*The method of Lagrange multipliers (see Appendix A, Problem 11) can be extended to the calculus of variations. To find the maxima and minima of an integral \(I\), subject to the condition that another integral \(J=0\), we have to find the stationary points of the integral \(I-\lambda J\) under variations of the function \(y(x)\) and of the parameter \(\lambda\). Apply this method to find the catenary, the shape in which a uniform heavy chain hangs between two fixed supports. [The required shape is the one that minimizes the total potential energy, subject to the condition that the total length is fixed. Show that this leads to a variational problem with $$ f\left(y, y^{\prime}\right)=(y-\lambda) \sqrt{1+y^{\prime 2}} $$ and hence to the equation $$ (y-\lambda) y^{\prime \prime}=1+y^{\prime 2} $$ Solve this equation by introducing the new variable \(u=y^{\prime}\) and solving for \(u(y) .]\)

The calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as integrals, such as the total potential energy of a system. Unlike ordinary calculus, which seeks to find extrema of functions with respect to variables, the calculus of variations seeks to find functions that minimize or maximize a functional.

The classic example in calculus of variations is finding the shortest path between two points, known as the geodesic. The underlying principle is that out of all possible paths, the one with the least length will satisfy a specific condition—captured by the Euler-Lagrange equation. This principle can be applied to various problems, such as the catenary problem, where the goal is to find the shape a heavy chain will form between two supports—a problem that's elegantly explained by the calculus of variations.

The Euler-Lagrange equation forms the core of the calculus of variations. It is a differential equation whose solutions are the functions that make a given functional stationary. A stationary function is one where the functional does not increase or decrease for small variations of the function.

To find such a function, we consider the functional's variation and set its derivative with respect to the variable to zero. The end result is the Euler-Lagrange equation:
\[\frac{\partial f}{\partial y} - \frac{d}{dx}\left(\frac{\partial f}{\partial y^{\prime}}\right) = 0\]
This equation is pivotal in determining the stationary points of a functional, and, in the context of our exercise, helps us deduce the shape of a catenary by yielding a differential equation that defines the relationship between the functions and their derivatives.

Solving differential equations is an essential part of calculus and one that plays a significant role in the calculus of variations. Differential equations describe relationships between functions and their rates of change; a solution to a differential equation is a function that satisfies the equation.

In the context of the catenary problem, we derive a second-order differential equation from the Euler-Lagrange equation. To solve this, we sometimes introduce a new variable, like \(u = y'\), which transforms our problem into a first-order differential equation. Solving this new equation can often be more straightforward and leads us back to our original function through integration. Once solved, this function represents the optimal shape or path dictated by the original variational problem.

The catenary shape optimization is a practical application of the calculus of variations. In this problem, we search for the shape that a uniform heavy chain or cable assumes when suspended by its ends and acted upon by gravity. The optimal shape minimizes the potential energy of the system while satisfying constraints such as a fixed length.

This optimization leads to a specific kind of curve, known as a catenary. Through the calculus of variations and the use of Lagrange multipliers, we find a variational problem with a certain functional. Once we apply the Euler-Lagrange equation and solve the resulting differential equation, the catenary curve is defined by the function \( y(x) \). This curve has important applications in engineering and architecture, illustrating how optimization problems in calculus can directly influence practical designs.