Finding functional extremals

Question

So my problem is as follows: find the extremals of the functional $$I[x_1(t), x_2(t)]=\int_{0.5}^1(\dot{x}_1^{2}-2x_{1}\dot{x}_2t)dt,$$
given: $$x_1(0.5)=2, \ \ \ \ x_2(0.5)=15, \ \ \ \ x_1(1)=1, \ \ \ \ x_{2}(1)=1.$$

Answer 1

Your Lagrange function is $$L(x_1,x_2,\dot{x_1}, \dot{x_2},t)=\dot{x_1}^2-2x_1\dot{x_2}$$ And the Euler-Lagrange equations are $$\frac{\partial L}{\partial x_1}-\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{x_1}}\right)=0$$ $$\frac{\partial L}{\partial x_2}-\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{x_2}}\right)=0$$ Which will give you a system of differential equations. And your second Euler-Lagrange equation will be a bit easier, because the Lagrange function is independent of $x_2$: $$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{x_2}}\right)=0$$ $$\frac{\partial L}{\partial \dot{x_2}}=\text{constant}$$