(r, r) = 0, (r, s) = -(s, r). (19)
Hence, if in (16) we put r = 1, 2, 3, ... m, and multiply by q`1,
q`2, ... q`_m respectively, and add, we find
d
--([@] + [Kappa]) = Q1q`1 + Q2q`2 + ..., (20)
dt
or, in the case of a conservative system
[@] + V + [Kappa] = const., (21)
which is the equation of energy.
The equation (16) includes Sec. 3 (17) as a particular case, the
eliminated co-ordinate being the angular co-ordinate of a rotating
solid having an infinite moment of inertia.
In the particular case where the cyclic momenta [kappa], [kappa]',
[kappa]", ... are all zero, (16) reduces to
d dP[@] dP[@]
-- ------ - ----- = Q_r. (22)
dt dPq`_r dPq_r
The form is the same as in Sec. 2, and the system now behaves, as regards
the co-ordinates q1, q2, ... q_m, exactly like the acyclic type there
contemplated. These co-ordinates do not, however, now fix the position
of every particle of the system. For example, if by suitable forces
the system be brought back to its initial configuration (so far as
this is defined by q1, q2, ..., q_m), after performing any evolutions,
the ignored co-ordinates [chi], [chi]', [chi]", ... will not in
general return to their original values.
If in Lagrange's equations Sec. 2 (10) we reverse the sign of the
time-element dt, the equations are unaltered. The motion is therefore
reversible; that is to say, if as the system is passing through any
configuration its velocities q`1, q`2, ..., q`_m be all
reversed, it will (if the forces be the same in the same
configuration) retrace its former path. But it is important to observe
that the statement does not in general hold of a gyrostatic system;
the terms of (16), which are linear in q`1, q`2, ..., q`_m,
change sign with dt, whilst the others do not. Hence the motion of a
gyrostatic system is not reversible, unless indeed we reverse the
cyclic motions as well as the velocities q`1, q`2, ..., q`_m.
For instance, the precessional motion of a top cannot be reversed
unless we reverse the spin.
Kinetostatics.
The _conditions of equilibrium_ of a system with latent cyclic motions
are obtained by putting q`1 = 0, q`2 = 0, ... q`_m = 0 in (16); viz.
they are
dP[Kappa] dP[Kappa]
Q1 = ---------, Q2 = ---------, ... (23)
dPq1 dPq2
These may of course be obtained independently. Thus if the system be
guided from (apparent) rest in the configuration (q1, q2, ... q_m) to
rest in the configuration q1 + [delta]q1, q2 + [delta]q2, ..., q_m +
[delta]q_m, the work done by the forces must be equal to the increment
of the kinetic energy. Hence