Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.
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But what are the curves' properties with respect to time when its positioned inverse to gravity? If you turn the curve upside down how do objects act when progressing downward from the center to the edges? Is it also the fastest path? Slowest? Does it have increased structural loading properties due to its unique connection to gravitational forces?
I might be wrong, but I think you missed something at 13:06.
The first term in x(t) shouldnt be R*t, should it? You want to make x(t) an length, not length*time 🤔
Maybe R*omega*t would be correct.
Nevertheless I absolutly enjoy your Videos. Your great! :)
I like this curve and this video. Very good job. I am no mathematician but the tautochrone is most interesting because it alludes to a temporal dynamics. It is not easy to see Time only as the parameter here and space or rather location in space the variable only. Could we not rotate both the space time plane with the time theta into an imaginary or other time dimensions? 2 or more times with variation between? Dimensions of curved time. I see it would imply temporal recurrence . Please forgive my ignorance.
I am NOT a mathematician, so this is probably going to sound stupid, but isn't the relationship you spoke of at the end represented as a straight line because of the constant called you explained earlier?
It has been a while since someone commented. So here's a challenge: How would you prove that blue-pi reaches the midpoint of the curve in the same time t regardless of where he starts? How could you compute t if the diameter of the circle tracing the curve was 1.
You got me thinking about another case: So far we're talking about brachistochrone with a homogeneous gravitational vector field. What would the brachistochrones look like under any arbitrary gravitational vector fields? And can we ever associate the vectors with the brachistochrones, like the vectors put weight literally on a tight string to make a brachistochrone?
Rewatching this, it occured to me that the waves of an earthquake move along curves as well. They dip into the earth before they surface. I have a strong suspicion that these curves might be brachistochrones as well, since the waves move faster the deeper they go.
Does this only work when point B is not somewhere nearly under point A? If point B is kinda below A and only a tine bit to the side then a brachistochrone wont work. How do you know how far to the side B has to be before a brachistochrone will work?
Edit: or maybe a brachistochrone with a huge radius would work? Also how do you know what radius the circle used should be to give the correct brachistochrone? Do you use the full semi circle of a smaller radius circle i do you get a bigger circle and not use all of the semi circle?
Another way to show the linearity: Imagine an "old time" clock with hands.... Put a pen on the second hand so that it would draw a circle in 60 seconds. Now place the clock under a 'strip chart recorder' whose width is the diameter of the clock. The strip chart moves at the same speed as the tangential speed of the second hand at 12 o-clock (i.e. the length of the second hand times the angular rate of change of the second hand).
So in 30 seconds, the clock's hand/pen moves through an angle of pi and goes from one point on top of the strip chart to another point on the bottom. The angle changes linearly with time even though (because of the centripetal acceleration of the pen) the pen draws a curved line (brachistochrone) on the strip chart.
The brachistochrone "maps" an accelerating reference frame (the pen) onto an inertial reference frame (the strip chart). But if you put yourself in the frame of reference of the pen, the brachistochrone maps the fastest way through time onto a (relatively) accelerating reference frame through space (the strip chart).
I am by no means a mathematician and my understanding of physics is rudimentary, but when you started talking about expressing the problem in term of time over theta, space time diagrams kept popping in my head. every curve or motion is actually a straight line in space time. I don't know maybe i am, reaching but i feel like there is a connection here.
I'm troubled, because the challenge seems actually really straightforward in solving just on insight, but I'm a depressive, so at the same time, I don't really think anybody would care if I did just write down what was in my head, at this point. It's probably been done by now, I'm a nobody, and that's a good bit of work for something unlikely to be actually desired anymore, but, uh, I mean, I'm all ears if I'm just being mopey; I could give it a shot, if it matters to anyone.
An answer to your question: the angle of the baristochrone represents where you are on it, and thus it shows ur distance on the curve. The graph can thus be changed to distance in function with time, where every point on the line created represents a point on the baristochrone relative to the starting point A where time is 0 and distance is a int x, and the ending point b. When observing the graph, we see that as distance and time both increase linearly (and thus speed is constant), the line created (which is the distance traveled on the baristochrone) increases linearly as well. This means that this graph of distance in funtion of time effectively turns the problem into “what is the shortest distance between A and B, when speed is constant” the answer is obviously a linear line.
What's the ratio of the length of the distance of a ball falling straight down and hitting the ball rolling, at the middle of an isochronous curve? And then, what's the ratio of acceleration in regards to "gravity" 😂
Does it have something to do with the fact that function minimums occur when derivatives of the function are zero? The second derivative of the linear t-theta space functions would be zero, so maybe these functions are some direct manipulation of the time- minimization curves. This manipulation would have to account for the additional derivative somehow. Just some random and probably incorrect thoughts.
Uhmm I am a bit lost here :D Can someone please explain to me if the velocity of the point moving along the brachistochrone is a constant, or is it changing along the way (assuming only gravity affects the object)?
Solution to the challenge:
Consider an infinitesimally small time interval dt, when the velocity of the pi-creature is v at angle theta with the vertical line.
In this time interval, the pi-creature descends a height of dy = v cos(theta) dt (1).
By conservation of kinetic energy, v^2 = 2gy (y being the total height the pi-creature has descended).
By Snell's law, sin(theta) = Cv (C is a constant).
