Write and solve the Euler equations to make the following integrals stationary. Change the independent variable, if needed, to make the Euler equation simpler. \(\int_{x_{1}}^{x_{2}} \sqrt{1+y^{2} y^{\prime 2}} d x\)

The calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals. A functional is a mapping from a space of functions to the real numbers. Unlike finding the maxima or minima of functions, in calculus of variations, you deal with finding the function that optimizes a given functional. For instance, in our problem, the functional is the integral \(\int_{x_1}^{x_2} \sqrt{1 + y'^2} dx\)\. The goal is to find the function y(x) that makes this integral stationary, which means it either reaches a maximum or minimum value. A common technique used in calculus of variations is the Euler-Lagrange equation, which provides the conditions that the function must satisfy to be optimal.

Lagrangian mechanics is a reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. It provides a powerful and elegant method to derive the equations of motion for a system. The Lagrangian of a system, usually denoted as L, is a function that summarizes the dynamics of the system. It is typically expressed as the difference between kinetic energy (T) and potential energy (U), so \[ L = T - U \] In the context of the calculus of variations and the given problem, the Lagrangian represents the integrand of the functional we want to optimize. For instance, in our problem, the Lagrangian is given by \(\f(y, y', x) = \sqrt{1 + y'^2}\)\.

Partial differentiation is the process of taking the derivative of a function with respect to one variable while keeping the other variables constant. It is a key tool in multivariable calculus and is used extensively in physics and engineering. In our problem, partial differentiation is used to compute the necessary derivatives for the Euler-Lagrange equation. For instance, the terms \(\frac{\f\f}{\f y}\)\ and \(\frac{\f\f}{\f y'}\)\ are computed. Since \(\f = \sqrt{1 + y'^2}\)\, \(\f\f / \f y = 0\)\ because \(\f\)\ doesn't explicitly depend on y. On the other hand, \(\f\f / \f y' = \frac{y'}{\f\sqrt{1 + y'^2}}\)\.

In calculus of variations, a stationary integral is one whose value does not change for small variations in the function that defines it. This means if you perturb the function slightly, the value of the integral remains the same. This concept is crucial for finding the optimal solution, as it corresponds to the principle of least action in physics. For instance, to find the stationary points of the integral \(\int_{x_1}^{x_2} \sqrt{1 + y'^2} dx\)\, we set up the Euler-Lagrange equation. When all necessary derivatives are computed and substituted back, we get the condition \(\frac{y''}{(1 + y'^2)^{\f3/2}} = 0\)\, leading to \(\f y'' = 0\)\. This tells us that the function y that makes the integral stationary has a second derivative equal to zero, implying it must be a linear function.