Page:Joseph Louis de Lagrange - Œuvres, Tome 11.djvu/380

362

MÉCANIQUE ANALYTIQUE

et il n’y aura plus qu’à substituer ces valeurs dans celles de ${\frac {da}{dt}},\,{\frac {db}{dt}},\ldots$ de l’article précédent.

En faisant ces substitutions et ordonnant les termes par rapport aux différences partielles de $\Omega ,$ on voit d’abord que le coefficient de ${\frac {\partial \Omega }{\partial a}}$ est nul dans la valeur de ${\frac {da}{dt}},$ que celui de ${\frac {\partial \Omega }{\partial b}}$ est nul dans la valeur de ${\frac {db}{dt}},\ldots .$

Ensuite, si, pour représenter la valeur de ${\frac {da}{dt}},$ on emploie la formule

${\frac {da}{dt}}=(a,b){\frac {\partial \Omega }{\partial b}}+(a,c){\frac {\partial \Omega }{\partial c}}+\ldots ,$

on aura

${\begin{aligned}(a,b)=&+{\frac {\partial a}{\partial \lambda }}{\frac {\partial b}{\partial \alpha }}+{\frac {\partial a}{\partial \mu }}{\frac {\partial b}{\partial \beta }}+{\frac {\partial a}{\partial \nu }}{\frac {\partial b}{\partial \gamma }}+\ldots \\&-{\frac {\partial a}{\partial \alpha }}{\frac {\partial b}{\partial \lambda }}-{\frac {\partial a}{\partial \beta }}{\frac {\partial b}{\partial \mu }}-{\frac {\partial a}{\partial \gamma }}{\frac {\partial b}{\partial \nu }}-\ldots ,\\(a,c)=&+{\frac {\partial a}{\partial \lambda }}{\frac {\partial c}{\partial \alpha }}+{\frac {\partial a}{\partial \mu }}{\frac {\partial c}{\partial \beta }}+{\frac {\partial a}{\partial \nu }}{\frac {\partial c}{\partial \gamma }}+\ldots \\&-{\frac {\partial a}{\partial \alpha }}{\frac {\partial c}{\partial \lambda }}-{\frac {\partial a}{\partial \beta }}{\frac {\partial c}{\partial \mu }}-{\frac {\partial a}{\partial \gamma }}{\frac {\partial c}{\partial \nu }}-\ldots ,\\\end{aligned}}$

 

Et, pour avoir la valeur de ${\frac {db}{dt}},$ il n’y aura qu’à changer dans ces formules $a$ en $b$ et $b$ en $a,$ en remarquant que l’on a

$(b,a)=-(a,b)\,;$

on aura ainsi

${\frac {db}{dt}}=-(a,b){\frac {\partial \Omega }{\partial a}}+(b,c){\frac {\partial \Omega }{\partial c}}+\ldots ,$

 

${\begin{aligned}(b,c)=&+{\frac {\partial b}{\partial \lambda }}{\frac {\partial c}{\partial \alpha }}+{\frac {\partial b}{\partial \mu }}{\frac {\partial c}{\partial \beta }}+{\frac {\partial b}{\partial \nu }}{\frac {\partial c}{\partial \gamma }}+\ldots \\&-{\frac {\partial b}{\partial \alpha }}{\frac {\partial c}{\partial \lambda }}-{\frac {\partial b}{\partial \beta }}{\frac {\partial c}{\partial \mu }}-{\frac {\partial b}{\partial \gamma }}{\frac {\partial c}{\partial \nu }}-\ldots ,\end{aligned}}$