Soap bubbles and films have always fascinated children and
adults, but they can also serve to solve complex mathematical
calculations. This is shown by a study carried out by two
professors at the University of Málaga, who have succeeded
in solving classic problems using just such an innovative
procedure.
"With the aid of soap films we have solved variational
mathematical problems, which appear in the formulation of many
physical problems", explains Carlos Criado, professor at the
University of Málaga, speaking to SINC. Together with his
colleague Nieves Álamo, he has just published his work in
the American Journal of Physics.
Soap films always adopt the shape which minimises their elastic
energy, and therefore their area, so that they turn out to be ideal
in the calculus of variations, "where we look for a function that
minimises a certain quantity (depending on the function)", adds the
researcher.
"Of course there are other ways to solve variational problems,
but it turns out to be surprising, fun and educative to obtain soap
films in the shape of brachistochrones, catenaries and
semicircles", Criado emphasises.
The professor offers the example of the famous problem of the
brachistochrone curve. What shape must a wire be in order that a
ball travels down it from one end to the other (at a different
height) as rapidly as possible? The answer is the brachistochrone
(from the Greek brachistos, the shortest, and cronos, time), the
curve of fastest descent.
New methods for old problems
The mathematician Johann Bernoulli found the answer centuries
ago when he realised that it was a cycloid (the curve described by
a point on a circle rolling along a line). That was the origin of
the calculus of variations, which was also used in other classic
problems, like that of the catenary (the shape of a chain suspended
by its endpoints) and the isoperimetric curve (a curve which
maximises the area it encloses).
The study shows that these calculations may be related to
Plateau's problem, that is, to find the shape adopted by a soap
film under certain boundary restrictions. Besides, the researchers
show how to design the experiments, constraining the soap films
between two surfaces in such a way as to obtain the appropriate
curves.
SOURCE
