Page 206 - Intro to Tensor Calculus

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The equations (2.2.44) and (2.2.45) represent the conservation of linear and angular momentum and can be
writteninthe forms
!
N N
d X (α) X (α)
m (α) ˙x r = F r (2.2.46)
dt
α=1 α=1
and
N ! N
d X (α) (α) X (α) (α)
m (α) e rst x s ˙ x t = e rst x s F t . (2.2.47)
dt
α=1 α=1
P (α) P (α)
By definition we have G r = m (α) ˙x r representing the linear momentum, F r = F r the total force
P (α) (α)
acting on the system of particles, H r = m (α) e rst x s x ˙ t is the angular momentum of the system relative
P (α) (α)
to the origin, and M r = e rst x s F t is the total moment of the system relative to the origin. We can
therefore express the equations (2.2.46) and (2.2.47) in the form

dG r
= F r (2.2.48)
dt
and
dH r
= M r . (2.2.49)
dt
The equation (2.2.49) expresses the fact that the rate of change of angular momentum is equal to the
moment of the external forces about the origin. These equations show that the motion of a system of
particles can be studied by considering the motion of the center of mass of the system (translational motion)
and simultaneously considering the motion of points about the center of mass (rotational motion).
We now develop some relations in order to express the equations (2.2.49) in an alternate form. Toward
this purpose we consider first the concepts of relative motion and angular velocity.

Relative Motion and Angular Velocity
Consider two different reference frames denoted by S and S. Both reference frames are Cartesian
coordinates with axes x i and x i , i =1, 2, 3, respectively. The reference frame S is fixed in space and is
called an inertial reference frame or space-fixed reference system of axes. The reference frame S is fixed
to and rotates with the rigid body and is called a body-fixed system of axes. Again, for convenience, it
is assumed that the origins of both reference systems are fixed at the center of mass of the rigid body.
Further, we let the system S have the basis vectors e i ,i =1, 2, 3, while the reference system S has the basis
b
vectors ˆ e i ,i =1, 2, 3. The transformation equations between the two sets of reference axes are the affine
transformations

x i = ` ji x j and x i = ` ij x j (2.2.50)
where ` ij = ` ij (t) are direction cosines which are functions of time t (i.e. the ` ij are the cosines of the
angles between the barred and unbarred axes where the barred axes are rotating relative to the space-fixed
unbarred axes.) The direction cosines satisfy the relations

and ` ij ` kj = δ ik . (2.2.51)
` ij ` ik = δ jk