Lagrangian is defined up to addition of a total derivative of function of positions and time. Now suppose we have a function $f(x,\dot x,t)$. How can one show (check) that $$\not\exists g(x,t):\; f(x,\dot x,t)=\frac{\text{d}}{\text{d}t}g(x,t)$$ ?
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Since $$\frac{d}{dt}g(x,t)=\frac{\partial g}{\partial x} \dot{x}+\frac{\partial g}{\partial t}$$ $g$ exists iff $f$ is linear in $\dot{x}$, $$f(x,\dot{x},t)=f_1(x,t)\dot{x}+f_2(x,t)$$ and moreover $$\frac{\partial f_1}{\partial t}=\frac{\partial f_2}{\partial x}$$ (if there are several $x$'s, there are more conditions, but they still say that the 2nd partial derivatives of $g$ are symmetric).
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