Find the extremals of the functional $$J[y, z] = \int_0^\frac{π}{2} ((y')^2 + (z')^2 + 2yz) \,dx$$
subject to the boundary conditions $y(0) = 0, y(\frac{π}{2})= 1, z(0) = 0, z(\frac{π}{2}) = 1$
Do I need to convert y and z to polar coordinates so they have the same variables? I do not have any examples like this in my textbook or notes.
Extremals should be found among the solutions of the Euler–Lagrange equations: $$ \left\{ \begin{array}{l} F'_y-\frac{d}{dx}F'_{y'}=0,\\ F'_z-\frac{d}{dx}F'_{z'}=0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} 2z-2y''=0,\\ 2y-2z''=0. \end{array} \right. $$ Here $F$ is the function under the integral. From that system one can find $y,z$ (there will be some constants involved). Those constants can be found using the boundary conditions.
