« Page:Poincaré - Sur la dynamique de l’électron.djvu/13 » : différence entre les versions
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<center><math>\frac{1}{\Delta}\frac{\partial A\Delta}{\partial\epsilon}=\frac{dA}{d\epsilon}+\sum\frac{dA\frac{\partial U}{\partial\epsilon}}{dx}</math></center> |
<center><math>\frac{1}{\Delta}\frac{\partial A\Delta}{\partial\epsilon}=\frac{dA}{d\epsilon}+\sum\frac{dA\frac{\partial U}{\partial\epsilon}}{dx}</math></center> |
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On a donc |
On a donc : |
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{{MathForm1|(17)|<math>\frac{1}{\Delta}\frac{\partial}{\partial\epsilon}\left(\rho\Delta\frac{\partial U}{\partial t}\right)=\frac{d\rho\frac{\partial U}{\partial t}}{ |
{{MathForm1|(17)|<math>\frac{1}{\Delta}\frac{\partial}{\partial\epsilon}\left(\rho\Delta\frac{\partial U}{\partial t}\right)=\frac{d\rho\frac{\partial U}{\partial t}}{\partial\epsilon}+\frac{d\left(\rho\frac{\partial U}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dx}+\frac{d\left(\rho\frac{\partial U}{\partial t}\frac{\partial V}{\partial\epsilon}\right)}{dy}+\frac{d\left(\rho\frac{\partial U}{\partial t}\frac{\partial W}{\partial\epsilon}\right)}{dz},</math>}} |
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{{MathForm1|(17<sup>bis</sup>)|<math>\frac{1}{\Delta}\frac{\partial}{\partial t}\left(\rho\Delta\frac{\partial U}{\partial\epsilon}\right)=\frac{d\rho\frac{\partial U}{\partial\epsilon}}{dt}+\frac{d\left(\rho\frac{\partial U}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dx}+\frac{d\left(\rho\frac{\partial V}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dy}+\frac{d\left(\rho\frac{\partial W}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dz}.</math>}} |
{{MathForm1|(17<sup>bis</sup>)|<math>\frac{1}{\Delta}\frac{\partial}{\partial t}\left(\rho\Delta\frac{\partial U}{\partial\epsilon}\right)=\frac{d\rho\frac{\partial U}{\partial\epsilon}}{dt}+\frac{d\left(\rho\frac{\partial U}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dx}+\frac{d\left(\rho\frac{\partial V}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dy}+\frac{d\left(\rho\frac{\partial W}{\partial t}\frac{\partial U}{\partial\epsilon}\right)}{dz}.</math>}} |
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Les 2<sup>ds</sup> membres de (17) et (17 |
Les 2<sup>ds</sup> membres de (17) et (17<sup>bis</sup>) doivent être égaux et, si l’on se souvient que |
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<center><math>\frac{\partial U}{\partial t}=\xi,\quad\frac{\partial U}{\partial\epsilon}\delta\epsilon=\delta U,\quad\frac{d\rho\xi}{d\epsilon}\delta\epsilon=\delta\rho\xi</math></center> |
<center><math>\frac{\partial U}{\partial t}=\xi,\quad\frac{\partial U}{\partial\epsilon}\delta\epsilon=\delta U,\quad\frac{d\rho\xi}{d\epsilon}\delta\epsilon=\delta\rho\xi</math></center> |
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