Page:Henri Poincaré - Les méthodes nouvelles de la mécanique céleste, Tome 1, 1892.djvu/201

189

EXPOSANTS CARACTERISTIQUES.

bien tous les déterminants contenus dans la matrice

$\left|\left|{\frac {d\mathrm {F} _{i}}{dx_{k}}}\right|\right|\qquad (i=1,\,2,\,\ldots ,\,p\,;\;\;k=1,\,\ldots ,\,n)$

seront nuls pour tous les points de la solution périodique génératrice.

Supposons, en effet, pour fixer les idées,

$n=4,\quad p=2.$

Nous aurons alors les équations suivantes

${\begin{array}{c}\left.{\begin{aligned}{\dfrac {d\mathrm {F} _{1}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{1}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{1}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{1}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\beta _{i}}}&=0,\\{\dfrac {d\mathrm {F} _{2}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{2}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{2}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{2}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\beta _{i}}}&=0\end{aligned}}\right\}\;(i=1,\,2,\,3,\,4)\\[1.5ex]\quad \qquad {\begin{aligned}{\dfrac {d\mathrm {F} _{1}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\tau }}+{\dfrac {d\mathrm {F} _{1}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\tau }}+{\dfrac {d\mathrm {F} _{1}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\tau }}+{\dfrac {d\mathrm {F} _{1}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\tau }}&=0,\\{\dfrac {d\mathrm {F} _{2}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\tau }}+{\dfrac {d\mathrm {F} _{2}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\tau }}+{\dfrac {d\mathrm {F} _{2}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\tau }}+{\dfrac {d\mathrm {F} _{2}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\tau }}&=0.\end{aligned}}\end{array}}$

De ces équations il est permis de conclure :

Ou bien que tous les déterminants contenus dans la matrice

$\left|\left|{\begin{array}{ccccc}{\dfrac {d\mathrm {F} _{1}}{dx_{1}}}&{\dfrac {d\mathrm {F} _{1}}{dx_{2}}}&{\dfrac {d\mathrm {F} _{1}}{dx_{3}}}&{\dfrac {d\mathrm {F} _{1}}{dx_{4}}}\\{\dfrac {d\mathrm {F} _{2}}{dx_{1}}}&{\dfrac {d\mathrm {F} _{2}}{dx_{2}}}&{\dfrac {d\mathrm {F} _{2}}{dx_{3}}}&{\dfrac {d\mathrm {F} _{2}}{dx_{4}}}\\\end{array}}\right|\right|$

sont nuls à la fois ; ou bien que tous les déterminants contenus dans la matrice

(1)

$\left|\left|{\begin{array}{ccccc}{\dfrac {d\psi _{1}}{d\beta _{1}}}&{\dfrac {d\psi _{1}}{d\beta _{2}}}&{\dfrac {d\psi _{1}}{d\beta _{3}}}&{\dfrac {d\psi _{1}}{d\beta _{4}}}&{\dfrac {d\psi _{1}}{d\tau }}\\{\dfrac {d\psi _{2}}{d\beta _{1}}}&{\dfrac {d\psi _{2}}{d\beta _{2}}}&{\dfrac {d\psi _{2}}{d\beta _{3}}}&{\dfrac {d\psi _{2}}{d\beta _{4}}}&{\dfrac {d\psi _{2}}{d\tau }}\\{\dfrac {d\psi _{3}}{d\beta _{1}}}&{\dfrac {d\psi _{3}}{d\beta _{2}}}&{\dfrac {d\psi _{3}}{d\beta _{3}}}&{\dfrac {d\psi _{3}}{d\beta _{4}}}&{\dfrac {d\psi _{3}}{d\tau }}\\{\dfrac {d\psi _{4}}{d\beta _{1}}}&{\dfrac {d\psi _{4}}{d\beta _{2}}}&{\dfrac {d\psi _{4}}{d\beta _{3}}}&{\dfrac {d\psi _{4}}{d\beta _{4}}}&{\dfrac {d\psi _{4}}{d\tau }}\end{array}}\right|\right|$

sont nuls à la fois, ainsi que leurs mineurs du premier ordre.