Built on Facts

Sunday Function

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.


  1. #1 IBY
    August 10, 2008

    What does the h stands for?

  2. #2 Palmer
    August 10, 2008

    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?

  3. #3 Matt Springer
    August 10, 2008

    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.

  4. #4 Ken Clark
    August 10, 2008

    Wow! Now that brings back some old memories of staticís class back in college, I always loved how nature tends to do things with such mathematical precision, in the midst of chaotic environment.

  5. #5 Jonathan Vos Post
    August 11, 2008

    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” …

  6. #6 GrayGaffer
    August 11, 2008

    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, …

  7. #7 IBY
    August 11, 2008

    Man, trigonometry makes every simple thing seem complicated. ^_^ Thanks for the response!

  8. #8 Ian
    August 11, 2008

    “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?!

  9. #9 Tom Jackson
    August 11, 2008

    “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?

  10. #10 Matt Springer
    August 11, 2008

    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.

  11. #11 venkat
    August 12, 2008

    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.

  12. #12 Chris Hertlein
    August 12, 2008

    Calculus of Variations! ::Fanfare plays::

    I remember this problem from my classical dynamics class.

  13. #13 Anonymous
    March 20, 2009

    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.

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