Page:Joseph Louis de Lagrange - Œuvres, Tome 11.djvu/488

470

NOTES.

coordonnées sont $x_{i},\,y_{i},\,z_{i},$ On a identiquement

(5)

$\left\{{\begin{aligned}&\sum m_{i}\left({\frac {\partial x_{i}}{\partial q_{m}}}{\frac {dx'_{i}}{dt}}+{\frac {\partial y_{i}}{\partial q_{m}}}{\frac {dy'_{i}}{dt}}+{\frac {\partial z_{i}}{\partial q_{m}}}{\frac {dz'_{i}}{dt}}\right)\\={\frac {d}{dt}}&\sum m_{i}\left(x'_{i}{\frac {\partial x_{i}}{\partial q_{m}}}+y'_{i}{\frac {\partial y_{i}}{\partial q_{m}}}+z'_{i}{\frac {\partial z_{i}}{\partial q_{m}}}\right)\\-&\sum m_{i}\left(x'_{i}{\frac {d}{dt}}{\frac {\partial x_{i}}{\partial q_{m}}}+y'_{i}{\frac {d}{dt}}{\frac {\partial y_{i}}{\partial q_{m}}}+z'_{i}{\frac {d}{dt}}{\frac {\partial z_{i}}{\partial q_{m}}}\right).\end{aligned}}\right.$

$x_{i},\,y_{i},\,z_{i}$ sont donnés, par hypothèse, en fonction de $q_{1},\,q_{2},\,\ldots ,\,q_{k}$ et $t\,;$ en différentiant les formules qui les expriment, on aura

(6)

$\left\{{\begin{aligned}x'_{i}=&{\frac {\partial x_{i}}{\partial t}}+{\frac {\partial x_{i}}{\partial q_{1}}}q'_{1}+{\frac {\partial x_{i}}{\partial q_{2}}}q'_{2}+\ldots +{\frac {\partial x_{i}}{\partial q_{k}}}q'_{k},\\y'_{i}=&{\frac {\partial y_{i}}{\partial t}}+{\frac {\partial y_{i}}{\partial q_{1}}}q'_{1}+{\frac {\partial y_{i}}{\partial q_{2}}}q'_{2}+\ldots +{\frac {\partial y_{i}}{\partial q_{k}}}q'_{k},\\z'_{i}=&{\frac {\partial z_{i}}{\partial t}}+{\frac {\partial z_{i}}{\partial q_{1}}}q'_{1}+{\frac {\partial z_{i}}{\partial q_{2}}}q'_{2}+\ldots +{\frac {\partial z_{i}}{\partial q_{k}}}q'_{k},\end{aligned}}\right.$

d’où l’on conclut

${\frac {\partial x_{i}}{\partial q_{m}}}={\frac {\partial x'_{i}}{\partial q'_{m}}},\qquad {\frac {\partial y_{i}}{\partial q_{m}}}={\frac {\partial y'_{i}}{\partial q'_{m}}},\qquad {\frac {\partial z_{i}}{\partial q_{m}}}={\frac {\partial z'_{i}}{\partial q'_{m}}}\,;$

on a, d’ailleurs,

${\frac {d}{dt}}{\frac {\partial x_{i}}{\partial q_{m}}}={\frac {\partial ^{2}x_{i}}{\partial q_{m}\partial t}}+{\frac {\partial ^{2}x_{i}}{\partial q_{1}\partial q_{m}}}q'_{1}+{\frac {\partial ^{2}x_{i}}{\partial q_{2}\partial q_{m}}}q'_{2}+\ldots +{\frac {\partial ^{2}x_{i}}{\partial q_{k}\partial q_{m}}}q'_{k},$

ce qui équivaut évidemment, d’après la valeur de $x'_{i}$ fournie par l’équation (6), à ${\frac {\partial x'}{\partial q_{m}}}.$ On obtiendrait de même

${\frac {d}{dt}}{\frac {\partial y_{i}}{\partial q_{m}}}={\frac {\partial y'_{i}}{\partial q_{m}}},\qquad {\frac {d}{dt}}{\frac {\partial z_{i}}{\partial q_{m}}}={\frac {\partial z'_{i}}{\partial q_{m}}}.$

Si nous avons égard à ces relations et si, de plus, nous posons

$\mathrm {T} ={\frac {1}{2}}\sum m_{i}\left(x_{i}^{_{'}2}+y_{i}^{_{'}2}+z_{i}^{_{'}2}\right),$

l’équation (4) deviendra

${\frac {d}{dt}}{\frac {\partial \mathrm {T} }{\partial q'_{m}}}-{\frac {\partial \mathrm {T} }{\partial q_{m}}}=\mathrm {Q} _{m}.$

On obtiendra $k$ équations de même forme en attribuant successivement à