Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/30

of which we can perfectly define any infinitesimal change in the configuration of the system; and let

${\dot {\omega }}_{1}={\frac {d\omega _{1}}{dt}},\,\,{\dot {\omega }}_{2}={\frac {d\omega _{2}}{dt}}\,\,{\text{etc.,}}$

where $d\omega _{1},d\omega _{2}$ are to be determined by the change in the configuration in the interval of time $dt$; and let

${\ddot {\omega }}_{1}={\frac {d{\dot {\omega }}_{1}}{dt}},\,\,{\ddot {\omega }}_{2}={\frac {d{\dot {\omega }}_{2}}{dt}}\,\,{\text{etc.}}$

It is evident that $U$ can be expressed in terms of ${\dot {\omega }}_{1},{\dot {\omega }}_{2}$, etc., ${\ddot {\omega }}_{1},{\ddot {\omega }}_{2}$, etc., and the quantities which express the configuration of the system, and that (since $\delta$ is used to denote a variation which does not affect the configuration or the velocities),

$\delta U={\frac {dU}{d{\ddot {\omega }}_{1}}}\delta {\ddot {\omega }}_{1}+{\frac {dU}{d{\ddot {\omega }}_{2}}}\delta {\ddot {\omega }}_{2}+{\text{etc.}}$

Moreover, since the quantities $p$ in the general formula are entirely determined by the configuration of the system

${\ddot {p}}={\frac {dp}{d{\omega }_{1}}}{\dot {\omega }}_{1}+{\frac {dp}{d{\dot {\omega }}_{2}}}{\dot {\omega }}_{2}+{\text{etc.,}}$

where ${\frac {dp}{d\omega _{1}}}$ denotes the ratio of simultaneous values of $dp$ and $d\omega _{1}$, when $d\omega _{2}$, etc. are equal to zero, and ${\frac {dp}{d\omega _{2}}}$, etc are to be interpreted on the same principle. Multiplying by $P$, and taking the sum with respect to the several forces, we have

${\mathfrak {S}}(P{\dot {p}})=\Omega _{1}{\dot {\omega }}_{1}+\Omega _{2}{\dot {\omega }}_{2}+{\text{etc.,}}$

If we differentiate with respect to $t$, and take the variation denoted by $\delta$, we obtain

${\mathfrak {S}}(P\delta {\ddot {p}})=\Omega _{1}\delta {\ddot {\omega }}_{1}+\Omega _{2}\delta {\ddot {\omega }}_{2}+{\text{etc.}}$

The general formula (12) is thus reduced to the form

$\left(\Omega _{1}-{\frac {dU}{d{\ddot {\omega }}_{1}}}\right)\delta {\ddot {\omega }}_{1}+\left(\Omega _{2}-{\frac {dU}{d{\ddot {\omega }}_{2}}}\right)\delta {\ddot {\omega }}_{2}+{\text{etc.}}\geqq 0.$

(26)

If the forces have a potential $V$, we may write

$\left({\frac {dV}{d\omega _{1}}}-{\frac {dU}{d{\ddot {\omega }}_{1}}}\right)\delta {\ddot {\omega }}_{1}+\left({\frac {dV}{d\omega _{2}}}-{\frac {dU}{d{\ddot {\omega }}_{2}}}\right)\delta {\ddot {\omega }}_{2}+{\text{etc.,}}$

(27)

where ${\frac {dV}{d\omega _{1}}}$ denotes the ratio of $dV$ and $d\omega _{1}$ when $d\omega _{2}$, etc. have the value zero, and the analogous expressions are to be interpreted on the same principle.