a combination of elementary functions we only do it in some very specific cases. Some of the reasons are - (1). The new function appears very frequently in other important areas. (2). The new function has some very interesting properties (3) There exists a representation of the function which makes it easily to do calculations with it etc Otherwise defining a new special function is pointless.
Engineers: haha simulation goes brrrrrrrr
let dt = 0.01
@@beneze3286 0.001 is absolute minimum tho?
@@missquprison yep. To quote Einstein: "any more than 1e-3 and you're not really gaining anything and to be honest 1e-2 is prolly alright"
Einstein didnt have computers that would do calculations in less than a seconds.
@@missquprison oh shit , my bad. James C Maxwell
As an engineer I have a really cool way to do this... Simulink
xD
"exact approximation"
yeye
4:55 „We can multiply both sides by two, because it's ≠0, because it's the successor of one“
*cries in characteristic 2*
LOL
In characteristic 2 there's no notion of a "successor", because a field with a finite characteristic cannot be ordered
0:13
Solves differential equations, but unable to read time so says 8 am
🤣🤣🤣
It is 8 am, if you're an engineer.
Insert Identity Here If you are an astronomer it is half past Tuesday week
JAJAJAJAJA
Ahhh, memories.... solved this equation in the late 90s, but haven't really used any advanced math since I graduated 20 years ago. Its really a sting if you know you once were able to do this, but can't remember how.
oh boi ^^'
If 20 years from now I never had to use this sort of math, I'd be ecstatic.
Right in the feels....
Preach it. I had a really full toolbox of math. Sadly they have all rusted solid to the point of being unrecognizable. Now I have a new toolbox filled with medicine. Getting old sucks.
"Exact solution of the nonlinear pendulum"
engies: yeehaw
"engies gtfo"
engies: *Genuine Anger*
there is no exact solution at all, because no closed solution exists for the complete elliptic integral of the first kind
@@kurtmandelbrot8485, i am not exactly a mathematician (we've just started integrals at school) and this is just a TF2 joke, but i've had one case of finding no closed form solutions (indefinite integral of x^x - not a school assignment, just for fun). The 2 things I have in my arsenal - integration by parts and substitution (thanks handbook) - did nothing. I looked it up and apparently there are no closed form solutions. Interesting stuff, I'll try to learn some more by myself
@@kurtmandelbrot8485 What do you mean? The complete elliptic integral of the first kind IS the solution.
@@MrAlRats yes it is a solution, but impossible to solve without numerical methods, and numerical methods are approximations
@@kurtmandelbrot8485 This is Mathematical problem not engineering. To design a bridge you dont need that. We learned the exact solutions in a course of theoretical mechanics at University.
I love when a differential equation has an exact solution especially one that describes real world phenomena. Super cool, thanks for sharing!
Finally! Elliptic integral! Yeah!
This is the one I waited for a long time!
:)) Finally got around doing it! =D
Same..me too .
Finally papa did it
Me: *reads the title
Also me: *sees “No approximations”
Me once again: *C O N F U S E D S C R E A M I N G*
easier way: use the well known result from the fundamental theorem that sin(φ)=φ for all φ. :)))
Fundamental theorem of engineering
That's exactly what i was doing, when he uploaded the video i was working on this question for my physics problem sheet😂
Not all, only small angles.
@@adriancordones all angles are small
@@adriancordones you need to say the magic words first: "let phi be small."
So cool! I just covered this in my physics class for the simple approximation with calculus, so it’s really nice to see the actual differential equation as a supplement! Nice work 🔥
This video is basically: How to find an exact solution to the nonlinear pendulum? Forget to find a function that solves the differential equation and just determine one constant!
Every explanation I found on Google does the same thing, they just find the period with respect to the length and that’s it. Maybe it’s obvious how that relates to theta(t) idk.
@@Reliquancy theta(t) = theta_0 * cos(2t*pi/T)
@@SpeakMouthWords, the exact solution is Jacobi amplitude function.
@@user-bw5xk9xv3w Of course, the sinusoidal solution would only be small angle - my mistake. What do you say to the guy above saying there's no closed form solution?
@@SpeakMouthWords probably he meant, that there is no solution in elementary functions. Of course the Jacobi amplitude is a closed form solution, strictly speaking.
0:13 It is 8 AM, with some error.
