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# A Minimalist Minimizes an Integral

In this issue we present a solution that is shorter than Johann Bernoulli’s famous optics-based idea of minimizing

$\begin{equation}\tag{1} \int_{\gamma OA}F(y)ds \end{equation}$

over smooth curves connecting two given points $$A$$ and $$B$$; here $$F(y) > 0$$ is a given function and $$ds$$ is an element of arc length. Bernoulli based his beautiful solution on the equivalence between Fermat’s principle and Snell’s law.

The following solution, in addition to being shorter, substitutes a mechanical analogy for Bernoulli’s optical one – and thus could have been given by Archimedes.

The sum

$\begin{equation} P_N = \sum F(y_k)\Delta s_k \\ \approx \int_{\gamma AB}F(y) ds \end{equation}$

can be interpreted mechanically as the potential energy of the system of rings and springs shown in Figure 1. Each of the $$N$$ rings slides without friction on its own line; the neighboring rings are coupled by constant-tension springs whose tensions are given by the discretized values of $$F_k = F(y_k)$$. If $$P_N$$ is minimal, each ring is in equilibrium, implying the balance of horizontal forces on the ring:

$\begin{equation} F(y_k)\mathrm{sin}~\theta_k = F(y_{k+1})\mathrm{sin}~\theta_{k+1}, \qquad k=1 \ldots N; \end{equation}$

in the continuous limit this gives

$\begin{equation} F(y) \mathrm{sin}~\theta = \mathrm{constant}, \end{equation}$

or, equivalently,

$\begin{equation} \frac{F(y)}{\sqrt{1+(y')^2}} = \mathrm{constant}. \end{equation}$

A discussion of this idea (along with some others in a similar spirit) can be found in . Figure 1. Each spring has a prescribed tension Fk independent of its length Δsk. The endpoints A and B are held fixed.

Stay tuned: A related topic is explored in the next Mathematical Curiosities column (July/August issue of SIAM News).

Acknowledgments: The work from which these columns are drawn is funded by NSF grant DMS-1412542.

References
 M. Levi, Classical Mechanics with Calculus of Variations and Optimal Control, AMS, Providence, Rhode Island, 2014.

Mark Levi ([email protected]) is a professor of mathematics at The Pennsylvania State University.