where ss1, ss2, ... are linear functions of [kappa], [kappa]', [kappa]",
..., say
ss_r = [alpha]_r[kappa] + [alpha]'_r[kappa]' + [alpha]"_r[kappa]" + ..., (13)
the coefficients [alpha]_r, [alpha]'_r, [alpha]"_r, ... being in
general functions of the co-ordinates q1, q2, ... q_m. Evidently ss_r
denotes that part of the momentum-component dPR/dPq`_r which is due to
the cyclic motions. Now
d dPR d / dP[@] \ d dP[@] dPss_r dPss_r
-- ------ = -- ( ------ + ss_r) = -- ------ + -----q`1 + -----q`2 + ..., (14)
dt dPq`_r dt \dPq`_r / dt dPq`_r dPq1 dPq2
dPR dP[@] dP[Kappa] dPss1 dPss2
----- = ----- - --------- + -----q`1 + -----q`2 + .... (15)
dPq_r dPq_r dPq_r dPq_r dPq_r
Hence, substituting in (8), we obtain the typical equation of motion
of a gyrostatic system in the form
d dP[@] dP[@] dP[Kappa]
-- ------ - ----- + (r, 1)q`1 + (r, 2)q`2 + ... + (r, s)q`_s + ... + --------- = Q_r, (16)
dt dPq`_r dPq_r dPq_r
where
dPss_r dPss_s
(r, s) = ----- - -----. (17)
dPq_s dPq_r
This form is due to Lord Kelvin. When q1, q2, ... q_m have been
determined, as functions of the time, the velocities corresponding to
the cyclic co-ordinates can be found, if required, from the relations
(7), which may be written
dP[Kappa] \
[Chi]` = --------- - [alpha]1q`1 - [alpha]2q`2 - ..., |
dP[kappa] |
|
dP[Kappa] > (18)
[Chi]`' = ---------- - [alpha]'1q`1 - [alpha]'2q`2 - ..., |
dP[kappa]' |
|
&c., &c. /