Rotating Systems 
195 
175-178] 
Let us put 
T R = (x 2 + y 2 + i 2 ) 
U — Sm {xy — yx) 
I = 2m ( x 2 + y 2 ) 
.(176*4), 
.(176-5), 
(176*6), 
so that T R is the kinetic energy relative to the rotating axes, TJ is the moment 
of momentum relative to the moving axes, and I is the moment of inertia. 
Then the values of T and M just obtained assume the forms 
177. The position of a rotating system may be supposed defined by a 
co-ordinates 0 1} 6 2 ,... 6 n _ 1 fixing the configuration of the system relative to 
the axes, so that the system has n co-ordinates in all. 
The equations of motion (169’5) take the form 
in which Q is the generalised force corresponding to the co-ordinate yjr, and so 
is the couple about the axis of z which acts upon the system. 
With the value of T given by equation (176’7), we have dT/dyfr =0, and 
9T/3a) = M, so that equation (177*1) reduces to 
which merely expresses that the rate of increase of the moment of momentum 
M is equal to the couple G. 
If a mass is rotating freely in space, G — 0, and M remains constant. If a 
mass is constrained to rotate at a constant angular velocity while M changes, 
a couple G will be necessary to maintain the uniformity of rotation, and the 
amount of this couple will be determined by equation (177*3). 
Mass constrained to rotate with Constant Angular Velocity. 
178. Let us first consider the problem when &> is kept constant. The 
value of any co-ordinate of position x will be a function of the n — 1 co 
ordinates 0 lt 0 2 ,... 0 n-i, so that 
T — T R + œil + \u> i I 
M = U + <oI 
(176-7). 
(176-8). 
The elimination of U gives 
T =T R + o)M — £<» 2 / 
(176-9). 
co-ordinate yfr fixing the position of the axes, such that yjr = a>, and n — 1 other 
dt d 0 s dt * 
dx dx 
and hence