*GREAT VIDEO! I CAN'T WAIT FOR THE NEXT VIDEO ON THE SOLUTION TO THE ELLIPTIC INTEGRAL OF THE FIRST KIND!!*
It's in cap and all bold, cause it's that great. I can't wait. This is something I've been waiting for for 3 years.
by changing the second order equation to a first order one, you have actually transformed the newton's second law to conversion of energy.
there is also a useful identity (or lemma) which is the same thing as you did and looks a little nicer in my opinion: d/dφ[[dφ/dt]²]=2[d²φ/dt²]
one of my teachers used to call this "lemma one" cause he used it a lot.
loved the video!
How did you write mathematical expression and superscript in TH-cam comment?
@@aniksamiurrahman6365 Superscripts actually exist as plain characters in the unicode standard. My phone's keyboard has shortcuts to them by long pressing a number, ²³⁴. Also some simple fractions, ½ ⅕ ⅔.
Typing them on a computer seems more tricky for some reason. Alt-codes is an old and primitve method. You can access special characters by holding down the alt key and typing some number combination on the numpad (at least on windows), but it requires you memorize a bunch of number combinations. I don't know if there are any simple virtual keyboards that let you define your own shortcuts. There really should be!
I'm not sure what else you mean by mathematical expression. If it's the φ that's just a greek letter. You can install languages and change the keyboard layout on any regular computer. On windows the shortcut to change layout is alt+shift.
@@aniksamiurrahman6365 ² is a Unicode character.
@@aniksamiurrahman6365 Should be on a phone.
Me: Boy, I really need to do some research on Dirac's equation for this term paper that's coming up.
*5 minutes later*
Me: Oh hey, a 20-minute math video that's completely unrelated. Is he gonna do the phi dot trick? Yeah, he's gonna do the phi dot trick.
This is hands down one the most beautiful things i ve seen in this website , so beautiful and elegant ❤️
Bless Papa
All hail the superior emperor of mathematical anal
This is probably the first time I thought about trying the solution in the general case, it worked.. thanks for motivating me to do so !
Now try adding some dissipation.
Excellent work! I only vaguely remember covering the exact solution at uni many years ago. This is a fantastic explanation that I would have crawled over broken glass to see.
Finally I can see it here. Cool
I am waiting for expending the eliptic integral. Best regards
exactly what i need, i'm wondering why the velocity derived from the energy equation is different from the velocity i get from differential equation and you're my savior this time XDDD definitely subcribe
I will never stop being uncomfortable with everybody in the comment section calling him "papa" lol.
:D
Whenever you read the words papa flammy, think of a papa smurf that can spit fire while doing math. Did you laugh ?
@@SjS_blue Yes
Nice one! Looking forward to the video about the expansion
Excellent! There are also explicit solutions of angle over time with Jacobian elliptic functions, equivalent differential equations for physical pendulum.
Yes, I’ve been waiting for this.
Beautiful solution. That was a work of art 👌
Just turned on mobile data. Papa flammy's notification was the first thing to pop up...
God wills it.
This is the way.
Can you make a video where you renormalize the path integral of the Einstin-Hilbert action? My first graders ask me everyday about it.
The solution of the elliptic integral is "exact" in the same spirit that the value of pi is exact, so the title of the video is justified. Nice job and keep it up!
I love diff eq videos, especially ones with physics motivation
I remember 3b1b mentioned this in his differential equations series and just writing down the equations took up the entire screen. This is going to be a fun one.
Request: Solve the system of non-linear differential equations describing the double pendulum. :)
This is amazing.
Would you mind trying to solve the non-homogeneous case of this problem as well? I guess we can just assume that the right hand side of your initial second order differential equation to be some constant F for example.
Anyways, great job!
"I like your funny words magic man"
Precisly what I wanted. Huge tks.
me: an engineer, likes this video, reads the title afterward, feels personally attacked, keeps the like cause the video is dope!
:DDD
Very cool video 👏.
It would be amazing to solve the same equation but taking into account the friction with the air
This is one of those “exercise left to the reader type situations” isn’t it.
Yeah it's one of "those" things😂😂
The authors just assume that you know all these stuff because *obviously* everyone took an advanced pure math course and got a PhD in it before becoming a physics major right?
@@shayanmoosavi9139 In fact "it gets so ugly I always get lost in it, so I'm not touching it. You're on your own."
This inthegeral is therivial!
Finally!!! I always wanted to know what the exact solution to this equation was :)
Me too I couldn’t find it anywhere. I wish this video came out when I was writing my honors math paper on Pendulums, last year.
Wow, the complete elliptic integral of the first kind. Thanks papa flammy.
