These may be interpreted as introducing partial constraints into a
previously free system. The variations [delta]q1, [delta]q2, ...
[delta]q_n in the expressions (6) and (7) of Sec. 2 which are to be
equated are no longer independent, but are subject to the relations
dPA dPA dPB dPB
---- [delta]q1 + ---- [delta]q2 + ... = 0, ---- [delta]q1 + ---- [delta]q2 + ... = 0, &c. (29)
dPq1 dPq2 dPq1 dPq2
Introducing indeterminate multipliers [lambda], mu, ..., one for each
of these equations, we obtain in the usual manner n equations of the
type
d dP[Tau] dP[Tau] dPA dPB
-- ------- - ------- = Q_r + [lambda] ----- + mu ----- + ..., (30)
dt dPq`_r dPq_r dPq_r dPq_r
in place of Sec. 2 (10). These equations, together with (28), serve to
determine the n co-ordinates q1, q2, ... q_n and the m multipliers
[lambda], mu, ....
When t does not occur explicitly in the relations (28) the system is
said to be _holonomic_. The term connotes the existence of integral
(as opposed to differential) relations between the co-ordinates,
independent of the time.
Again, it may happen that although there are no prescribed relations
between the co-ordinates q1, q2, ... q_n, yet from the circumstances
of the problem certain geometrical conditions are imposed on their
_variations_, thus
A1[delta]q1 + A2[delta]q2 + ... = 0, B1[delta]q1 + B2[delta]q2 + ... = 0, &c., (31)
where the coefficients are functions of q1, q2, ... q_n and (possibly)
of t. It is assumed that these equations are not integrable as regards
the variables q1, q2, ... q_n; otherwise, we fall back on the previous
conditions. Cases of the present type arise, for instance, in ordinary
dynamics when we have a solid rolling on a (fixed or moving) surface.
The six co-ordinates which serve to specify the position of the solid
at any instant are not subject to any necessary relation, but the
conditions to be satisfied at the point of contact impose three
conditions of the form (31). The general equations of motion are
obtained, as before, by the method of indeterminate multipliers, thus
d dP[Tau] dP[Tau]
-- ------- - ------- = Q_r + [lambda]A_r + muB_r + ... (32)
dt dPq`_r dPq_r
The co-ordinates q1, q2, ... q_n, and the indeterminate multipliers
[lambda], mu, ..., are determined by these equations and by the
velocity-conditions corresponding to (31). When t does not appear
explicitly in the coefficients, these velocity-conditions take the
forms