The Catenary

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Catenary is the curve formed by a hanging chain or flexible cable, as in this photo:

Catenary ropes

The first person to wonder what mathematical function that a catenary takes on was Galileo. He guessed, quite reasonably, that the catenary formed a parabola. Had he tested his assumption by experiment, he would have found that no parabola can be drawn that will exactly match the hanging chain shape. In any case, the mathematics of Galileo’s time was not up to the task of finding the catenary function – calculus is required for this.

In 1691, the Swiss mathematician  Johann Bernoulli used the newly invented calculus to derive the catenary shape. In modern notation, the function for the catenary is:

\[ y = c \frac{e^{x/c} + e^{-x/c}}{2} \tag{1} \]

where c is a constant to be determined from the chain length and  the location of the end points. Bernoulli’s solution involved one the first uses of a differential equation. The fraction on the right occurs often enough in math and physics that it has been given a name: hyperbolic cosine, or cosh (why it is called that is interesting in itself, but I will not get into that here).

So, mathematicians usually write formula (1) as:

\[ y = c \cdot \cosh{\left(\frac{x}{c}\right)} \tag{2} \]

Obviously, the hyperbolic cosine is the average value of an increasing and a decreasing exponential function:

cosh graph

Arches

Arches allow wide spaces to be spanned without using massive beams. They are particularly efficient with materials like stone and concrete, which are much stronger in compression than in tension. It was the ancient Romans who  first made widespread, systematic use of the arch in construction – neither the Greeks, Chinese, or Sumerians constructed buildings with arches. 

Roman arch, Tebessa, Algeria.

However, virtually all Roman arches were, like the one above, in the shape of a semicircle. In a semi-circular arch, not all of the force is transmitted along the line of the arch; there is a sideways component of the force which tends to cause the arch to buckle. No Roman engineer seems to have ever wondered if some curve other than a circle would work better.

Again, it was Galileo who speculated about what mathematical curve would be best, given that we want the arch’s weight to be transmitted along the line of the arch. Such an arch would not buckle sideways, even it were very thin, because there would be no sideward force to cause the arch to bend and collapse.

Proving again that even a genius can be mistaken, he guessed that the cycloid might be the curve that would do the trick. He was actually doubly wrong – first, the correct shape is the catenary (i.e., the hyperbolic cosine).  Secondly, it must be the same curve, because the arch needs to meet the same requirement as does the hanging chain.

chainVSarch

In the drawing above, both the chain and the arch must transmit a force that is along the path of the curve. The arch is in compression, instead of tension, because gravity is acting on the curve in the opposite sense. The English scientist Robert Hooke realized, in 1671, that the two curves were the same, though this was before the mathematical shape was worked out.

One of the most famous catenary arches is the Gateway Arch in St. Louis:

St Louis Gateway Arch

The formula for the arch is inscribed on its base:

\[ y = -127.7 \, \cosh{\left( \frac{x}{127.7} \right)} + 757.78\:\:\: \textrm{(units are in feet)} \]

Minimal Surfaces

Consider the lines shown in red below:

surface area for catenary
For each line, imagine rotating the line around the x-axis. The line sweeps out a surface, and I have computed the surface area for each.

In 1744, Leonard Euler wondered what shape a line connecting the two points would need to be in order for the surface area to be a minimum. He solved the problem, and, as you will have guessed, the answer is a catenary, like the fourth curve above. This was a non-trivial problem; ordinary calculus is not sufficient to deal with it. Instead, it requires a branch of mathematics called Calculus of Variations, which Euler was one of the discovers of.

These minimal surfaces actually occur in nature, most famously in soap films. Suppose we take two wire rings whose planes are parallel, and dip them into a soap solution. We get a soap film like this:

soap film

Because of surface tension, there is energy stored in the surface of the film. The film forms a shape where this energy is a minimum, which means the surface area is a minimum.

Reference

For those with some knowledge of calculus, the derivation of the catenary equation is not difficult to follow. One good explanation is here:
http://astrowww.phys.uvic.ca/~tatum/classmechs/class18.pdf

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