Special Relativity Force and Linearization

zachzach

Homework Statement

Consider a particle moving at close to the speed of light with $$v \approx c \ \hat{z}$$. A small oscillatory force $$F(t)$$ acts on the particle. Consider $$F(t)$$ to be a first order (eg. linear) perturbation which will not effect $$v_o$$, only $$v_1$$, the first order component of v. Linearize the equation of motion and find the acceleration when $$F(t)$$ is perpendicular and when $$F(t)$$ is parallel to $$v_o$$.

Homework Equations

$$\vec{F}(t) = \dfrac{d}{dt}(\gamma m \vec{v})$$

The Attempt at a Solution

My trouble is with linearizing the equation.

$$\vec{F} = m\vec{v} \dfrac{d \gamma}{dt} + m \gamma \dfrac{d\vec{v}}{dt}$$

Then linearize each term so:

$$\vec{v} \dfrac{d \gamma}{dt} = \vec{v}_o\dot{\gamma}_o + \dfrac{d}{d \vec{v}} [\vec{v}\dot{\gamma} ] \mid_{\vec{v}= {\vec{v}_o, \dot{\gamma}= \dot{\gamma_o}}}(\vec{v} - \vec{v}_o) + \dfrac{d}{d \dot{\gamma}}[\vec{v} \dot{\gamma}] \mid_{\vec{v}= {\vec{v}_o,\dot{ \gamma}= \dot{\gamma}_o}} (\dot{\gamma} - \dot{\gamma}_o) = \vec{v}_o\dot{\gamma_o} +\dot{\gamma}_o (\vec{v} - \vec{v}_o) + \vec{v}_o(\dot{\gamma} - \dot{\gamma_o})$$

And then do the next term similarly. Is this the correct way to linearize?

You also need to expand $$\gamma$$ (and its derivative) in powers of the velocity.