How to check $\mathbf v' · \mathbf V$ and $\mathbf v'^2$ are time derivatives of some other functions? From Landau, Lifshitz Mechanics p.127
$$
\renewcommand{\vec}[1]{\mathbf{#1}}
L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U
$$
He states that "$\vec{V}^2(t)$ can be written as the total derivative with respect to $t$ of some other function.". I understood that but the point that I could not understand is why we can't write the terms $\vec{v'}\cdot\vec{V}$ and $\vec{v}'^2$ as the total derivative with respect to $t$ of some other function?
$\vec{v}'$ is the velocity of particle measured in non-inertial frame.
$\vec{V}$ is translational velocity of non-inertial frame with respect to inertial frame.
 A: The main point is that: 


*

*On one hand, ${\bf r}^{\prime}(t)$ is an dynamical active variable of the system. Its time evolution is determined by the equations of motion of the system, i.e. Newton's 2nd law. Here ${\bf v}^{\prime}:=\dot{\bf r}^{\prime}$.

*On the other hand, ${\bf R}(t)$ is a passive (background) variable with pre-defined time-dependence. Here ${\bf V}:=\dot{\bf R}$.
We could in principle define e.g.
$$\tag{1} F_1~:=~\frac{m}{2}\int_0^t \! dt^{\prime}~
\dot{\bf r}^{\prime}(t^{\prime})^2,\qquad 
\dot{F}_1~=~\frac{m}{2}\dot{\bf r}^{\prime 2}, $$
$$\tag{2} F_2~:=~\frac{m}{2}\int_0^t\! dt^{\prime}~
\dot{\bf r}^{\prime}(t^{\prime})\cdot \dot{\bf R}(t^{\prime}),\qquad 
\dot{F}_2~=~\frac{m}{2}\dot{\bf r}^{\prime}\cdot\dot{\bf R}  ,$$
$$\tag{3} F_3~:=~\frac{m}{2}\int_0^t \! dt^{\prime}~
\dot{\bf R}(t^{\prime})^2,\qquad 
\dot{F}_3~=~\frac{m}{2}\dot{\bf R}^2. $$
The problem with e.g. the definition (1) is that $F_1$ would be a non-local quantity. It would depend on the dynamical active variable ${\bf r}(t^{\prime})$ in the past $t'<t$. See also this Phys.SE post. 
On the other hand, there is no problem with locality in the definition (3) of $F_3$ because the passive variable ${\bf R}(t)$ is pre-defined for all times $t$. In other words, $F_3$ is just another passive background variable. 
References:


*

*L.D. Landau & E.M. Lifshitz, Mechanics, vol. 1, (1976), p. 127.

