Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on the Lagrangian of a free particle. At the very top of page 7, he equates two Lagrangian by 
$$\mathrm{L}'=\mathrm{L}(v'^2)=\mathrm{L}(v^2+2v \cdot \epsilon+ \epsilon^2).$$
Then he goes to to say that you can expand this in powers of $ \epsilon$ and neglect all terms above the first order to get 
$$\mathrm{L}(v'^2)=\mathrm{L}(v^2)+\frac{\partial \mathrm{L}}{\partial v^2}2v \cdot \epsilon.$$
How do you expand in terms of $\epsilon$ to get this result? This Lagrangian seems to me to be a function of three variables and I am looking at the multivariable Taylor expansion and I am not seeing how Landau could get his result.
 A: Landau defines $v'\equiv v+\epsilon$. Then
$$ L(v'^2)= L[ (v+\epsilon)^2] = L(v^2+2v\cdot\epsilon+\epsilon^2)\sim L(v^2+2v\cdot \epsilon) $$
In the last step we have omited the second order terms in $\epsilon$.
So we have $L(v^2+2v\cdot\epsilon)$ which is a function like $f(a+x)$. We expand it as
$$ f(a+x)\sim f(a)+f'(a)x $$
at first order. In our case, we have $a=v^2$ and $x=2v\cdot\epsilon$, so
$$ L(v'^2)\sim L(v^2 + 2v\cdot\epsilon)  \sim L(v^2) + \frac{\partial L}{\partial v^2} 2v\cdot\epsilon$$
A: Consider $L(v'^2)$ as a function of the variable $\epsilon$ it is
\begin{equation}
L(v'^2)=F(\epsilon)
\end{equation}
then use the conventional Taylor expansion around $\epsilon=0$
\begin{equation}
F(\epsilon)=F(0)+\frac{dF(0)}{d\epsilon}\cdot\epsilon+\mathcal{O}(\epsilon^2)
\end{equation}
then using the fact that $dF/d\epsilon=(\partial F/\partial v'^2)(dv'^2/d\epsilon)=(\partial F/\partial v'^2)2v'$ and that $v'$ reduces to $v$ when you evaluate at $\epsilon=0$, you get
\begin{equation}
L(v'^2)=L(v^2)+\frac{\partial L}{\partial v^2}2v\cdot\epsilon+\mathcal{O}(\epsilon^2)
\end{equation}
which is the desired relation.
