Mathematics of Physics and Engineering

Rigid Bodies 79

the principal axes frame, we have DLcM{t) = LCMx{t) i 4- LCMy(t) j +
LCMz(t)k. On the other hand, since the matrix i£M does not depend
on time, we use (2.2.33) to conclude that the column vector tCM of the
components of DLcM{t) satisfies tCM = IQM^ • Finally, we compute
the cross product in (2.2.34) by writing the vectors in the principal axes
frame and using the relation (2.2.33) for the components C*CM{t) of LCM
in that frame. The result is the three Euler equations describing the
rotation of the rigid system about the center of mass:

< i;yul+u*xul{rxx-rzz) = T*CMy, (2.2.35)

These equations were first published in 1765 by a Swiss mathematician
LEONHARD EULER (1707-1783). Leonhard (or Leonard) Euler was the
most prolific mathematician ever: extensive publication of his works con-
tinued for 50 years after his death and filled 80+ volumes; he also had
13 children. He introduced many modern mathematical notations, such
as e for the base of natural logs (1727), f(x) for a function (1734), E for
summation (1755), and i for the square root of —1 (1777).

EXERCISE 2.2.13? (a) Verify that (2.2.34) is indeed equivalent to (2.2.35).
(b) Write (2.2.27) in an inertial frame and verify that the result is a par-
ticular case of (2.2.35).
With a suitable definition of the numbers Ix, Iy, Iz and the vector
TCM, equations (2.2.35) also describe the motion of a rigid body. We study
rigid bodies in the following section.

2.2.3 Rigid Bodies

Any collection of points, finite or infinite, can be a rigid system: if two
points in the collection have trajectories rl(t), r2(t) in some frame, then
the rigidity condition ||Ti(i) — T2()|| = llri(0) — >2(0)|| must hold for every
two points in the collection.
Intuitively, a rigid body is a rigid system consisting of uncountably
many points, each with infinitesimally small mass. Mathematically, a rigid
body is described in a frame O by a mass density function p = p{r), so
that the volume Ay of the body near the point with the position vector ro
has, approximately, the mass Am = p(r) AV. Even more precisely, if the