Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/27

any function of the velocity, the terms due to that resistance in the general formula of motion may be expressed in the form

$\delta \left[\phi (v){\frac {\dot {x}}{v}}{\ddot {x}}+\phi (v){\frac {\dot {y}}{v}}{\ddot {y}}+\phi (v){\frac {\dot {z}}{v}}{\ddot {z}}\right],$

where $v$ denotes the velocity and $\phi (v)$ the resistance. But

${\frac {{\dot {x}}{\ddot {x}}}{v}}+{\frac {{\dot {y}}{\ddot {y}}}{v}}+{\frac {{\dot {z}}{\ddot {z}}}{v}}={\frac {dv}{dt}}={\dot {v}}.$

The terms due to the resistance reduce, therefore, to

$\delta [\phi (v){\dot {v}}],$

or,$\delta {\frac {d}{dt}}f(v),$

where $f$ denotes the primitive of the function denoted by $\phi$.

Discontinuous Changes of Velocity.—Formula (9), which relates to discontinuous changes of velocity, is capable of similar transformations.

If we set$w^{2}=\Delta {\dot {x}}^{2}+\Delta {\dot {y}}^{2}+\Delta {\dot {z}}^{2},$

the formula reduces to

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

(10)

where $X,Y,Z$ are to be regarded as constant. If ${\mathfrak {S}}({\text{P}}dp)$ represents the sum of the moments of the impulsive forces, and we regard ${\text{P}}$ as constant, we have

$\delta [{\mathfrak {S}}({\text{P}}\Delta {\dot {p}})-\textstyle \sum \displaystyle ({\tfrac {1}{2}}mw^{2})]\leqq 0.$

(19)

The expressions affected by $\delta$ in these formulæ have a greater value than they would receive from any other changes of velocity consistent with the constraints of the system.

Deduction of other Properties of Motion.

The principles which have been established furnish a convenient point of departure for the demonstration of various properties of motion relating to maxima and minima. We may obtain several such properties by considering how the accelerations of a system, at a given instant, will be modified by changes of the forces or of the constraints to which the system is subject. Let us suppose that the forces $X,Y,Z$ of a system receive the increments $X',Y',Z'$, in consequence of which, and of certain additional constraints, which do not produce any discontinuity in the velocities, the components of acceleration ${\dot {x}},{\dot {y}},{\dot {z}}$ receive the increments ${\ddot {x}},{\ddot {y}},{\ddot {z}}$. The expression

${\begin{aligned}\textstyle \sum \displaystyle [(X+X')({\ddot {x}}+{\ddot {x}}')&+(Y+Y')({\ddot {y}}+{\ddot {y}}')+(Z+Z')({\ddot {z}}+{\ddot {z}}')\\&-{\tfrac {1}{2}}m{({\ddot {x}}+{\ddot {x}}')^{2}+({\ddot {y}}+{\ddot {y}}')^{2}+({\ddot {z}}+{\ddot {z}}')^{2}}]\end{aligned}}$

(20)

will be the greatest possible for any values of ${\ddot {x}},{\ddot {y}},{\ddot {z}}$ consistent with the constraints. But this expression may be divided into three parts,

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

(21)

$\textstyle \sum \displaystyle [X{\ddot {x}}'+Y{\ddot {y}}'+Z{\ddot {z}}'-m({\ddot {x}}{\ddot {x}}'+{\ddot {y}}{\ddot {y}}'+{\ddot {z}}{\ddot {z}}')],$

(22)

and$\textstyle \sum \displaystyle [X'{\ddot {x}}'+Y'{\ddot {y}}'+Z'{\ddot {z}}'-{\tfrac {1}{2}}m({\ddot {x}}^{'2}+{\ddot {y}}^{'2}+{\ddot {z}}^{'2})].$

(23)