Page:Joseph Louis de Lagrange - Œuvres, Tome 11.djvu/383

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SECONDE PARTIE. — SECTION V.

Mais les équations différentielles de l’article 3 donnent

${\frac {d{\cfrac {\partial \mathrm {Z} }{\partial \xi '}}}{dt}}={\frac {\partial \mathrm {Z} }{\partial \xi }},\qquad {\frac {d{\cfrac {\partial \mathrm {Z} }{\partial \psi '}}}{dt}}={\frac {\partial \mathrm {Z} }{\partial \psi }},\qquad \ldots .$

Donc on aura

$\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}={\frac {\partial \mathrm {Z} }{\partial \xi }}\Delta \mathrm {K} ,\qquad \Delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}={\frac {\partial \mathrm {Z} }{\partial \psi }}\Delta \mathrm {K} ,\qquad \ldots .$

Ainsi la formule générale de l’article 11 deviendra par ces substitutions, et après la division par $\Delta \mathrm {K} ,$

${\frac {\partial \Omega }{\partial \mathrm {K} }}dt=\xi '\delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}+\psi '\delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}+\varphi '\delta {\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots -{\frac {\partial \mathrm {Z} }{\partial \xi }}\delta \xi -{\frac {\partial \mathrm {Z} }{\partial \psi }}\delta \psi -{\frac {\partial \mathrm {Z} }{\partial \varphi }}\delta \varphi -\ldots .$

Or, on a

${\begin{aligned}\xi '\delta &{\frac {\partial \mathrm {Z} }{\partial \xi '}}+\psi '\delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}+\varphi '\delta {\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots \\&=\delta \left(\xi '{\frac {\partial \mathrm {Z} }{\partial \xi '}}+\psi '{\frac {\partial \mathrm {Z} }{\partial \psi '}}+\varphi '{\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots \right)-{\frac {\partial \mathrm {Z} }{\partial \xi '}}\delta \xi '-{\frac {\partial \mathrm {Z} }{\partial \psi '}}\delta \psi '-{\frac {\partial \mathrm {Z} }{\partial \varphi '}}\delta \varphi '-\ldots \,;\end{aligned}}$

et comme $\mathrm {Z}$ est censée fonction de $\xi ,\,\psi ,\,\varphi ,\,\ldots$ et de $\xi ',\,\psi ',\,\varphi ',\,\ldots ,$ on aura

${\begin{aligned}\delta \mathrm {Z} =&+{\frac {\partial \mathrm {Z} }{\partial \xi }}\ \delta \xi \ +{\frac {\partial \mathrm {Z} }{\partial \psi }}\ \delta \psi \ +{\frac {\partial \mathrm {Z} }{\partial \varphi }}\ \delta \varphi \ +\ldots \\&+{\frac {\partial \mathrm {Z} }{\partial \xi '}}\delta \xi '+{\frac {\partial \mathrm {Z} }{\partial \psi '}}\delta \psi '+{\frac {\partial \mathrm {Z} }{\partial \varphi '}}\delta \varphi '+\ldots .\end{aligned}}$

Donc l’équation précédente deviendra

${\frac {\partial \Omega }{\partial \mathrm {K} }}dt=\delta \left(\xi '{\frac {\partial \mathrm {Z} }{\partial \xi '}}+\psi '{\frac {\partial \mathrm {Z} }{\partial \psi '}}+\varphi '{\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots -\mathrm {Z} \right),$

dont le second membre doit être une fonction des constantes arbitraires, indépendante de $t.$

21. En effet, si l’on change $\mathrm {Z}$ en $\mathrm {T-V}$ et $\xi ',\,\psi ',\,\varphi ',\,\ldots$ en ${\frac {d\xi }{dt}},\,{\frac {d\psi }{dt}},$ ${\frac {d\varphi }{dt}},\,\ldots$ (art. 3), il est facile de voir que la quantité

$\xi '{\frac {\partial \mathrm {Z} }{\partial \xi '}}+\psi '{\frac {\partial \mathrm {Z} }{\partial \psi '}}+\varphi '{\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots -\mathrm {Z}$