On page 4 of Landau & Lifshitz's _Mechanics_ they say

> $$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right).$$  Expanding this expression in powers of $\epsilon$ and neglecting terms above the first order, we obtain
>
>$$L\left({v^\prime}^2\right) = L\left(v^2\right) + \frac{\partial L}{\partial v^2}2~\bf{v \cdot} \epsilon\,.$$

and then on the next line:

> The second term on the right of this equation is a total time derivative only if it is a linear function of the velocity $\bf v$

What is the meaning of this statement?
This being a total time derivative would say $ \frac{\partial L}{\partial v^2}2\bf{v \cdot} \epsilon = C(t)$.  Where do we get a linear function of $\bf v$ from this?

EDIT: The first part can be found at https://physics.stackexchange.com/q/83101/ 
https://physics.stackexchange.com/q/128139/ , & https://physics.stackexchange.com/q/158380/