[SOLVED] Lagrangian formalism (demonstration)

My question is about the multiplicity of the Lagrangian to a Physics system.

I pretend to demonstrate the following proposition:

For a system with $n$ degrees of freedom, written by the Lagrangian $L$, we have that: $$L'=L+\frac{d F(q_1,...,q_n,t)}{d t}$$ also satisfies Lagrange's equations, where $F$ is any arbitrary function, but differentiable.

I saw the resolution to this problem and I found that another interesting proposition was needed to complete the demonstration, and it was:

$$\frac{\partial \dot{F}}{\partial \dot{q}}=\frac{\partial F}{\partial q}$$

Why is this true? How could I say that the relation between the temporal variation of $F$ and the temporal variation of $q$, is equivalent to the relation between $F$ and $q$.

Maybe this is a silly question, but I don't get it in my head. I can try examples and see that it's true, but I can't figure it out why.

@user12029 2015-04-06 00:20:07

Hint: $\frac{\partial }{\partial \dot{q}}\dot{F}=\frac{\partial}{\partial \dot{q}}\left(\frac{\partial F}{\partial q}\dot{q}+\frac{\partial F}{\partial \dot{q}}\ddot{q}+\frac{\partial F}{\partial t}\right)$

What does $F=F(q,t)$ imply about $\frac{\partial}{\partial \dot{q}} F$?