The Cauchy-Schwarz Inequality via Springs
By Mark Levi
Figure 1. Potential energy decreases as the system relaxes to the equilibrium. This is the Cauchy-Schwarz inequality for Hookean springs and the Holder’s inequality for “polynomial” springs with \(F= k L ^p\). Figure courtesy of Mark Levi.
Here I present a different physical implementation of the idea in
[1] and [2]; the mathematical portion is exactly the same except for the notations, but I still present it for the sake of self-sufficiency. Let us connect \(n\) springs end-to-end, as shown in Figure 1 for \(n=3\). Initially, we forcibly hold the connections at some arbitrary positions. Then we release them and let the system settle into the equilibrium configuration. In the process, potential energy decreases:
\[P_{\rm old}\geq P_{\rm new}.\tag1\]
The equality holds if and only if the system was already in equilibrium at the outset. I claim that \((1)\) is the Cauchy-Schwarz inequality (in disguise) if the springs are Hookean, i.e., if the tension of the \(i\)th spring is in direct proportion to its length: \(F_i=k_i L_i\). Indeed, since a Hookean spring’s potential energy is \(\frac{1}{2} k L ^2 = \frac{1}{2} \lambda F ^2\)—where \(\lambda = k ^{-1}\) is the spring’s “laxness”—\((1)\) amounts to
\[\sum \lambda_i F_i ^2\geq {\overline F} ^2 \sum \lambda_i,\tag2\]
where \(\overline F\) is the common tension of all the springs when the system is in equilibrium. But
\[{\overline F} = \frac{\sum \lambda_i F_i}{\sum \lambda_i}\]
is the weighted average; I omit the verification of this fact, which involves showing that “laxnesses” add for springs connected in series.1 Substituting this expression into \((2)\) gives
\[\sum \lambda_i \sum \lambda _i F_i ^2 \geq \biggl( \sum \lambda_i F_i\biggl) ^2.\]
By setting \(\lambda_i = x_i ^2\) and \(\lambda_i F_i ^2 = y_i ^2\), we get the familiar form of the Cauchy-Schwarz inequality. And if the springs are non-Hookean, with \(F = k L ^p\) and \(p \neq 1\), then \((1)\) amounts to a Holder inequality via an essentially verbatim repetition of the argument in [3].
1 This is an analog of resistance additivity in electric circuits.
References
[1] Levi, M. (2019, November). The Cauchy-Schwarz inequality and a paradox/puzzle. SIAM News, 52(9), p. 11.
[2] Levi, M. (2020). A water-based proof of the Cauchy-Schwartz inequality. Am. Math. Month., 127(6), 572.
[3] Levi, M., & Tokieda, T. (2020). A communicating-vessels proof of Holder’s inequality. Am. Math. Month. In press.
Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.