A particle in certain Inertial Frame of Reference IFR is described by the Lagrangian:
$L = av^2$
where $\vec{v} = v_1\hat{e}_1 + v_2\hat{e}_2 + v_3\hat{e}_3$ is the velocity vector of the particle.
The velocity of the particle in a different Inertial Frame of Reference IFR' has a velocity
$\vec{v'} = \vec{v} + \vec{V}$
where $\vec{V}$ is the constant relative velocity between the two Inertial Frames of Reference.
In this new IFR' the Lagrangian will be:
$L' = av'^2 = a(\vec{v} + \vec{V})^2$
Show that $L$ and $L'$ yield the same equations of motion - i.e. $\delta S' = \delta S$. Use the result of
Problem 2, i.e. show that $L'$ can be written as $L' = L + \frac{d}{dt}f(\vec{x}, t)$