A _cyclic_ or _gyrostatic_ system is characterized by the following
properties. In the first place, the kinetic energy is not affected if
we alter the absolute values of certain of the co-ordinates, which we
will denote by [chi], [chi]', [chi]", ..., provided the remaining
co-ordinates q1, q2, ... q_m and the velocities, including of course
the velocities [.[chi]], [.[chi]]', [.[chi]]", ..., are unaltered.
Secondly, there are no forces acting on the system of the types [chi],
[chi]', [chi]", .... This case arises, for example, when the system
includes gyrostats which are free to rotate about their axes, the
co-ordinates [chi], [chi]', [chi]", ... then being the angular
co-ordinates of the gyrostats relatively to their frames. Again, in
theoretical hydrodynamics we have the problem of moving solids in a
frictionless liquid; the ignored co-ordinates [chi], [chi]', [chi]",
... then refer to the fluid, and are infinite in number. The same
question presents itself in various physical speculations where
certain phenomena are ascribed to the existence of _latent motions_ in
the ultimate constituents of matter. The general theory of such
systems has been treated by E.J. Routh, Lord Kelvin, and H.L.F.
Helmholtz.
Routh's equations.
If we suppose the kinetic energy [Tau] to be expressed, as in
Lagrange's method, in terms of the co-ordinates and the velocities,
the equations of motion corresponding to [chi], [chi]', [chi]'', ...
reduce, in virtue of the above hypotheses, to the forms
d dP[Tau] d dP[Tau] d dP[Tau]
-- --------- = 0, -- --------- = 0, -- --------- = 0, ..., (1)
dt dP[chi]` dt dP[chi]`' dt dP[chi]`"
whence
dP[Tau] dP[Tau] dP[Tau]
-------- = [kappa], --------- = [kappa]', --------- = [kappa]", ..., (2)
dP[chi]` dP[chi]`' dP[chi]`"
where [kappa], [kappa]', [kappa]", ... are the constant momenta
corresponding to the cyclic co-ordinates [chi], [chi]', [chi]", ....
These equations are linear in [.[chi]], [.[chi]]', [.[chi]]", ...;
solving them with respect to these quantities and substituting in the
remaining Lagrangian equations, we obtain m differential equations to
determine the remaining co-ordinates q1, q2, ... q_m. The object of
the present investigation is to ascertain the general form of the
resulting equations. The retained co-ordinates q1, q2, ... q_m may be
called (for distinction) the _palpable_ co-ordinates of the system; in
many practical questions they are the only co-ordinates directly in
evidence.