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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

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

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

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

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:

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

in the continuous limit this gives

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

or, equivalently,

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

A discussion of this idea (along with some others in a similar spirit) can be found in [1].

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

Mark Levi (levi@math.psu.edu) is a professor of mathematics at The Pennsylvania State University.