Page:Becquerel - Le Principe de relativité et la théorie de la gravitation, 1922.djvu/121

101

chapitre IX. — dynamique de la relativité.

(6-9) peuvent s’écrire

(7-9)

${\frac {d}{dt}}\!\left({\frac {m_{0}}{\alpha }}{\frac {dx}{dt}}\right)\!=\mathrm {F} _{x};\;\;{\frac {d}{dt}}\!\left({\frac {m_{0}}{\alpha }}{\frac {dy}{dt}}\right)\!=\mathrm {F} _{y};\;\;{\frac {d}{dt}}\!\left({\frac {m_{0}}{\alpha }}{\frac {dz}{dt}}\right)\!=\mathrm {F} _{z}.\!$

Nous avons, en effet, en posant $\beta ={\frac {v}{c}},$

(8-9)

$\left\{{\begin{aligned}{\frac {d}{dt}}\left({\frac {m_{0}}{\alpha }}{\frac {dx}{dt}}\right)&={\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {dx}{dt}}\right)\\&\qquad ={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {d^{2}x}{dt^{2}}}+{\frac {m_{0}\beta }{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}{\frac {d\beta }{dt}}{\frac {dx}{dt}},\\[0.75ex]{\frac {d}{dt}}\left({\frac {m_{0}}{\alpha }}{\frac {dy}{dt}}\right)&={\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {dy}{dt}}\right)\\&\qquad ={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {d^{2}y}{dt^{2}}}+{\frac {m_{0}\beta }{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}{\frac {d\beta }{dt}}{\frac {dy}{dt}},\\[0.75ex]{\frac {d}{dt}}\left({\frac {m_{0}}{\alpha }}{\frac {dz}{dt}}\right)&={\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {dz}{dt}}\right)\\&\qquad ={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {d^{2}z}{dt^{2}}}+{\frac {m_{0}\beta }{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}{\frac {d\beta }{dt}}{\frac {dz}{dt}},\end{aligned}}\right.$

à l’instant $t$ considéré, d’une part,

${\begin{aligned}{\frac {dx}{dt}}&=c\beta ,&c{\frac {d\beta }{dt}}&={\frac {d^{2}x}{dt^{2}}}\end{aligned}}$

et, d’autre part,

${\frac {dy}{dt}}={\frac {dz}{dt}}=0\,;$

nous avons donc, d’une part,

${\begin{aligned}{\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {dx}{dt}}\right)&=m_{0}\left[{\frac {1}{\sqrt {1-\beta ^{2}}}}+{\frac {\beta ^{2}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}\right]{\frac {d^{2}x}{dt^{2}}}\\&={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}{\frac {d^{2}x}{dt^{2}}}\\[0.75ex]&={\frac {m_{0}}{\alpha ^{3}}}{\frac {d^{2}x}{dt^{2}}}\,;\end{aligned}}$

d’autre part,

${\begin{aligned}{\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {dy}{dt}}\right)&={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {d^{2}y}{dt^{2}}}={\frac {m_{0}}{\alpha }}{\frac {d^{2}y}{dt^{2}}},\\{\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {dz}{dt}}\right)&={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}{\frac {d^{2}z}{dt^{2}}}={\frac {m_{0}}{\alpha }}{\frac {d^{2}z}{dt^{2}}}\,;\end{aligned}}$

ce qui démontre l’identité des équations (6-9) et (7-9).

Sous la forme (7-9) absolument symétrique par rapport aux coordonnées, les équations sont indépendantes de l’orientation des axes, c’est-à-dire qu’elles sont valables quelle que soit