Is the Variation of the Action Correct Under a Boost?

[spaghetti3451]

Homework Statement

Calculate the variation of the action $S = \int L dt = \int \frac{1}{2} m v^{2} dt$ under a boost $\vec v \rightarrow \vec v + \vec v_{0}$.

Homework Equations

The Attempt at a Solution

$\delta S$
$= \int \delta L\ dt$
$= \int \frac{1}{2} m\ \delta(\vec v^{2})\ dt$
$= \int \frac{1}{2} m\ \delta(v^{2} + vv_{0}cos \theta + v_{0}^{2})\ dt$
$= \int \frac{1}{2} m\ (2 v\ \delta v + v_{0}\ cos \theta\ \delta v )\ dt$
$= \bigg( \int (mv + v_{0} cos \theta)\ dt\ \bigg)\ \delta v$

Am I correct so far?

 

[theodoros.mihos]

$\delta{v^2} = 2\mathbf{v}\cdot\delta{\mathbf{v}}$