Take the derivative of both equations above:
2v dv = 2g dy (2)
cos(theta) d(theta) = Cdv (3)
Putting (1) (2) (3) together, we have d(theta) = Cg dt, so the theta-t function is a straight line :D
My attempt at an intuitive answer to why theta(t) = k:
The gravitational field is essentially constant (and downward) close to the surface of the earth. Imagine, in the void of free space, this vertical field and then take a large sphere and give it a nudge along an axis perpendicular to the direction of the field. On this axis, in the absence of any other fields or forces, a straight line path will be the shortest distance between any two points (measured on this axis) as well as the shortest time. Since this observed phenomenon is true for the *large* sphere we must deduce as an a posteriori assumption that it is true for all matter *inside* the sphere as well. If any atom or chunk inside failed to meet this requirement (that it was on the path of least *time*), then it would fail for the object as a whole as well. In the absence of any of the information about forces in the perpendicular direction then the large sphere must move in a straight line from A to B, at a constant speed (this ignores the degenerate case where it is given no initial energy and does not move). Given the geometry of a sphere, the angle made by any surface point as a function of time will be constant.
I’ve heard that another property of this curve is that when you use it to connect points a and b, and point on the curve between a and b, when rolled towards b, with arrive at point b at the same time.
Nature is imbued with this property of doing the most efficient thing, as Bernoulli's solution shows. There was some other new technology I saw recently in which they realised the naturally-occurring way was the most efficient....I remember, it was the best way to merge (and therefore greatly amplify) several laser beams into one, was to focus them all on some jelly-fish protein. It was a Dnews YouTube clip hosted by Trace (who hates jellyfish).
Lol, actually, yes this "is" a shortest path problem. It's just not a shortest path in space but shortest in time. If any sort of shortest path is asked, you could start by drawing a straight line. The axis of time is already given by the question itself, so the next question is: what makes sense as the other axis? Well we need theta there to be able to construct a curve. Then there are plenty of ways from there to find the equation of the curve in the x,y-plane.
This was my first intuition because I've learnt to often swap time and space. They're both just number-lines. As humans we tend to treat things (or each other) differently once we categorize them differently.
What I find beautiful about math is that some of the most elegant solutions just treat numbers for what they are treating them equally.
Its a straight line because the fastest way to get to point a to b in this case is actually a straight line. Since you want to maximize the angle theta with respect to time as this translates to the circle going across the surface the fastest.
Saw your video yesterday, really great :)
*Here is the answer to your challenge*:
particle moves such that speed v is proportional to sin(theta).
Differentiate it: rate of change of change of speed is proportional to cos(theta)* rate of change of theta.
But rate of change of speed is component of gravity along the curve = g cos(theta).
Hence *rate of change of theta = constant*.
I’m not sure about this, but about your challenge I feel like there’s some comparison to energy being both 1/2mv^2 and mgh, where the former compares to the acceleration from gravity on the particle, and the latter compares to the constant rolling speed.
Just like you can have multiple circles passing between two points, I'm curious if the same argument stands for Brachistochrones. Can I have multiple Brachistochrone curves between two fixed points? If yes, then which among those will be the fastest?
What is shortest time between one meter height y=1 and one meter long x=1 distance curve quarter of cycle or 45 degree straight slope exponent of 1? or maybe other exponent k ? it took 5 months to calculate and Newton solved in one day brachistochrone curve, very doubtful :/
Snell's Law in Quantum Mechanics comes from conservation of horizontal momentum (which holds because a collection of parallel transparent media is translation-symmetric); what this *really* means for the brachistochrone, I don't know, but there's something about it echoing in that constant-speed wheel.
Maybe this isn’t a very mathematic way of putting it but my answer tot he challenge is: since the curve is the fastest way to go from A to B, it must have a constant speed. If it didn’t, there would be another way to reshape the curve that would even out the speed to make a faster way from A to B.
Thank you so much for these videos, I lost a night of sleep when I discovered them. Can I use one of the diagrams in this videos in an educative Master's thesis about (general) plane curves? If that's no problem, I will 'force' my audience to subscribe to your channel of course!
Question: if there are 30 students and the teacher gives the 1.student back 30 test results,
Then student n.1 distributes them until he finds his own test result.then he gives the remaining tests to the next student who doesn't has his result jet. This continues until everybody has there result. How often are the remaining results passed on to the next student, on average?
A nice paper assisting in answering the question presented may be found here, https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/1401.2660&ved=0ahUKEwjro6GGk4nYAhWo44MKHRZGD4UQFgggMAA&usg=AOvVaw3F3Vn7Czn8whSGNgsCykyf . This is a paper titled, "The straight line, the catenary, the
brachistochrone, the circle, and Fermat", by Raul Rojas
Freie Universit¨at Berlin
January 2014. The conclusion is awesome.
Congratulations for the great video.. but I didn't get the question at the end; and since my reply is to a 19 months old question.. mm, let's see:
- Since "Y (to the power-2) is constant to T", I don't find it difficult to imagine that it won't take a straight line shape; given T an axis on that diagram.. we are basically converting an exponential graph to logarithmic.
The idea that nature looks at all possible paths and chooses the best one is surprisingly close to Feynman's model of quantum field theory, which was a crucial missing piece in what is now the most well proved theory in history. On that theory, light (and indeed all particles) do in fact take every possible path. But the paths that are close to the optimal path resonate with each other so much that it makes it nearly guaranteed that if measured, light will be observed along that optimal path while the other paths destructively interfere with each other making it nearly zero chance that you will ever see light moving in some odd curve or other non-straight path.
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