Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/386

378

FRAGMENTS.

IV.

Autre fragment sur la rotation d’un système quelconque.

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Ainsi l’on a, en général (p. 215),

${\begin{aligned}d\xi ^{2}+d\eta ^{2}+d\zeta ^{2}=&(c\,d\mathrm {Q} -b\,d\mathrm {R} +da)^{2}\\&+(a\,d\mathrm {R} -c\,d\mathrm {P} +db)^{2}+(b\,d\mathrm {P} -a\,d\mathrm {Q} +dc)^{2}.\end{aligned}}$

Si les forces accélératrices ne dépendent que de la situation respective des corps, elles ne seront fonctions que de $a,\,b,\,c.$ Faisant

${\begin{aligned}\scriptstyle \mathrm {T} =&\scriptstyle {\frac {1}{2}}\left[\scriptstyle {\frac {d\mathrm {P} ^{2}}{dt^{2}}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} \left(b^{2}+c^{2}\right)+{\frac {d\mathrm {Q} ^{2}}{dt^{2}}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} \left(a^{2}+c^{2}\right)+{\frac {d\mathrm {R} ^{2}}{dt^{2}}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} \left(a^{2}+b^{2}\right)\right]\\&\scriptstyle -{\frac {d\mathrm {Q} }{dt}}{\frac {d\mathrm {R} }{dt}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} \,bc-{\frac {d\mathrm {P} }{dt}}{\frac {d\mathrm {R} }{dt}}\displaystyle \mathrm {S} \scriptstyle {\mathrm {m} \,ac-{\frac {d\mathrm {P} }{dt}}{\frac {d\mathrm {Q} }{dt}}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} \,ab\\&\scriptstyle +{\frac {d\mathrm {P} }{dt}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} {\frac {b\,dc-c\,db}{dt}}+{\frac {d\mathrm {Q} }{dt}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} {\frac {c\,da-a\,dc}{dt}}\\&\scriptstyle +{\frac {d\mathrm {R} }{dt}}\displaystyle \mathrm {S} \scriptstyle \mathrm {m} {\frac {a\,db-b\,da}{dt}}+\displaystyle \mathrm {S} \scriptstyle \mathrm {m} {\frac {da^{2}+db^{2}+dc^{2}}{2dt^{2}}}.\end{aligned}}$

On aura, relativement aux variables $d\mathrm {P} ,\,d\mathrm {Q} ,\,d\mathrm {R} ,$

${\frac {\delta \mathrm {T} }{\delta d\mathrm {P} }}\delta d\mathrm {P} +{\frac {\delta \mathrm {T} }{\delta d\mathrm {Q} }}\delta d\mathrm {Q} +{\frac {\delta \mathrm {T} }{\delta d\mathrm {R} }}\delta d\mathrm {R} =0,$

savoir (art. 15, p. 217)

${\begin{aligned}{\frac {\delta \mathrm {T} }{\delta d\mathrm {P} }}(d\delta \mathrm {P} +d\mathrm {Q} \delta \mathrm {R} -d\mathrm {R} \delta \mathrm {Q} )&+{\frac {\delta \mathrm {T} }{\delta d\mathrm {Q} }}(d\delta \mathrm {Q} +d\mathrm {R} \delta \mathrm {P} -d\mathrm {P} \delta \mathrm {R} )\\&+{\frac {\delta \mathrm {T} }{\delta d\mathrm {R} }}(d\delta \mathrm {R} +d\mathrm {P} \delta \mathrm {Q} -d\mathrm {Q} \delta \mathrm {P} )=0\,;\end{aligned}}$

d’où l’on tire les équations

${\begin{alignedat}{4}&d{\frac {\delta \mathrm {T} }{\delta d\mathrm {P} }}&-&{\frac {\delta \mathrm {T} }{\delta d\mathrm {Q} }}d\mathrm {R} &+&{\frac {\delta \mathrm {T} }{\delta d\mathrm {R} }}d\mathrm {Q} &=&0,\\&d{\frac {\delta \mathrm {T} }{\delta d\mathrm {Q} }}&-&{\frac {\delta \mathrm {T} }{\delta d\mathrm {P} }}d\mathrm {P} &+&{\frac {\delta \mathrm {T} }{\delta d\mathrm {R} }}d\mathrm {R} &=&0,\\&d{\frac {\delta \mathrm {T} }{\delta d\mathrm {R} }}&-&{\frac {\delta \mathrm {T} }{\delta d\mathrm {P} }}d\mathrm {Q} &+&{\frac {\delta \mathrm {T} }{\delta d\mathrm {Q} }}d\mathrm {P} &=&0,\end{alignedat}}$