Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/17

I.

ON THE FUNDAMENTAL FORMULÆ OF DYNAMICS.

[American Journal of Mathematics, vol. ii. pp. 49-64, 1879.]

Formation of a new Indeterminate Formula of Motion by the Substitution of the Variations of the Components of Acceleration for the Variations of the Coordinates in the usual Formula.

The laws of motion are frequently expressed by an equation of the form

$\sum [(X-m{\ddot {x}})\delta x+(Y-m{\ddot {y}})\delta y+(Z-m{\ddot {z}})\delta z]=0,$

(1)

in which

$x,y,z$	its rectangular coordinates,
${\ddot {x}},{\ddot {y}},{\ddot {z}}$	the second differential coefficients of the coordinates with respect to the time,
$X,Y,Z$	the components of the forces acting on the particle,
$\delta x,\delta y,\delta z$	any arbitrary variations of the coordinates which are simultaneously possible, and
$\sum$	a summation with respect to all the particles of the system.

It is evident that we may substitute for $\delta x,\delta y,\delta z$ any other expressions which are capable of the same and only of the same sets of simultaneous values.

Now if the nature of the system is such that certain functions $A,B$, etc. of the coordinates must be constant, or given functions of the time, we have

$\sum \left({\frac {dA}{dx}}\delta x+{\frac {dA}{dy}}\delta y+{\frac {dA}{dz}}\delta z\right)=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (2)
$\sum \left({\frac {dB}{dx}}\delta x+{\frac {dB}{dy}}\delta y+{\frac {dB}{dz}}\delta z\right)=0,$
etc.

These are the equations of condition, to which the variations in the general equation of motion (1) are subject. But if $A$ is constant or a determined function of the time, the same must be true of ${\dot {A}}$ and ${\ddot {A}}$. Now

${\dot {A}}=\sum \left({\frac {dA}{dx}}{\dot {x}}+{\frac {dA}{dy}}{\dot {y}}+{\frac {dA}{dz}}{\dot {z}}\right)$

and ${\ddot {A}}=\sum \left({\frac {dA}{dx}}{\ddot {x}}+{\frac {dA}{dy}}{\ddot {y}}+{\frac {dA}{dz}}{\ddot {z}}\right)+H,$