Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/25

the masses of the particles from those which contain the forces, we have

$\textstyle \sum \displaystyle (X\delta {\ddot {x}}+Y\delta {\ddot {y}}+Z\delta {\ddot {z}})-\textstyle \sum \displaystyle [{\tfrac {1}{2}}m\delta ({\ddot {x}}^{2}+{\ddot {y}}^{2})+{\ddot {z}}^{2}]\leqq 0,$

(10)

or, if we write $u$ for the acceleration of a particle,

$\textstyle \sum \displaystyle (X\delta {\ddot {x}}+Y\delta {\ddot {y}}+Z\delta {\ddot {z}})-\delta \textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})\leqq 0.$

(11)

If, instead of terms of the form $X\delta x$ or in addition to such terms, equation (1) had contained terms of the form $P\delta p$, in which $p$ denotes any quantity determined by the configuration of the system, it is evident that these would give terms of the form $P\delta {\ddot {p}}$ in (6), (10) and (11). For the considerations which justified the substitution of $\delta {\ddot {x}},\delta {\ddot {y}},\delta {\ddot {z}}$ for $\delta x,\delta y,\delta z$ in the usual formula were in no respect dependent upon the fact that $x,y,z$ denote rectangular coordinates, but would apply equally to any other quantities which are determined by the configuration of the system.

Hence, if the moments of all the forces of the system are represented by the sum

${\mathfrak {S}}(Pdp),$

the general formula of motion may be written

${\mathfrak {S}}(P\delta {\ddot {p}})-\delta \textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})\leqq 0$

(12)

If the forces admit of a force-function $V$, we have

$\delta {\ddot {V}}-\delta \textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})\leqq 0,$

or$\delta [{\ddot {V}}-\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})]\leqq 0.$

(13)

But if the forces are determined in any way whatever by the configuration and velocities of the system, with or without the time, $X,Y,Z$ and $P$ will be unaffected by the variation denoted by $\delta$, and we may write the formula of motion in the form

$\delta \textstyle \sum \displaystyle (X{\ddot {x}}+Y{\ddot {y}}+Z{\ddot {z}}-{\tfrac {1}{2}}mu^{2})\leqq 0,$

(14)

or$\delta [{\mathfrak {S}}(P{\ddot {p}})-\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})\leqq 0.$

(15)

If the forces are determined by the configuration alone, or the configuration and the time, $\delta {\dot {X}}=0,\delta {\dot {Y}}=0,\delta {\dot {Z}}=0$, and the

general formula may be written

$\delta \left[{\frac {d}{dt}}\textstyle \sum \displaystyle (X{\ddot {x}}+Y{\ddot {y}}+Z{\ddot {z}})-\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})\right]\quad \leqq 0,$

(16)

or$\delta \left[{\frac {d}{dt}}{\mathfrak {S}}(P{\dot {p}})-\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})\right]\quad \leqq 0.$

(17)

The quantity affected by $\delta$ in any one of the last five formulæ has not only a maximum value, but absolutely the greatest value consistent with the constraints of the system. This may be shown in reference to (15) by giving to ${\dot {p}},{\dot {x}},{\dot {y}},{\dot {z}}$, contained explicitly or implicitly in the expression affected by $\delta$, any possible finite