Find the extremal of the following function:
$$J(x,y) = \int_{0}^{\frac{\pi}{2}} \left[|x'|^{2}+|y'|^{2}+2xy\right]dt$$ subject to $$x(0)=y(0)=0,~x(\pi/2) = y(\pi/2) = 1$$
let $L[x,y,x',y'] = |x'|^{2}+|y'|^{2}+2xy$, since L is independent of t, we use the beltrami equation:
$$|x'|^{2}+|y'|^{2} + 2xy - x'(2x') - y'(2y') = C \implies |x'|^{2}+ |y'|^{2}-2xy = C \implies$$ $$(x'-y')^{2} = C \implies$$ $$x'-y' = C \implies$$ $$\int\left[\frac{dx}{dt} - \frac{dy}{dt}\right] ~dt = C\int dt \implies$$ $$x(t)+C_1 - y(t) - C_2 = Ct + C_3 \implies$$ $$x(t)-y(t)=Ct+C_4$$ $$x(0)-y(0)=C_4 = 0$$ $$x(\pi/2)-y(\pi/2) = \frac{c\pi}{2} = 1 \implies c = \frac{2}{\pi} \implies$$ $$x(t)-y(t) = 1$$
I feel like this is wrong; i was expecting two functions which would describe their movement in t, ie $x(t) = ...., y(t)=....$ where have i gone wrong? i imagine if it is wrong, it's at the ODE. Thank you.
You made a mistake. Note the corresponding Euler-Lagrange equations are $$ \frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial x'}=0, \frac{\partial L}{\partial y}-\frac{d}{dt}\frac{\partial L}{\partial y'}=0. $$ Since $$ L=(x')^2+(y')^2+2xy $$ one has $$ 2y-\frac{d}{dt}(2x')=0, 2x-\frac{d}{dt}(2y')=0$$ or $$ x''=y, y''=x. $$ So $$ x=c_1e^t+c_2e^{-t}+c_3\cos t+c_4\sin t, y=c_1e^t+c_2e^{-t}-c_3\cos t-c_4\sin t. $$ Using the boundary condtions $$x(0)=y(0)=0,~x(\pi/2) = y(\pi/2) = 1$$ one has $$ x(t)=y(t)=\frac{e^{\frac{\pi}{2}-t}(e^{2t}-1)}{e^\pi-1}. $$