I trust you're having a relaxing Sunday? Mathematical physics can be relaxing too, especially when you just look at it. We're just going to look at this one. In fact, this is a literal mathematical instantiation of Sunday relaxation.
If you fix a wire or a rope at two points and let it hang naturally, it forms a shape called a catenary.
It looks a lot like a parabola, and it turns out that in fact as long as the suspension points aren't too close together compared to the length of the rope, it's a very good approximation. For this particular graph the error from a purely parabolic approximation would be nowhere greater than 4%.
What's physically interesting about this curve is that the sum total of the potential energy of each little part of this rope is a minimum. There's no possible way to arrange the curve to have a lower potential energy. Nature does it automatically.
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What does the h stands for?
The function is not Cosine, it is the hyperbolic cosine, abbreviated "cosh".
cosh x = (e^x + e^-x)/2
Is it possible to get latex enable in comments?
Unfortunately I don't have enough access to the site internal code to put in latex myself, but I'll start agitating for the powers-that-be to work on it.
Wow! Now that brings back some old memories of statics class back in college, I always loved how nature tends to do things with such mathematical precision, in the midst of chaotic environment.
That's why The St. Louis Arch closely approximates an inverted catenary.
For details, see:
Weisstein, Eric W., "Catenary." From MathWorld--A Wolfram Web Resource.
Even more interesting, google: "brachistochrone" ...
A looong time ago I was given this wonderful book 'A Book of Curves' by E. H. Lockwood, Cambridge University Press, 1961. I was going to copy you from the 'Tractrix and Catenary' section, but a little Googling found it available online in several formats at the Internet Archive 'http://www.archive.org/details/bookofcurves006299mbp'. So there you have it all, for free no less. Not the same as having the original hard-cover in hand, but nonetheless a fascinating read for this audience.
The emphasis is on actually drawing these curves, not just the mathematics. For me, a play-along read as a teenager with too much time on his hands. So a challenge to program, or to implement in Mathlab, ...
Man, trigonometry makes every simple thing seem complicated. ^_^ Thanks for the response!
"Is it possible to get latex enable in comments?"
I'm all for latex...oh! Wait a minute - wrong blog....
Seriously, what's the difference between a catenary and an arc?!
"What's physically interesting about this curve is that the sum total of the potential energy of each little part of this rope is a minimum."
Are there any static systems that _don't_ settle at minimum potential energy?
Set a block of lead on a block of wood, and the minimum potential would of course be for the blocks to be the other way around. It's still a local minimum in the potential though. Thermodynamics can also complicate things, as for instance if a heavy gas is mixed in a somewhat lighter gas.
Unless i'm taking the wrong approach, finding this function f involves solving for f by minimizing a finite integral, whose integrand involves f and f'. i can't think of an analytical way to solve it right now.
Calculus of Variations! ::Fanfare plays::
I remember this problem from my classical dynamics class.
Just FYI, the differential equation involved is 1+(y')^2=yy". One needs to integrate the gravitational potential energy along the rope and subject the integrand to Euler-Lagrange equation to find the minimal value for the integral (potential). It's a good physics problem.