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A Minimalist Minimizes an Integral

By Mark Levi

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 [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. 

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