where t, t′ are the prescribed times of passing through the given initial and final configurations. The proof of (12) is simple; we have
δ ∫t′t (T − V)dt = ∫t′t (δΤ − δV)dt = ∫t′t {Σm (ẋδẋ + ẏδẏ + z˙δz˙) − δV} dt
= [ Σm (ẋδx + ẏδy + z˙δz)]t′t − ∫t′t {Σm (ẍδx + ÿδy + z¨δz) + δV} dt.
(13)
The integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d’Alembert’s principle.
The fact that in (12) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. Thus, to deduce Lagrange’s equations, we have
| ∫t′t (δΤ − δV) dt = ∫t′t { | ∂T | δq˙1 + | ∂T | δq1 + ... − | ∂V | δq1 − ... } dt |
| ∂q˙1 | ∂q1 | ∂q1 |
| = [p1δq1 + p2δq2 + ...]t′t − ∫t′t { [ṗ1 − | ∂T | + | ∂V | ) δq1 + (ṗ2 − | ∂T | + | ∂V | ) δq2 + ...} dt. |
| ∂q1 | ∂q1 | ∂q2 | ∂q2 |
(14)
The integrated terms vanish at both limits; and in order that the remainder of the right-hand member may vanish it is necessary that the coefficients of δq1, δq2, ... under the integral sign should vanish for all values of t, since the variations in question are independent, and subject only to the condition of vanishing at the limits of integration. We are thus led to Lagrange’s equation of motion for a conservative system. It appears that the formula (12) is a convenient as well as a compact embodiment of the whole of ordinary dynamics.