Euler-Lagrange equation - Exercises

Derive the Euler Lagrange equation(s) for the functional \(\)S[x_1,x_2] = \int_a^b\L(t,x_1,x_2,\dot{x}_1,\dot{x}_2) \d t\(\)

Prove that if \(\L\) has no explicit dependence on the independent variable, that is, \(\L = \L(x,\dot{x})\), then the following identity holds. \(\)\frac{\d}{\d t}\left(\dot{x}\frac{\partial\L}{\partial\dot{x}}- \L\right) = 0\(\)

Find the Euler Lagrange equation for the functional