Question

Find a first integral of the Euler equation for the Problem if the length of the wire is given.

Short answer

First integral of the Euler equation when the length of the wire is given as:

Step-by-step solution

Definition of Euler equation

In mathematics, Euler's equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure, and density of a moving fluid. It is based on Newton's Second Law of Motion which states that if the external force is zero, linear momentum is conserved.

Given parameters

The length of the wire is given.

Step 3:Find first integral of motion

By taking reference from the problem 26 and solving the equation,

$I={\int }_c\frac{ds}{r}+\lambda {\int }_{c}ds$in polar coordinates and the integral need to maximize has the following form:

$\begin{array}{rcl}I& =& {\int }_c\frac{ds}{r}+\lambda {\int }_{c}ds\\ ds& =& \sqrt{1+r^2\theta {\text{'}}^2}d\theta \\ const& =& {\int }_{c}ds\\ F& =& \left(\frac{1}{r}+\lambda \right)\sqrt{1+r^2\theta {\text{'}}^2}d\theta ........\left(2\right)\end{array}$

Step 4:Use Beltrami identity to find the first integral of motion

Now it has been noticed that functional F does not explicitly on θ. Therefore, use Beltrami identity to find the first integral of motion.

Simplify

Now further simplify equation(2),

$C=\left(\frac{1}{r}+\lambda \right)\frac{1}{\sqrt{1+r^2\theta {\text{'}}^2}}.........\left(3\right)$

Hence using Beltrami identity first integral of motion is