Page:Das Prinzip der Relativität und die Grundgleichungen der Mechanik.djvu/6

If we now consider the kinetic force L as a function of ξ, η, ζ, and if we set the abbreviation: $\xi ^{2}+\eta ^{2}+\zeta ^{2}=\varrho ^{2}$, it follows:

$L=mc^{2}{\sqrt {1+{\frac {\varrho ^{2}}{m^{2}c^{2}}}}}+const$

and the Hamiltonian equations of motion become:

${\begin{matrix}{\frac {d\xi }{dt}}&=&X,\quad &{\frac {d\eta }{dt}}&=&Y,\quad &{\frac {d\zeta }{dt}}&=&Z,\\\\{\frac {dx}{dt}}&=&{\frac {\partial L}{\partial \xi }},\quad &{\frac {dy}{dt}}&=&{\frac {\partial L}{\partial \eta }},\quad &{\frac {dz}{dt}}&=&{\frac {\partial L}{\partial \zeta }}.\end{matrix}},$

All these relations are valid for the reference system (x, y, z, t) used here, as well as for any other reference system (x', y', z', t'), which is connected with it by equations 1).