Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/31

If the variations $\delta \omega _{1},\delta \omega _{2}$, etc. are capable both of positive and of negative values, we must have

${\frac {dU}{d{\ddot {\omega }}_{1}}}=\Omega _{1},$${\frac {dU}{d{\ddot {\omega }}_{2}}}=\Omega _{2},$etc.,

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or${\frac {dU}{d{\ddot {\omega }}_{1}}}={\frac {dV}{\omega _{1}}},$${\frac {dU}{d{\ddot {\omega }}_{2}}}={\frac {dV}{\omega _{2}}},$etc.

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To illustrate the use of these equations in a case in which $d\omega _{1},d\omega _{2}$, etc. are not exact differentials, we may apply them to the problem of the rotation of a rigid body of which one point is fixed. If $d\omega _{1},d\omega _{2},d\omega _{3}$ denote infinitesimal rotations about the principal axes which pass through the fixed point, $\Omega _{1},\Omega _{2},\Omega _{3}$, will denote the moments of the impressed forces about these axes, and the value of $U$ will be given by the formula

${\begin{aligned}2U&=(a+b+c)({\dot {\omega }}_{1}^{2}+{\dot {\omega }}_{2}^{2}+{\dot {\omega }}_{3}^{2})^{2}-({\dot {\omega }}_{1}^{2}+{\dot {\omega }}_{2}^{2}+{\dot {\omega }}_{3}^{2})(a{\dot {\omega }}_{1}^{2}+b{\dot {\omega }}_{2}^{2}+c{\dot {\omega }}_{3}^{2})\\&+2(b-c){\dot {\omega }}_{2}^{2}{\dot {\omega }}_{3}^{2}{\dot {\omega }}_{1}^{2}+2(c-a){\dot {\omega }}_{3}^{2}{\dot {\omega }}_{1}^{2}{\dot {\omega }}_{2}^{2}+2(a-b){\dot {\omega }}_{1}^{2}{\dot {\omega }}_{2}^{2}{\dot {\omega }}_{3}^{2}\\&+(b+c){\ddot {\omega }}_{1}^{2}+(c+a){\ddot {\omega }}_{2}^{2}+(a+b){\ddot {\omega }}_{3}^{2},\end{aligned}}$

where $a,b$, and $c$ are constants, $a+b,b+c,c+a$ being the moments of inertia about the three axes. Hence,

${\frac {dU}{d{\ddot {\omega }}_{1}}}=(b-c){\dot {\omega }}_{2}{\dot {\omega }}_{3}+(b+c){\ddot {\omega }}_{1},$	${\frac {dU}{d{\ddot {\omega }}_{2}}}=(c-a){\dot {\omega }}_{3}{\dot {\omega }}_{1}+(c+a){\ddot {\omega }}_{2},$
${\frac {dU}{d{\ddot {\omega }}_{3}}}=(a-b){\dot {\omega }}_{1}{\dot {\omega }}_{2}+(a+b){\ddot {\omega }}_{3};$

and the equations of motion are

${\ddot {\omega }}_{1}={\frac {(c-b){\dot {\omega }}_{2}{\dot {\omega }}_{3}+\Omega _{1}}{c+b}},$ ${\ddot {\omega }}_{2}={\frac {(a-c){\dot {\omega }}_{3}{\dot {\omega }}_{1}+\Omega _{2}}{a+c}},$ ${\ddot {\omega }}_{3}={\frac {(b-a){\dot {\omega }}_{1}{\dot {\omega }}_{2}+\Omega _{3}}{b+a}}\cdot$