δS = −Hδτ + Σm (ẋ′δx′ + ẏ′δy′ + z˙′δz′) − Σm (ẋδx + ẏδy + z˙δz),
(3)
where H (= T + V) is the constant value of the energy in the free motion of the system, and τ (= t′ − t) is the time of transit. In generalized co-ordinates this takes the form
δS = −Hδτ + p′1δq′1 + p′2δq′2 + ... − p1δq1 − p2δq2 − ....
(4)
Now if we select any two arbitrary configurations as initial and final, it is evident that we can in general (by suitable initial velocities or impulses) start the system so that it will of itself pass from the first to the second in any prescribed time τ. On this view of the matter, S will be a function of the initial and final co-ordinates (q1, q2, ... and q′1, q′2, ...) and the time τ, as independent variables. And we obtain at once from (4)
| p′1 = | ∂S | , p′2 = | ∂S | , ... , |
| ∂q′1 | ∂q′2 |
(5)
| p1 = − | ∂S | , p2 = − | ∂S | , ... , |
| ∂q1 | ∂q2 |
and