How do I show that f(x) is independent for the variationproblem?

Question

The variationproblem where f(x) is a reel differential function defined in the interval [0,1].

Answer 1

You can use Euler–Lagrange equation: $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot x}\right) = \frac{\partial L}{\partial x} $$ In your case $$ L = -x^2 - \dot x^2 + f(x)\dot x $$ $$ \frac{d}{dt}\big(-2\dot x + f(x)\big) = -2x + \frac{df}{dx}\dot x \\\Longrightarrow -2\ddot x + \frac{df}{dx}\dot x = -2x + \frac{df}{dx}\dot x \\\Longrightarrow \ddot x = x $$

Or, you can see that $$ \int_0^1 f(x)\dot x\,dt = \int_0^1 f(x)\,dx $$ is a constant.