Ich bin so dankbar für dieses Video, ich bin im zweiten Semester und diese Differentialgleichung ist da in einer Übungsaufgabe und wir sind alle völlig verloren gewesen QwQ
This is how a PRO teaches, this is all I've been waiting for,...
algebras, calculus, and now....Physics(well of course the language of physics is mathematics) so mathematician can be possibly learn physics,
Nice one papa flammy, that was really really EXCELLENT and BRILLIANT 😮👏👏👏 round of applause for Papa flammy
Well all the great physicists were also mathematicians. For example Newton, Laplace, Lagrange, Gauss, Euler, Fermat, etc.
@@shayanmoosavi9139 well if we are great at mathematics, we can also now explore the world of physics
@@unknownpalooza8475 yeah you definitely need to learn physics. You'll be blown away to see all of the abstract stuff that you learned in action :)
@@shayanmoosavi9139 yes you're definitely right :)
Really love these Physics stuff
*Summary*
- *The meat of the argument is that first step* : multiply through by derivatives that allow you to convert the existing ones to a form that makes the whole equation one big derivative wrt the independent variable... then do the *first integration*
- after that, the *second integration* can be done by the usual methods: separate the variables and definite integrate both sides...
The straight up madman!
Engineering student here. I actually like this kind of thing.
The prophecy was true ! There is a chosen one among the egineering students... ! It is said that he will be the first to ask himself where things come from and be tired of look-up tables..... It's time for the king to rise !
Bro, do you even brrrrr?
This is interesting because when doing the approximation we see that the period of the pendulum is independent of the iniitial angle. But from your formula that doesn't seem to be the case. So when one considers it mathematically, one of the core principles of physics is wrong.
...when did it become a core principle of physics?
Great video. Thank you
7:47 that was incredibly smooth!
Gran finale!👍🏻👍🏻👍🏻
You rock, Flammy!
I love the amount of chaotic energy in this video.
Brilliant derivation. Enjoyed it.
thx
That intro meme is really reaching back into the dusty recesses of my memory
For those who doesn't like special functions, you can see that 1/sqrt(1-k²sin²(v)) converges for |k|
Why am I watching this in my freetime? And why can I follow this mans explanations? What’s wrong with my life?
Me: *reads the title of the video*
Also me: THE FORBIDDEN OSCILLATOR!
A nice trick to look at is that If you have phi . initial /=/ 0, you can use conersvaition of energy to find the maximum phi to get to phi. =0. Then the time it gets to return to that place... Just shift your solution by that as the updward motion should be reflectice of downward motion.
(If it doesn't have enough energy to go in circles).
Thank youuuu very much
You are my savior!!! Heavenly thanks 🙏🏼
Love it!
This is the vídeo I was looking for
Thanks!
I actually suggested this in one ur video comments. What i suggested was about the period of the mathematical pendulum when ur displacement angle pi/2. The solition for this is expressed by betta function which is interesting too
Its been a while since I was taught calculus (trying to reteach it to myself), but at 16:18 when you "let sin(φ/2) = k sin(θ)".
When is this technique valid?
The range (for real numbers at least) of the sine function is -1 to 1. So would this just be a valid technique if k was between [-1,1]?
You used k twice to represent two different values so at first I thought k take any value, but I see now it will be between [-1,1], but you should still check before making that type of substitution right?
But if k was allowed to be any value, but that just affect my domain for the solution (like it would only be valid for when ksin(θ) was between [-1,1])?
You are a fantastic lecturer!
thank you
I’ll stick with the small angle approximation
:D
Then just do what all practical people do and use the universal replacements that work for all the right equations:
∫=∑, ∞=1e6, π=3.14, dt=1e-3.
This is enough even for the so called special functions that are just shortcuts for their simple formulas. It is enough to model everything you need.
If the replacements do not work or if an equation has any independant variables apart from "t" then it is a clear indication that the equation is just wrong to begin with and is not even worth solving. And don't listen to any mathematicians - they are all crazy.
@@Kirillissimus square root of -5? That doesn't exist! And don't come here with your special ι or whatever, that's cheating! If you can say they exist you just have to multiply by this "imaginary number" then i can also say that π/shit=0 so long as you let this "shit" be an "imaginary number"
@@xXJ4FARGAMERXx
This one is simple: π/∞=3.14/1e6=3.14e-6 < dt --> π/∞=0. Some "magic numbers" are here for a reason. One just has to remember that magic does not really exist.
