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 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   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),  Moreover, since the quantities $$p$$ in the general formula are entirely determined by the configuration of the system  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  If we differentiate with respect to $$t$$, and take the variation denoted by $$\delta$$, we obtain  The general formula (12) is thus reduced to the form  If the forces have a potential $$V$$, we may write  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.