Johann Bernoulli maintained correspondence with 110 scholars of his time, with approximately 2,500 letters involving academic discussions. Many of these have become precious documents in the history of science. For instance, his correspondence with his brother Jacob, as well as with Leibniz and Huygens, regarding the catenary, the brachistochrone (cycloid), and the isoperimetric problem is well-known. Although they were in constant debate—particularly the sharp and often hurtful criticisms between Johann and Jacob that frequently caused unpleasantness between the brothers—these disputes undoubtedly promoted the development of science. The brachistochrone problem, in particular, led to the birth of the calculus of variations.
Interestingly, in 1690, Johann Bernoulli’s brother, Jacob Bernoulli, posed the catenary problem to the mathematical community, seeking a solution:
Fix both ends of a necklace and let it hang naturally in a gravitational field. Find the curve equation of the necklace.
The suspended steel cables above a suspension bridge, a spider web hanging with water droplets, and the wires between utility poles are all catenaries. Galileo was the first to notice the catenary and conjectured that it was a parabola. In 1691, Leibniz, Huygens, and Johann Bernoulli each independently obtained the correct answer using the then-recently developed calculus.
Based on the knowledge we have acquired in the "Natural Extremum" series, let us attempt to solve this problem. Undoubtedly, the "hanging chain" has reached an equilibrium state, at which point we can consider its gravitational potential energy to have reached a minimum. Although this feels somewhat intuitive, our life experience tells us it is inevitable. The principle of minimum potential energy states that a system is in equilibrium when its potential energy is minimized, and a catenary can have only one equilibrium state.
Since we assume its gravitational potential energy is minimized, the problem becomes finding the curve of a fixed length that has the minimum gravitational potential energy. Like the previous "brachistochrone" problem, this falls within the scope of variational calculus. Therefore, the brachistochrone and the catenary problems are essentially consistent in nature.
Let the shape of the curve be y=y(x), where the x-axis represents the ground. Assume the chain is uniform with a linear density of 1. The mass of each small segment ds = \sqrt{dy^2 + dx^2} is dm = ds = \sqrt{dy^2 + dx^2}. Simultaneously, its height is y, so the gravitational potential energy of this segment is dE_p = gy \, dm = gy \sqrt{dy^2 + dx^2} = gy \sqrt{\dot{y}^2 + 1} dx. Similarly, since we are purely exploring the shape of the curve, we let g=1. Thus, the problem becomes finding the function y(x) that minimizes: \int_{x_1}^{x_2} y\sqrt{\dot{y}^2 + 1} dx
In the sixth article of this series, we derived a formula (1):
v^2(1+\dot{y}^2) = \text{Const} (1)
and pointed out that when the expression for v only explicitly contains y, the function that minimizes t = \int_{x_1}^{x_2} \frac{\sqrt{\dot{y}^2+1}dx}{v} is calculated by equation (1).
For this problem, we can let v = \frac{1}{y}. Substituting this into (1), we get: \frac{1+\dot{y}^2}{y^2} = C This can be transformed into dx = \frac{1}{\sqrt{C}} \frac{d(\sqrt{C}y)}{\sqrt{(\sqrt{C}y)^2 - 1}}. Integrating this (using integration formulas and placing the curve in the first quadrant so that x and y are positive): x = \frac{1}{\sqrt{C}} \ln|\sqrt{C}y + \sqrt{Cy^2 - 1}| + C_2 = \frac{1}{\sqrt{C}} \mathop{\mathrm{arcosh}}(\sqrt{C}y) + C_2
Mathematicians prefer using the form on the far right (hyperbolic functions) to describe this shape. By making appropriate settings (i.e., translating the curve and changing the starting point) such that C_2 = 0, we can simplify it to: y = \frac{1}{\sqrt{C}} \cosh(\sqrt{C}x) This is the final catenary equation. The last step is simply to fit the length of the chain. I leave this step for the reader to consider. ^_^
Readers can also think about a relatively more complex problem:
If there were a sufficiently large hanging chain, with a length as long as the Earth’s radius, what would the shape of the chain be? That is, the gravitational field is no longer uniform, and only the Earth’s gravitational pull is considered.
Discussion thread: http://bbs.spaces.ac.cn/topic.php?id=4
The "Natural Extremum" series is now drawing to a close. Next, we will perform some purely analytical work. Although this may be tedious, it is unavoidable. Analytical work allows us to understand the essence of the problem more clearly and guides us in the right direction, preventing us from becoming overly intoxicated by superficial beauty. Through the following analysis, we will understand the general idea of finding extrema, from which we can derive a fundamental equation of variational calculus—the Euler-Lagrange equation.