In Example 10. 34, the geodesic equations for geodesics in the plane in polar coordinates were found as

\[\begin{align*} r\left(\frac{d \phi}{d s}\right)^{2}-\frac{d^{2} r}{d s^{2}} & =0 \\ \frac{d}{d s}\left(r^{2} \frac{d \phi}{d s}\right) & =0 \tag{10.103} \end{align*}\]

a. Solve this set of equations for \(r=r(s)\) and \(\phi=\phi(s)\).

b. Prove that the solutions obtained in part

a. are the familiar straight lines in the plane.

Data from Example 10.34

Geodesics for polar coordinates in the plane.

For polar coordinates, the line element is given by
\[d s^{2}=d r^{2}+r^{2} d \phi^{2}\]