Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/28

The first part is evidently constant with reference to variations of ${\ddot {x}}',{\ddot {y}}',{\ddot {z}}'$, and may, therefore, be neglected. With respect to the second part we observe that by the general formula of the motion we have

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

for all values of $\delta {\ddot {x}},\delta {\ddot {y}},\delta {\ddot {z}}$ which are possible and reversible before the addition of the new constraints. But values proportional to ${\ddot {x}}',{\ddot {y}}',{\ddot {z}}'$ and of the same sign, are evidently consistent with the original constraints, and when the components of acceleration are altered to ${\ddot {x}}+{\ddot {x}}',{\ddot {y}}+{\ddot {y}}',{\ddot {z}}+{\ddot {z}}'$ variations of these quantities proportional to and of the same sign as $-{\ddot {x}}',-{\ddot {y}}',-{\ddot {z}}'$ are evidently consistent with the original constraints. Now if these latter variations were not possible before the accelerations were modified by the addition of the new forces and constraints, it must be that some constraint was then operative which afterwards ceased to be so. The expression (22) will, therefore, be equal to zero, provided only that all the constraints which were operative before the addition of the new forces and constraints, remain operative afterwards.^[1] With this limitation, therefore, the expression (23) must have the greatest value consistent with the constraints. This principle may be expressed without reference to rectangular coordinates. If we write u' for the relative acceleration due to the additional forces and constraints, we have

$u^{'2}={\ddot {x}}^{'2}+{\ddot {y}}^{'2}+{\ddot {z}}^{'2},$

and expression (23) reduces to

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

(24)

If the sum of the moments of the additional forces which are considered is represented by ${\mathfrak {S}}(Qdq)$ (the $q$ representing quantities determined by the configuration of the system), we have

$\textstyle \sum \displaystyle (X'{\dot {x}}+Y'{\dot {y}}+Z'{\dot {z}})={\mathfrak {S}}(Q{\dot {q}}).$

We may distinguish the values of ${\frac {d^{2}q}{dt^{2}}}$ immediately before and immediately after the application of the additional forces and constraints by the expressions ${\ddot {q}}$ and ${\ddot {q}}+{\ddot {q}}'$. With this understanding, we have, by differentiation of the preceding equation,

$\textstyle \sum \displaystyle [{\dot {X}}'{\dot {x}}+{\dot {Y}}'{\dot {z}}+{\dot {Z}}'{\dot {z}}+X'({\ddot {x}}+{\ddot {x}}')+Y'({\ddot {y}}+{\ddot {y}}')+Z'({\ddot {z}}+{\ddot {z}}')]$ $={\mathfrak {S}}[{\dot {Q}}{\dot {q}}+Q({\ddot {q}}+{\ddot {q}}')];$

1. As an illustration of the significance of this limitation, we may consider the condition afforded by the impenetrability of two bodies in contact. Let us suppose that if subject only to the original forces and constraints they would continue in contact, but that, under the influence of the additional forces and constraints, the contact will cease. The impenetrability of the bodies then ceases to be operative as a constraint. Such cases form an exception to the principle which is to be established. But there are no exceptions when all the original constraints are expressed by equations.