Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/22

form the subject of inquiry. Its position in the immediate future is naturally specified by

$x+{\dot {x}}dt+{\tfrac {1}{2}}{\ddot {x}}dt^{2}+etc.,$$y+{\dot {y}}dt+{\tfrac {1}{2}}{\ddot {y}}dt^{2}+etc.,$$z+{\dot {z}}dt+{\tfrac {1}{2}}{\ddot {z}}dt^{2}+etc.,$

and we may regard the variations of these expressions as corresponding to the $\delta x,\delta y,\delta z$ of the statical problem. It is evidently sufficient to take account of the first term of these expressions of which the variation is not zero. Now, $x,y,z$, as has already been said, are to be regarded as constant. With respect to the terms containing ${\dot {x}},{\dot {y}},{\dot {z}}$, two cases are to be distinguished, according as there is, or is not, a finite change of velocity at the instant considered.

Let us first consider the most important case, in which there is no discontinuous change of velocity. In this case, ${\dot {x}},{\dot {y}},{\dot {z}}$ are not to be regarded as variable (by $\delta$), and the variations of the above expressions are represented by

${\tfrac {1}{2}}\delta {\ddot {x}}dt^{2},$${\tfrac {1}{2}}\delta {\ddot {y}}dt^{2},$${\tfrac {1}{2}}\delta {\ddot {z}}dt^{2},$

which are, therefore, to be substituted for $\delta x,\delta y,\delta z$ in the general formula of equilibrium (8) to adapt it to the conditions of a dynamical problem. By this substitution (in which the common factor ${\tfrac {1}{2}}dt^{2}$ may of course be omitted), and the addition of the terms expressing the reaction against acceleration, we obtain formula (6).

But if the circumstances are such that there is (or may be) a discontinuity in the values of ${\dot {x}},{\dot {y}},{\dot {z}}$ at the instant considered, it is necessary to distinguish the values of these xpressions before and after the abrupt change. For this purpose, we may apply ${\dot {x}},{\dot {y}},{\dot {z}}$ to the original values, and denote the changed values by ${\dot {x}}+\Delta {\dot {x}},{\dot {y}}+\Delta {\dot {y}},{\dot {z}}+\Delta {\dot {z}}$. The value of $x$ at a time very shortly subsequent to the instant considered, will be expressed by ${\dot {x}}+({\dot {x}}+\Delta x)dt+etc.,$, in which we may regard $\Delta {\dot {x}}$ as subject to the variation denoted by $\delta$. The variation of the expression is therefore $\delta \Delta {\dot {x}}dt$. Instead of $-m{\ddot {x}}$, which expresses the reaction against acceleration, we need in the present case $-m\Delta {\dot {x}}$ to express the reaction against the abrupt change of velocity. A reaction against such a change of velocity is, of course, to be regarded as infinite in intensity in comparison with reactions due to acceleration, and ordinary forces (such as cause acceleration) may be neglected in comparison. If, however, we conceive of the system as acted on by impulsive forces (i.e., such as have no finite duration, but are capable of producing finite changes of velocity, and are measured numerically by the discontinuities of velocity which they produce in the unit of mass), these forces should be combined with the reactions due to the discontinuities of velocity in the general formula which determines these discontinuities. If the impulsive forces are specified by $X,Y,Z$, the formula will be

$[(X-m\Delta {\dot {x}})\delta \Delta {\dot {x}}+(Y-m\Delta {\dot {y}})\delta \Delta {\dot {y}}+(Z-m\Delta {\dot {z}})\delta \Delta {\dot {z}}]\geqq 0.$

(9)