Question

Write and solve the Euler equations to make stationary the integrals. $$\int_{a}^{b} \sqrt{\frac{y^{\prime 2}}{y^{2}}+1} d x$$

Short answer

The Euler equation for this integral is \(\frac{-y^{\text{'}}^{2}}{y^{3}F} - \frac{d}{dx}\left(\frac{y^{\text{'}} / y^{2}}{F}\right) = 0\.

Step-by-step solution

Identify the Integral

We are given the integral to be minimized: display math\(\frac{I[y]} = \int_{a}^{b} \sqrt{\frac{y^{\text{'}}^{2}}{y^{2}}+1} dx\).

Formulate the Euler-Lagrange Equation

The Euler-Lagrange equation is given by display math\(\frac{\partial F}{\partial y} - \frac{d}{dx} \left( \frac{\partial F}{\partial y^{\text{'}}} \right) = 0\).Here, \(F\) is the integrand \(F = \sqrt{\frac{y^{\text{'}}^{2}}{y^{2}}+1}\).

Compute \(\frac{\partial F}{\partial y}\)

Compute the partial derivative of \(F\) with respect to \(y\): display math\(\frac{\partial F}{\partial y} = \frac{1}{2} \left( \frac{\frac{y^{\text{'}}^{2}}{y^{2}}+1 \right)^{-\frac{1}{2}} \left(-2 \frac{y^{\text{'}}^{2}}{y^{3}}\right) = \frac{-y^{\text{'}}^{2}}{y^{3}F}\).

Compute \(\frac{\partial F}{\partial y^{\text{'}}}\)

Compute the partial derivative of \(F\) with respect to \(y^{\text{'}}\): display math\(\frac{\partial F}{\partial y^{\text{'}}} = \frac{1}{2} \left(\frac{y^{\text{'}}^{2}}{y^{2}} + 1\right)^{-\frac{1}{2}} \left(2 \frac{y^{\text{'}}}{y^{2}}\right) = \frac{y^{\text{'}} / y^{2}}{F}\).

Compute \(\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\text{'}}}\right)\)

Differentiate \(\frac{\partial F}{\partial y^{\text{'}}}\) with respect to \(x\):display math\(\frac{d}{dx} \left( \frac{y^{\text{'}} / y^{2}}{F} \right)\).This requires the product rule and the chain rule. Let \(u = \frac{y^{\text{'}}}{y^{2}}\) and \(v = F\).

Set Up the Euler-Lagrange Equation

We now have:display math\(\frac{-y^{\text{'}}^{2}}{y^{3}F} - \frac{d}{dx}\left(\frac{y^{\text{'}} / y^{2}}{F}\right) = 0\).This simplifies to finding functions \(y\) that satisfy the equation.

Key concepts

Euler-Lagrange equation

The Euler-Lagrange equation is essential in finding functions that make an integral stationary. This comes up often in physics and calculus, especially in problems involving minimization or optimization. Imagine you have an integral that you want to minimize or maximize. For example, consider an integral that represents the path of least energy taken by a particle.

The Euler-Lagrange equation is typically written as:
\(\frac{\partial F}{\partial y} - \frac{d}{dx} \left( \frac{\partial F}{\partial y^{\text{'}}} \right) = 0\).

Here, \(F\) is your integrand, which can depend on \(x\), the function \(y\), and its derivative \(y'\). The strength of this formula stems from its ability to convert a complex problem into a differential equation that can be solved.

Let's break down how this applies to our problem:

• We first identify \(F = \sqrt{\frac{y^{\text{'}}^{2}}{y^{2}}+1}\).
• Next, compute \(\frac{\partial F}{\partial y}\) and \(\frac{\partial F}{\partial y^{\text{'}}}\).
• Finally, we use the Euler-Lagrange equation to solve for \(y\).

Integral calculus

Integral calculus is a part of mathematical analysis that studies integrals and their properties. In simpler terms, integration is the reverse process of differentiation. While differentiation measures the rate of change, integration measures the accumulation of quantities.

Understanding how integration works is key to solving many problems in physics and engineering. For example, in our given integral \[ I[y] = \int_{a}^{b} \sqrt{\frac{y^{\text{'}}^{2}}{y^{2}}+1} \, dx \], we aim to find the function \(y(x)\) which minimizes (or maximizes) this integral.

Here are some basic steps in solving integral calculus problems:

• Identify the function to be integrated (the integrand).
• Determine the limits of the integral (from \(a\) to \(b\)).
• Use the Fundamental Theorem of Calculus to evaluate definite integrals.

For our case, after simplifying the integral using the Euler-Lagrange equation, we eventually solve it to find the desired function \(y\). This showcases the importance of understanding both integration and the foundational concepts behind it.

Calculus of Variations

Calculus of variations is a field of mathematical analysis that deals with optimizing functionals, typically integrals. Unlike traditional calculus where you deal with functions, in calculus of variations, your goal is to find functions that optimize (minimize or maximize) a given functional.

For example, in our problem, we need to find a function \(y(x)\) that makes the integral \[ I[y] = \int_{a}^{b} \sqrt{\frac{y^{\text{'}}^{2}}{y^{2}}+1} \, dx \] stationary.

• First, you identify the functional you want to optimize.
• Then, you set up the Euler-Lagrange equation based on the identified functional.
• Finally, you solve the resulting differential equation to find the function \(y(x)\).

Calculus of variations is not just theoretical; it has practical applications in physics, engineering, and economics. It helps to solve problems like finding the shortest path, the least energy trajectory, or maximizing profit. In essence, it's a powerful tool for solving real-world optimization problems.