(r, s) = -(s, r), (r, r) = 0 (19)
The remaining dynamical equations, equal in number to the co-ordinates
[chi], [chi]', [chi]", ..., yield expressions for the forces which
must be applied in order to maintain the velocities [chi]`, [chi]`',
[chi]`", ... constant; they need not be written down. If we follow the
method by which the equation of energy was established in Sec. 2, the
equations (17) lead, on taking account of the relations (19), to
d
--([@] - [Tau]0) = Q1q`1 + Q2q`2 + ... + Q_nq`_n, (20)
dt
or, in case the forces Q_r depend only on the co-ordinates q1, q2, ...
q_n and are conservative,
[@] + V - [Tau]0 = const. (21)
The conditions that the equations (17) should be satisfied by zero
values of the velocities q`1, q`2, ... q`_n are
dP[Tau]0
Q_r = - --------, (22)
dPq_r
or in the case of conservative forces
dP
------ (V - [Tau]0) = 0, (23)
dPq_r
i.e. the value of V - [Tau]0 must be _stationary_.
Rotating axes.
We may apply this to the case of a system whose configuration relative
to axes rotating with constant angular velocity ([omega]) is defined
by means of the n co-ordinates q1, q2, ... q_n. This is important on
account of its bearing on the kinetic theory of the tides. Since the
Cartesian co-ordinates x, y, z of any particle m of the system
relative to the moving axes are functions of q1, q2, ... q_n, of the
form Sec. 1 (1), we have, by (15)
2[@] = [Sigma]m(x` squared + y` squared + z` squared), 2[Tau]0 = [omega] squared[Sigma]m(x squared + y squared), (24)
/ dPy dPx \
a_r= [Sigma]m( x----- - y----- ), (25)
\ dPq_r dPq_r/
whence
dP(x, y)
(r, s) = 2[omega].[Sigma]m ------------. (26)
dP(q_s, q_r)
The conditions of relative equilibrium are given by (23).
It will be noticed that this expression V - [Tau]0, which is to be
stationary, differs from the true potential energy by a term which
represents the potential energy of the system in relation to
fictitious "centrifugal forces." The question of stability of relative
equilibrium will be noticed later (Sec. 6).
It should be observed that the remarkable formula (20) may in the
present case be obtained directly as follows. From (15) and (14) we
find
d[Tau] d
------ = --([@] + [Tau]0) + [omega].[Sigma]m(xy" - yx")
dt dt
d
= --([@] - [Tau]0) + [omega].[Sigma](xY - yX). (27)
dt
This must be equal to the rate at which the forces acting on the
system do work, viz. to
[omega][Sigma](xY - yX) + Q1q`1 + Q2q`2 + ... + Q_nq`_n,
where the first term represents the work done in virtue of the
rotation.
Constrained systems.
We have still to notice the modifications which Lagrange's equations
undergo when the co-ordinates q1, q2, ... q_n are not all
independently variable. In the first place, we may suppose them
connected by a number m ( < n) of relations of the type