(1)
The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property
δA = 0,
(2)
provided the total energy have the same constant value in the varied motion as in the actual motion.
If t, t′ be the times of passing through the initial and final configurations respectively, we have
δA = δ ∫t′t Σm (ẋ² + ẏ² + z˙²) dt
= ∫t′t δΤdt + 2Τ′δt′ + 2Τδt,
(3)
since the upper and lower limits of the integral must both be regarded as variable. This may be written