Isochronous Curves
en.wikipedia.org
Isochronous Curves
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Re: Isochronous Curves
#2Variation calculus is a great lens through which to look at many physics and CS problems. It is IMO under-taught and under-utilized, especially in the machine learning discipline.
The framework is basically minimization in functional space (as opposed to R^n or a subset thereof, the more common case).
The (to me) surprising thing is that finding an extremum in functional space (i.e. infinite-dimensional space) can be reduced to solving a differential equation, via Euler–Lagrange.
The first problem feels intractable, while the second is "just" numerical integration.
Re: Isochronous Curves
#3Adam Savage builds a brachistochrone curve, which is the same curve as an Isochronous, just with potentially different starting/ending points, with Vsauce's Michael Stevens. It's a pretty bizarre phenomenon: https://youtu.be/skvnj67YGmw
Re: Isochronous Curves
#4Is this off the back of a video of a 1970s Open University style presenter showing the properties of isochromus curves - i loved watching it and realised that these are national treasures of programs - and also was quite stunned by the assumption one would have a round empty tobacco tin lying around to do the experiment with - times do chnage :-)
Re: Isochronous Curves
#5Is this off the back of a video of a 1970s Open University style presenter showing the properties of isochromus curves - i loved watching it and realised that these are national treasures of programs - and also was quite stunned by the assumption one would have a round empty tobacco tin lying around to do the experiment with - times do chnage :-)
I suspect it is... It turned up in my YouTube suggestions yesterday. Here's a link:
Re: Isochronous Curves
#6Variation calculus is a great lens through which to look at many physics and CS problems. It is IMO under-taught and under-utilized, especially in the machine learning discipline. The framework is basically minimization in functional space (as opposed to R^n or a subset thereof, the more common case). The (to me) surprising thing is that finding an extremum in functional space (i.e. infinite-dimensional space) can be…
It's funny how different backgrounds bring you to having different ideas of what is easy and hard. As a mathematician, to me the easy problem is finding the minimum of the functional (where "easy" means "we can at least try", certainly not "trivial"; some minimum problems are actually relatively easy, some others are very difficult and open), while proving existence of PDEs is exactly why calculus of variations (and many other theories) were invented in the first place.
Re: Isochronous Curves
#7Is this off the back of a video of a 1970s Open University style presenter showing the properties of isochromus curves - i loved watching it and realised that these are national treasures of programs - and also was quite stunned by the assumption one would have a round empty tobacco tin lying around to do the experiment with - times do chnage :-)
I suspect it is... It turned up in my YouTube suggestions yesterday. Here's a link: https://youtu.be/eBc827pwKf0
It came up on my suggestions few hours ago.
Re: Isochronous Curves
#8Adam Savage builds a brachistochrone curve, which is the same curve as an Isochronous, just with potentially different starting/ending points, with Vsauce's Michael Stevens. It's a pretty bizarre phenomenon: https://youtu.be/skvnj67YGmw
They should build tautochronous pendulums. Or just add a spring bouncer on the end of the tracks so to enjoy multiple simultaneous bounces of different heights.
Re: Isochronous Curves
#9Variation calculus is a great lens through which to look at many physics and CS problems. It is IMO under-taught and under-utilized, especially in the machine learning discipline. The framework is basically minimization in functional space (as opposed to R^n or a subset thereof, the more common case). The (to me) surprising thing is that finding an extremum in functional space (i.e. infinite-dimensional space) can be…
I remember a physics problem where teacher explained that a ball on different paths, even if the height differences would sum to zero would arrive at different times. I often think about this when planning car drives because I believe you could make better use of gravity and use less fuel.
Re: Isochronous Curves
#10Variation calculus is a great lens through which to look at many physics and CS problems. It is IMO under-taught and under-utilized, especially in the machine learning discipline. The framework is basically minimization in functional space (as opposed to R^n or a subset thereof, the more common case). The (to me) surprising thing is that finding an extremum in functional space (i.e. infinite-dimensional space) can be…
It's funny how different backgrounds bring you to having different ideas of what is easy and hard. As a mathematician, to me the easy problem is finding the minimum of the functional (where "easy" means "we can at least try", certainly not "trivial"; some minimum problems are actually relatively easy, some others are very difficult and open), while proving existence of PDEs is exactly why calculus of variations (and…
I didn't mean to imply that PDE's were an easy problem.
They clearly aren't, especially when it comes to proving some sort of formal property about them (which is something a mathematician would worry about, but which is rarely an engineer's first concern: they'd only worry about that type of thing when numerical integration starts producing "crazy" results).
However, there is a very intuitive way to compute an approximate solution to a system of PDE's, namely increase time in very tiny steps and solve the resulting system of equations each time, rinse and repeat.
Now, when you deal with arbitrary functions, there are so many ways to "represent" them: as various kind of series, as algebraic composition of elementary functions, as solutions to implicit equations, as solutions to PDE's, as polynomial approximations, as neural networks, etc ...
These representations are usually dense in "regular" functional spaces and you do get a lot of mileage out of them.
Whichever way you chose to represent functions in your functional space, the problem of finding an extrema over that representation is very much not obvious in the calculus of variation setting (or even that the extremum will be itself be representable, for that matter), whereas the conversion to a system of PDE gives you a clear path to victory - as long as the representation can be differentiated.