| dT | = | d | (⅋ + T0) + ω·Σm (xÿ − yẍ) = | d | (⅋ − T0) + ω·Σ (xY − yX). |
| dt | dt | dt |
(27)
This must be equal to the rate at which the forces acting on the system do work, viz. to
ωΣ (xY − yX) + Q1q˙1 + Q2q˙2 + ... + Qnq˙n,
where the first term represents the work done in virtue of the rotation.
We have still to notice the modifications which Lagrange’s equations undergo when the co-ordinates q1, q2, ... qn Constrained systems. are not all independently variable. In the first place, we may suppose them connected by a number m (< n) of relations of the type
A (t, q1, q2, ... qn) = 0, B (t, q1, q2, ... qn) = 0, &c.
(28)
These may be interpreted as introducing partial constraints into a previously free system. The variations δq1, δq2, ... δqn in the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations