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Special Relativity Force and Linearization

  • Thread starter zachzach
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Homework Statement



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


Homework Equations



[tex] \vec{F}(t) = \dfrac{d}{dt}(\gamma m \vec{v})[/tex]


The Attempt at a Solution



My trouble is with linearizing the equation.

[tex] \vec{F} = m\vec{v} \dfrac{d \gamma}{dt} + m \gamma \dfrac{d\vec{v}}{dt}[/tex]

Then linearize each term so:

[tex]\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})[/tex]

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

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
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You also need to expand [tex]\gamma[/tex] (and its derivative) in powers of the velocity.
 

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