As for the so called "imaginary numbers" - don't even get me started. They are just a misleding way to define circular rotation, an extra dimension and vector operations all thoroughly mixed togehter and seasoned with a tiny drop of pure insanity. Mathematicians only love them because of the compact notation and today with digital computers available everywhere to keep everything together automagically and to shuffle all the mess around you don't really need any of that nonsense.
@@Kirillissimus (didn't expect you to reply, "a surprise to be sure, but a welcome one.")
Wow, im studing this for the final paper of my course.
what a beaultiful and elegant solution
Not to be offensive but, as an ex-mathematician(?) watching this video feels like
Papa: *explains one line so fast*
Me: *not getting it*
Papa: *writes the next line*
Me: *immediately understands whats going on* oh it's actually just that
Everyone else: "x is 2"
Jens: "x IS NOTHING BUT 2"
Your physics intuitive thinking is correct ! Why not using 'elliptic substitution' to confirm directly into the nonlinear second order ordinary differential equation ? A very good trial !
I really should get around to learn trig identities by heart, huh.
is it possible to derive a formula for the angle phi as a function of time t explicitly?
is there a way to find a function for the motion, velocity and acceleration from here?
why 86 dislikes this guy does nice... well done bruh greetings from colombia u r the greatest
Great video man 😀, and it is very clear I am 12 years old and I was learning that to finish physics college lesson.
Let's imagine a pendulum swings so fast that it does full 360 degree rotations. In this case there is no point at which the velocity is zero so do your initial conditions not exclude this scenario?
it can only do a 360 degree rotation if you give it a push initially but this problem assumes that at the begining the only force acting is gravity
This video was brilliant.
Hello. What is the physical interpretation of second kind of elliptic integral? And also when both elliptic integrals (I mean first and second kind) are in the same asymptotic equation? For what physical situations we can see them?
every analyst, any country: "this is nothing, but.."
That's such a chud special function. I remember in a PDE course a friend had to skip a couple of classes due to drug stuff, and when he resume his regular schedule, the professor had introduced elliptical integrals and shit. My friend's face was the best things that happened to me during the undergrad days.
:D
For those of you confused by the part where he discusses the bounds of integration and where the period comes into play, don't worry, I was too. On the one side you have bounds of the form t1,t2 but on the other you have bounds of the form φ(t1),φ(t2). This is the part where, if you go about it a different way (play around with fixing those bounds in different ways and you'll see what I mean) you'll come across the problem of trying to invert that elliptic integral as a function of the upper bound to get φ(t), which will lead you to the definition of the Jacobi amplitude function am(u,k). You can then write φ(t) out explicitly in terms of the amplitude function and some sine and inverse sine functions. I'll pull a 21st century Fermat here and say "The solution is too long to here include in unicode"
Now add friction😎
Love this vid papa flammy keep up the good work
And also non-constant g :))
It's easier to solve with friction
Also take into account the heat generated into the pendulum because of the friction. Note that the heat changes over time ... Thermodynamics problem 😎
Watched this thinking all the time there's no way this bad boii ain't getting an elliptical integral.
Isn't this just the mathematician way of sweeping under the carpet?
That's some high quality content
"Brilliant like our sponsor Brilliant." Brilliant sponsor break papa.
So, I was going through these calculations for a related problem, and I noticed you can also try to integrate twice 1/sin x, giving you dilogarithms that simplify into a sum of odd numbered exponents and some other stuff. It doesn't seem to yield any practical expression since some x is stuck inside sines and logs. Is there a way to do it via this route anyway? Did I miss something? I prefer having a simple infinite series (dilogs are extremely easy to write as infinite series) because then I can plug it into python without needing some magic functions
I have a small question; why do you use a partial derivative wrt time? phi is a function of time, this is an ODE rather than a PDE correct? is it the same or am I missing sth?
There is a way more elegant solution set out in Ian Stewart’s book “Does God Play Dice in a chapter titled “The One Way Pendulum”. Conservation of energy means that potential energy plus kinetic energy equals a constant. The solution then drops out using basic trig functions. Stewart goes on to consider a pendulum with so much energy it goes round and round in one direction ( a propellor), for which there is a neat model in phase space that looks like a U-bend in a toilet. Well worth reading.
He "uses conservation of energy": "phi dot squared" is just 2/m times kinetic energy.
Finally, elliptic integarahl!
That is useful for Physics in k and movement of pendulum
Thanks, finally.
Finally an *exacc* video.
Im using this video for reference for home experiment project for training in selection process for APhO team :)