we shall have
_
/t'
[delta] | (R - V)dt = 0, (16)
_/t
provided that the variation does not affect the cyclic momenta
[kappa], [kappa]', [kappa]", ..., and that the configurations at times
t and t' are unaltered, so far as they depend on the palpable
co-ordinates q1, q2, ... q_m. The initial and final values of the
ignored co-ordinates will in general be affected.
To prove (16) we have, on the above understandings,
_ _
/t' /t'
[delta] | (R - V)dt = | ([delta][Tau] - [kappa][delta][chi]` - ... -[delta]V)dt
_/t _/t
_
/t' /dP[Tau] dP[Tau] \
= | ( -------[delta]q`1 + ... + -------[delta]q1 + ... - [delta]V )dt, (17)
_/t \ dPq`1 dPq1 /
where terms have been cancelled in virtue of Sec. 5 (2). The last member
of (17) represents a variation of the integral
_
/t'
| ([Tau] - V)dt
_/t
on the supposition that [delta]X = 0, [delta]X' = 0, [delta]X" = 0,
... throughout, whilst [delta]q1, [delta]q2, [delta]q_m vanish at
times t and t'; i.e. it is a variation in which the initial and final
configurations are absolutely unaltered. It therefore vanishes as a
consequence of the Hamiltonian principle in its original form.
Larmor has also given the corresponding form of the principle of least
action. He shows that if we write
_
/
A = |(2[Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ...)dt, (18)
_/
then
[delta]A = 0, (19)
provided the varied motion takes place with the same constant value of
the energy, and with the same constant cyclic momenta, between the
same two configurations, these being regarded as defined by the
palpable co-ordinates alone.