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

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.

The length of the wire is given.

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:

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

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

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