[delta]A = [tau][delta]H + p'1[delta]q'1 + p'2[delta]q'2 + ...
- p1[delta]q1 - p2[delta]q2 - .... (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
dPA dPA \
p'1 = -----, p'2 = -----, ..., |
dPq'1 dPq'2 |
> (10)
dPA dPA |
p1 = - ----, p2 = - ----, ..., |
dPq1 dPq2 /
dPA
and [tau] = --- (11)
dPH
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).
The preceding theorems are easily adapted to the case of cyclic
systems. We have only to write
_ _
/t' /t'
S = | (R - V)dt= | (T - [kappa][chi]` - [kappa]'[chi]`' - ... - V)dt (12)
_/t _/t
in place of (1), and
_
/
A = | (2T - [kappa][chi]` - [kappa]'[chi]`' - ...)dt, (3)
_/
in place of (8); cf. Sec. 7 ad fin. It is understood, of course, that in
(12) S is regarded as a function of the initial and final values of
the palpable co-ordinates q1, q2, ... q_m, and of the time of transit
[tau], the cyclic momenta being invariable. Similarly in (13), A is
regarded as a function of the initial and final values of q1, q2, ...
q_m, and of the total energy H, with the cyclic momenta invariable. It
will be found that the forms of (4) and (9) will be conserved,
provided the variations [delta]q1, [delta]q2, ... be understood to
refer to the palpable co-ordinates alone. It follows that the
equations (5), (6) and (10), (11) will still hold under the new
meanings of the symbols.