Rigid Systems of Points 73(1738-1822), who also discovered the planet Uranus (1781) and infra-red
radiation (around 1800).
2.2.2 Rigid Systems of Points
A system 5 of point masses rrij, 1 < j < n, is called rigid if the distance
between every two points rrii, rrij never changes. Let O be a reference frame
and let Vj{t) be the position in that frame of rrij at time t. The rigidity
condition can be stated as
\\rj{t)-ri{t)\\=dii foralU, i,j = l,...,n, (2.2.21)where the dij are constants.
EXERCISE 2.2.5? Verify that condition (2.2.21) is independent of the choice
of the frame.
EXERCISE 2.2.6.° Let S be a rigid system in motion. Prove that the norm
\rj — rCM|| and the dot product Tj • TCM remain constant over time for all
j = 1,..., n, that is, the center of mass of a rigid system is fixed relative to
all rrij. Hence, the augmented system mi,... ,mn, M, with M located at
the center of mass, is also a rigid system.
We will now derive the equations of motion for a rigid system. If we con-
sider a motion as a linear transformation of space, then condition (2.2.21)
implies that the motion of a rigid system is an isometry. The physical real-
ity also suggests that this motion is orientation-preserving, that is, if three
vectors in a rigid system form a right-handed triad at the beginning of the
motion, they will be a right-handed triad throughout the motion.
EXERCISE 2.2.7. (a)B Show that an orientation-preserving orthogonal
transformation is necessarily a rotation. (b)c Using the result of part (a)
and the result of Problem 1.9 on page 412 conclude that the only possible
motions of a rigid system are shifts (parallel translations) and rotations.
Let O be a frame with a Cartesian coordinate system (z, j, k). Let S
be a rigid system moving relative to O. Denote by rcM(t) the position of
the center of mass of S in the frame O. We start by introducing two frames
connected with the system. Let OCM be the parallel translation of the
frame O to the center of mass. Thus, OCM moves with the center of mass
of the system S but does not rotate relative to O. Let 0\ be the frame with
00\ — rcM{t) and with the cartesian basis (?i, Jj, k) rotating with the