Question

Write and solve the Euler equations to make the following integrals stationary. Change the independent variable, if needed, to make the Euler equation simpler. \(\int_{0}^{\theta_{2}} \sqrt{r^{\prime 2}+r^{2}} d \theta, \quad r^{\prime}=d r / d \theta\)

Short answer

The Euler-Lagrange equation is \[ r''(r^2 + r'^2) = r(r'^2 + r^2) \].

Step-by-step solution

Define the functional

Identify the integrand in the given integral as the function we need to make stationary. Here, the integrand is \(F(r, r', \theta) = \sqrt{r'^2 + r^2}\).

Compute the partial derivatives of F

Calculate the partial derivatives of \(F\) with respect to \(r\) and \(r'\): \[ \frac{\partial F}{\partial r} = \frac{r}{\sqrt{r'^2 + r^2}} \] and \[ \frac{\partial F}{\partial r'} = \frac{r'}{\sqrt{r'^2 + r^2}} \].

Formulate the Euler-Lagrange equation

Write the Euler-Lagrange equation: \[ \frac{d}{d\theta} \left( \frac{\partial F}{\partial r'} \right) - \frac{\partial F}{\partial r} = 0 \].

Substitute the derivatives into the Euler-Lagrange equation

Substitute the partial derivatives found in Step 2 into the Euler-Lagrange equation: \[ \frac{d}{d\theta} \left( \frac{r'}{\sqrt{r'^2 + r^2}} \right) - \frac{r}{\sqrt{r'^2 + r^2}} = 0 \].

Simplify the Euler-Lagrange equation

Differentiate the derivative term and simplify the equation: \[ \frac{r''(r'^2 + r^2) - r'r(r'r'')} {(r'^2 + r^2)^{3/2}} - \frac{r}{\sqrt{r'^2 + r^2}} = 0 \].

Derive the final simplified form of the differential equation

After simplification, the equation becomes: \[ r''(r^2 + r'^2) = r(r'^2 + r^2) \]. This can be solved further to find \(r\).

Key concepts

Calculus of Variations

The calculus of variations is a field of mathematical analysis used to find functions that optimize or make a functional stationary. A functional is a mapping from a space of functions to the real numbers. For our exercise, we are dealing with an integral functional that depends on a function and its derivative.

The main goal is to find the function that makes this integral functional stationary—which means that small variations in the function do not change the value of the functional significantly. In our case, we are given the integral \( \int_{0}^{\theta_{2}} \sqrt{r^{\prime 2}+r^{2}} d \theta \) and need to find the function \( r(\theta) \) that makes it stationary.

Partial Derivatives

Partial derivatives measure how a function changes as its input variables change. In the context of the Euler-Lagrange equation, partial derivatives of the integrand (denoted as \( F \)) are essential.

For our problem, the integrand is \( F(r, r', \theta) = \sqrt{r'^2 + r^2} \). We need its partial derivatives concerning \( r \) and \( r' \). Here, they are:

\

Differential Equations

A differential equation is an equation that involves the derivatives of a function. Solving a differential equation means finding a function that satisfies the given equation.

Applying the Euler-Lagrange equation to our function \( F(r, r', \theta) = \sqrt{r'^2 + r^2} \): \[ \frac{d}{d\theta} \left( \frac{\partial F}{\partial r'} \right) - \frac{\partial F}{\partial r} = 0 \]

Substituting in the partial derivatives, we arrive at our differential equation: \[ \frac{d}{d\theta} \left( \frac{r'}{\sqrt{r'^2 + r^2}} \right) - \frac{r}{\sqrt{r'^2 + r^2}} = 0 \]

Stationary Functionals

A functional is stationary if its value does not change for small variations in the function it depends on. In the Euler-Lagrange equation, achieving this involves finding a function that satisfies the differential equation derived from the functional.

For our problem, making the integral \( \int_{0}^{\theta_{2}} \sqrt{r^{\prime 2}+r^{2}} d \theta \) stationary means that the derived differential equation \[ r''(r^2 + r'^2) = r(r'^2 + r^2) \] must hold true. Solving this equation would give us the exact form of \( r(\theta) \) that achieves stationary behavior.

In summary, the concepts of calculus of variations, partial derivatives, differential equations, and stationary functional are interconnected. Together, they provide a structured approach to determining the conditions under which an integral functional becomes stationary.