I'm trying to find relativistic transformations of force which has two components [tex]F_{x}[/tex] and [tex]F_{y}[/tex] and velocities [tex]v_{x}[/tex] and [tex]v_{y}[/tex]. I'm not sure if I have right idea so I would be grateful if someone could check it out.(adsbygoogle = window.adsbygoogle || []).push({});

First of all, we have second Newton's law in special relativity (in this case for reference frame S)

[tex]

F_{x}=\frac{d}{dt}\left(\frac{m_{0}v_{x}}{\sqrt{1-\frac{v_{x}^2+v_{y}^2}{c^2}}}\right)

[/tex]

[tex]

F_{y}=\frac{d}{dt}\left(\frac{m_{0}v_{y}}{\sqrt{1-\frac{v_{x}^2+v_{y}^2}{c^2}}}\right)

[/tex]

When I derive everything, I get equations including velocities [tex]v_{x}[/tex], [tex]v_{y}[/tex] and accelerations [tex]a_{x}[/tex] and [tex]a_{y}[/tex].

Then I use second Newton's law in other reference frame, S' moving with velocity [tex]V[/tex] relative to frame S.

[tex]

F_{x}'=\frac{d}{dt}\left(\frac{m_{0}v_{x}'}{\sqrt{1-\frac{v_{x}'^2+v_{y}'^2}{c^2}}}\right)

[/tex]

[tex]

F_{y}'=\frac{d}{dt}\left(\frac{m_{0}v_{y}'}{\sqrt{1-\frac{v_{x}'^2+v_{y}'^2}{c^2}}}\right)

[/tex]

Then I use Lorentz transformations to find [tex]a_{x}\rightarrow a_{x}'[/tex], [tex]a_{y}\rightarrow a_{y}'[/tex] and [tex]v_{x}\rightarrow v_{x}'[/tex], [tex]v_{y}\rightarrow v_{y}'[/tex].

When I derive formulas for [tex]F_{x}'[/tex] and [tex]F_{y}'[/tex] and plug [tex]a_{x}'[/tex],[tex]a_{y}'[/tex], [tex]v_{x}'[/tex] and [tex]v_{y}'[/tex] I should get [tex]F_{x}\rightarrow F_{x}'[/tex] and [tex]F_{y}\rightarrow F_{y}'[/tex].

I'm not sure if this is the right way to do it because I don't have any literature concerning this. I would like to hear your opinion.

Sorry for eventually bad English and thanks in advance!

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# Transformation of Force

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