Page 209 - Intro to Tensor Calculus

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The equation (2.2.61) implies that ω i and ω ij are dual tensors and

1
ω i = e ijk ω jk .
2
Also the velocity of a point which is rotating about the origin relative to the S frame of reference is v i (S)=
e ijk ω j x k which can also be written in the form v m (S)= −ω mk x k . Since the barred axes rotate with the rigid
body, then a particle in the barred reference frame will have v m (S) = 0, since the coordinates of a point
in the rigid body will be constants with respect to this reference frame. Consequently, we write equation
˙
(2.2.59) in the form 0 = v m (S)+ ` mi ` ji x j which implies that
˙ ˙
v m (S)= −` mi` ji x j = −ω mk x k or ω mj = ω mj (S, S)= ` mi ` ji .

This equation is interpreted as describing the angular velocity tensor of S relative to S. Since ω ij is a tensor,
it can be represented in the barred system by

ω mn (S, S)= ` im ` jn ω ij (S, S)
˙
= ` im ` jn ` is ` js
(2.2.62)
˙
= δ ms ` jn ` js
˙
= ` jn ` jm
By differentiating the equations (2.2.51) it is an easy exercise to show that ω ij is skew-symmetric. The
second order angular velocity tensor can be used to write the equations (2.2.59) and (2.2.60) in the forms

v m (S)= v m (S)+ ω mj (S, S)x j
(2.2.63)
v m (S)= v m (S)+ ω jm (S, S)x j
The above relations are now employed to derive the celebrated Euler’s equations of motion of a rigid body.

Euler’s Equations of Motion

We desire to find the equations of motion of a rigid body which is subjected to external forces. These
equations are the formulas (2.2.49), and we now proceed to write these equations in a slightly different form.
Similar to the introduction of the angular velocity tensor, given in equation (2.2.61), we now introduce the
following tensors
1. The fourth order moment of inertia tensor I mnst which is related to the second order moment of
inertia tensor I ij by the equations

1 1
I mnst = e jmn e ist I ij or I ij = I pqrs e ipq e jrs (2.2.64)
2 2
2. The second order angular momentum tensor H jk which is related to the angular momentum vector
H i by the equation
1
H i = e ijk H jk or H jk = e ijk H i (2.2.65)
2
3. The second order moment tensor M jk which is related to the moment M i by the relation

1
M i = e ijk M jk or M jk = e ijk M i . (2.2.66)
2