- #1
N-Gin
- 56
- 0
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.
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!
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!