Page:Das Prinzip der Relativität und die Grundgleichungen der Mechanik.djvu/3

its simple relation to the potential energy would be lost. For since according to the principle of relativity, also in the "primed" frame of reference the equations:

$m{\ddot {x'}}=X',\ m{\ddot {y}}'=Y',\ m{\ddot {z}}'=Z'$

should be valid in general, thus the relation between X, Y, Z and X', Y', Z' would result in very complicated equations, which one has to derive by means of valid relations between ${\ddot {x}}$ and ${\ddot {x}}'$ etc. according to 1), and that excludes a simple physical meaning of these quantities.

To learn about the general relation between acceleration and moving force, it is advisable to start with a special case in which one knows the connection between the components of the moving force in both reference frames; one such case is the effect of an electromagnetic field in vacuum on a charged mass point m with the quantity of electricity e. Thus for the electric and magnetic field strengths in both reference frames 1) we have the relations^[1]

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

4)

We imagine that the particle is located at the origin of the coordinates of the "unprimed" system (x, y, z, t) and has the velocity components ${\dot {x}},{\dot {y}},{\dot {z}}$ with respect to this system, and ask for the equations of motion. This question can be answered unequivocally by the fact that we first think of the mass points as the origin of a new reference frame, which moves against the original frame with the constant velocity components ${\dot {x}},{\dot {y}},{\dot {z}}$. The x-axis of this system may coincide with the direction of the velocity q of the mass point, whose size is expressed by 3). Then the mass point is at rest in the new reference frame and

1. A. Einstein, l. c., S. 909.