dP[Kappa] dP[Kappa] dP[Kappa]
[chi]` = ---------, [chi]`' = ----------, [chi]`" = ---------- ..., (25)
dP[kappa] dP[kappa]' dP[kappa]"
[Kappa] may also be expressed as a homogeneous quadratic function of
the cyclic velocities [.[chi]], [.[chi]]', [.[chi]]", ... Denoting it
in this form by [Tau]0, we have
[delta]([Tau]0 + [Kappa] = 2[delta][Kappa] = [delta]([kappa] [chi]` + [kappa]'[chi]`' + [kappa]"[chi]`" + ...). (26)
Performing the variations, and omitting the terms which cancel by (2)
and (25), we find
dP[Tau]0 dP[Kappa] dP[Tau]0 dP[Kappa]
-------- = - ---------, -------- = - ---------, ..., (27)
dPq1 dPq1 dPq2 dPq2
so that the formulae (23) become
dP[Tau]0 dP[Tau]0
Q1 = - --------, Q2 = - --------, ... (28)
dPq1 dPq2
A simple example is furnished by the top (MECHANICS, Sec. 22). The cyclic
co-ordinates being [psi], [phi], we find
( mu - [nu] cos [theta]) squared [nu] squared
2[@] = A[theta]` squared, 2[Kappa] = ----------------------- + -----,
A sin squared [theta] C
2[Tau]0 = A sin squared[theta][psi]` squared + C([phi]` + [psi] cos [theta]) squared, (29)
whence we may verify that dP[Tau]0/dP[theta] = - dP[Kappa]/dP[theta]
in accordance with (27). And the condition of equilibrium
dP[Kappa] dPV
--------- = - --------- (30)
dP[theta] dP[theta]
gives the condition of steady precession.