
412 Classical Mechanics
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constant in time. The line dened by this constant vector
is called the invariable line. The kinetic
energy T
rot
is a second constant of the motion for the torque-free motion. Multiply the rst of the
Euler Equations 12.82 by ω
1
, the second by ω
2
, and the third by ω
3
, and add, we get
d
dt
IIIN
1
2
11
2
22
2
33
2
11 22
ωωωω
++
()
=++
. (12.93)
The left-hand side is the time rate of change of the kinetic energy; the right-hand side is the rate
at which the applied torque works. Thus, in the absence of external torque, the kinetic energy is
constant.
The rotational kinetic energy T
rot
can b ...