Page:Zur Dynamik bewegter Systeme.djvu/15

Before we perform the integration, we derive the relevant equations for the velocity components ${\dot {y}}$ and ${\dot {z}}$. In addition to the differential equations (6) with respect to the primed system we have to use:

${\frac {d}{dt'}}{\frac {\partial H'}{\partial {\dot {x}}'}}={\mathfrak {F}}'_{x'},\quad {\frac {d}{dt'}}{\frac {\partial H'}{\partial {\dot {y}}'}}={\mathfrak {F}}'_{y'},\quad {\frac {d}{dt}}{\frac {\partial H'}{\partial {\dot {z}}'}}={\mathfrak {F}}'_{z'}$

(20)

the relations between the primed and unprimed components of the moving force ${\mathfrak {F}}$. To find them, we consider a special case, namely, an infinitely small diathermanous solid body charged with electricity e, in an arbitrary, evacuated electromagnetic field. Then, for the unprimed system:

${\mathfrak {F}}_{x}=e{\mathfrak {E}}_{x}+{\frac {e}{c}}({\dot {y}}{\mathfrak {H}}_{x}-{\dot {z}}{\mathfrak {H}}_{y})$

${\mathfrak {F}}_{y}=e{\mathfrak {E}}_{y}+{\frac {e}{c}}({\dot {z}}{\mathfrak {H}}_{x}-{\dot {x}}{\mathfrak {H}}_{y})$

${\mathfrak {F}}_{z}=e{\mathfrak {E}}_{z}+{\frac {e}{c}}({\dot {x}}{\mathfrak {H}}_{y}-{\dot {y}}{\mathfrak {H}}_{x})$,

where ${\mathfrak {E}}$ denotes the electric, ${\mathfrak {H}}$ the magnetic field intensity. The same equations apply according to the relativity principle, when all the variables, except e and c, were provided with primes. This leads with respect to the relations (13) and the relations:^[1]

${\begin{matrix}{\mathfrak {E}}'_{x'}=&{\mathfrak {E}}_{x}&{\mathfrak {H}}'_{x'}=&{\mathfrak {H}}_{x}\\\\{\mathfrak {E}}'_{y'}=&{\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {E}}_{y}-{\frac {v}{c}}{\mathfrak {H}}_{z}\right)\quad &{\mathfrak {H}}'_{y'}=&{\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {H}}_{y}+{\frac {v}{c}}{\mathfrak {E}}_{z}\right)\\\\{\mathfrak {E}}'_{z'}=&{\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {E}}_{z}+{\frac {v}{c}}{\mathfrak {H}}_{y}\right)\quad &{\mathfrak {H}}'_{z'}=&{\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {H}}_{z}-{\frac {v}{c}}{\mathfrak {E}}_{y}\right)\end{matrix}}$

the following equations between the primed and unprimed force components:

${\mathfrak {F}}'_{x'}={\mathfrak {F}}_{x}-{\frac {v{\dot {y}}}{c^{2}-v{\dot {x}}}}{\mathfrak {F}}_{y}-{\frac {v{\dot {z}}}{c^{2}-v{\dot {x}}}}{\mathfrak {F}}_{z}$,

(21)

${\mathfrak {F}}'_{y'}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\mathfrak {F}}_{y},\ {\mathfrak {F}}'_{z}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\mathfrak {F}}_{z}.$

(22)

The last two relations (22) we accept as generally valid; this give in combination with (6) and (20):

${\frac {d}{dt'}}{\frac {\partial H'}{\partial y'}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {d}{dt}}{\frac {\partial H}{\partial {\dot {y}}}}$.

1. A. Einstein, Ann. d. Phys. (4), 17, p. 909, 1905.