Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/29

whence it appears that $\textstyle \sum \displaystyle (X'{\ddot {x}}'+Y'{\ddot {y}}'+Z'{\ddot {z}}')$ differs from ${\mathfrak {S}}(Q{\ddot {q}}')$ only by quantities which are independent of the relative acceleration due to the additional forces and constraints. It follows that these relative accelerations are such as to make

${\mathfrak {S}}(Q{\ddot {q}}')-\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{'2})$

(25)

a maximum.

It will be observed that the condition which determines these relative accelerations is of precisely the same form as that which determines absolute accelerations.

An important case is that in which new constraints are added but no new forcea The relative accelerations are determined in this case by the condition that $\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{'2})$ is a minimum. In any case of motion, in which finite forces do not act at points, lines or surfaces, we may first calculate the accelerations which would be produced if there were no constraints, and then determine the relative accelerations due to the constraints by the condition that $\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{'2})$ is a minimum. This is Gauss's principle of least constraint.^[1]

Again, in any case of motion, we may suppose $u$ to denote the acceleration which would be produced by the constraints alone, and $u'$ the relative acceleration produced by the forces; we then have

$\textstyle \sum \displaystyle [m({\ddot {x}}{\ddot {x}}'+{\ddot {y}}{\ddot {y}}'+{\ddot {z}}{\ddot {z}}')]=0,$

whence, if we write $u''$ for the resultant or actual acceleration,

$\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{2})+\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{'2})=\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{''2}).$

Moreover, differentiating (25), we obtain

${\mathfrak {S}}(Q\delta {\ddot {q}}')-\textstyle \sum \displaystyle [m({\ddot {x}}'\,\delta {\ddot {x}}'+{\ddot {y}}'\,\delta {\ddot {y}}'+{\ddot {z}}'\,\delta {\ddot {z}}')]=0,$

whence, since $\delta {\ddot {q}}',\delta {\ddot {x}}',\delta {\ddot {y}}',\delta {\ddot {z}}'$ may have values proportional to ${\ddot {q}}',{\ddot {x}}',{\ddot {y}}',{\ddot {z}}',$

${\mathfrak {S}}(Q{\ddot {q}}')=2\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{'2}).$

These relations are similar to those which exist with respect to vis viva and impulsive forces.

Particular Equations of Motion.

From the general formula (12), we may easily obtain particular equations which will express the laws of motion in a very general form.

Let $d\omega _{1},d\omega _{2},$ etc. be infinitesimals (not necessarily complete differentials) the values of which are independent, and by means

1. This principle may be derived very directly from the general formula (6), or vice versa, for $\textstyle \sum \displaystyle ({\tfrac {1}{2}}mu^{'2})$ may be put in the form

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

   the variation of which, with the sign changed, is identical with the first member of (6).