δA = τδH + p′1δq′1 + p′2δq′2 + ... − p1δq1 − p2δq2 − ....
(9)
This formula (it may be remarked) contains the principle of “least action” as a particular case. Selecting, as before, any two arbitrary configurations, it is in general possible to start the system from one of these, with a prescribed value of the total energy H, so that it shall pass through the other. Hence, regarding A as a function of the initial and final co-ordinates and the energy, we find
| p′1 = | ∂A | , p′2 = | ∂A | , ... , |
| ∂q′1 | ∂q′2 |
(10)
| p1 = − | ∂A | , p2 = − | ∂A | , ... , |
| ∂q1 | ∂q2 |
and
| τ = | ∂A | . |
| ∂H |
(11)
A is called by Hamilton the characteristic function; it represents, of course, the “action” of the system in the free motion (with prescribed energy) between the two configurations. Like S, it satisfies a partial differential equation, obtained by substitution from (10) in (7).