SIAM News Blog
SIAM News

# Lagrange Multiplier as Depth or Pressure

With bicycle season beginning in the Northeastern U.S., I would like to describe a small bike-related observation. It actually has nothing to do with a bicycle’s mechanics; it simply occurred to me when I was riding my bike last fall. While climbing up a steep incline and observing a stream by the roadside, I asked myself the following question: The water in any collection of connected vessels settles to the state of least potential energy; what is a mathematical expression of this obvious fact? For cylindrical vessels it turned out to be the Cauchy-Schwarz inequality, as described in the November 2019 issue of SIAM News (and in [2]). For polynomially tapered vessels, the expression yields Hölder’s inequality [3].

These inequalities are therefore special cases of what every child knows: water levels equalize in communicating vessels. What other theorems are hiding behind this simple fact? In this month’s column I provide one simple consequence; it would be interesting to discover more.

Incidentally, all of this—the Cauchy-Schwarz and Hölder inequalities, as well as the observation below—are, in the final analysis, consequences of the law of conservation of energy. Indeed, assume for a moment that water in communicating vessels settles at different levels. Then build a trough from the higher level to the lower one. The water will flow down this trough, and forever so due to the assumption, providing a free source of energy — a contradiction proving that the water settles at the same level, and also that this level minimizes potential energy.

As an aside, quite a few other geometrical theorems result from the impossibility of the perpetual motion machine [1].

### Problem 1

Here is another problem that can be solved by the communicating vessels idea.

Given $$n$$ functions $$f_k : {\mathbb R} _+ \rightarrow {\mathbb R}_+$$, $$k=1, \ldots n$$, minimize the sum

$F(x_1, \ldots , x_n ) = \sum_{k=1}^n\int_{0}^{x_k} x f_k (x) dx , \tag1$

subject to the constraint

$G(x_1, \ldots , x_n ) = \sum_{k=1}^n\int_{0}^{x_k} f_k (x) dx = 1. \tag2$

To interpret this problem1 physically, imagine $$n$$ vessels (as in Figure 1) with valves closed and the $$k$$th vessel filled with water of depth $$x_k$$. The sum $$(1)$$ is thus the system’s total potential energy (we choose the units in which the water density and gravitational accelerations are one unit). And $$(2)$$ prescribes the total volume of water. Now as we open the valves in Figure 1, the potential energy $$F$$ settles to its least value, which also corresponds to equal levels:

$x_k=x_l \ \ \ 1\leq k, l\leq n. \tag3$

The total volume $$G$$ remains unchanged during the redistribution. This solves the problem: the minimizer is given by $$(3)$$ and $$(2)$$.

Figure 1. $$f_k(x)$$ is the area of the horizontal cross-section at height $$x$$ of the $$k$$th vessel. Figure courtesy of Mark Levi.

To verify the answer, the Lagrange multipliers method $$\nabla F = \lambda\nabla G$$ yields

$\require{cancel} x_k \cancel {f(x_k)} = \lambda \cancel {f(x_k)},$

so that $$x_k= \lambda$$ for all $$k=1, \ldots, n$$. As we already know, the levels of water equalize. But we now discover that the Lagrange multiplier $$\lambda$$ is the common water level, or equivalently, the water pressure at the bottom of the vessels

As a side remark, this problem generates the aforementioned inequalities for special choices of $$f_k$$.

### Problem 2 (A Generalization)

Let $$p_k: {\mathbb R} _+ \rightarrow {\mathbb R}_+$$, $$1\leq k \leq n$$ be monotone increasing functions, and let $$f_k$$ be as it was before. Minimize

$F_1( {\bf{x}} ) = \sum_{k=1}^n\int_{0}^{x_k} p_k(x) f_k (x) dx,$

subject to the same previous constraint $$(2)$$. The Lagrange multiplier method $$\nabla F_1 = \lambda\nabla G$$ produces

$p_k(x_k) = \lambda.$

I leave it as a puzzle to build a thought-experimental “analog computer” that results in this answer and gives a physical interpretation of $$\lambda$$.

1 To be more precise, we must assume that $$f_k$$ are such that the constraint $$(2)$$ is even satisfiable. I also probably should have said in fine print that $$f_k \in L^1$$, but I’ll leave out these distracting details.

References
[1] Levi, M. (2009). The mathematical mechanic: Using physical reasoning to solve problems. Princeton, NJ: Princeton University Press.
[2] Levi, M. (2020). A water-based proof of the Cauchy-Schwarz Inequality. Am. Math. Month., in press.
[3] Levi, M., & Tokieda, T. (2020). A communicating-vessels proof of Hölder’s inequality. Am. Math Month., in press.

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