Page 210 - Intro to Tensor Calculus

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Now if we multiply equation (2.2.49) by e rjk ,thenitcan be writteninthe form

dH ij
= M ij . (2.2.67)
dt
Similarly, if we multiply the equation (2.2.42) by e imn , then it can be expressed in the alternate form

H mn = e imn ω j I ji = I mnst ω st
and because of this relation the equation (2.2.67) can be expressed as
d
(I ijst ω st )= M ij . (2.2.68)
dt
We write this equation in the barred system of coordinates where I pqrs will be a constant and consequently
its derivative will be zero. We employ the transformation equations
I ijst = ` ip ` jq ` sr ` tk I pqrk

ω ij = ` si ` tj ω st
M pq = ` ip ` jq M ij
and then multiply the equation (2.2.68) by ` ip ` jq and simplify to obtain
d
` ip ` jq ` iα ` jβ I αβrk ω rk = M pq .
dt
Expand all terms in this equation and take note that the derivative of the I αβrk is zero. The expanded
equation then simplifies to

dω rk
I pqrk +(δ αu δ pv δ βq + δ pα δ βu δ qv ) I αβrk ω rk ω uv = M pq . (2.2.69)
dt
Substitute into equation (2.2.69) the relations from equations (2.2.61),(2.2.64) and (2.2.66), and then multiply
by e mpq and simplify to obtain the Euler’s equations of motion
dω i
I im − e tmj I ij ω i ω t = M m . (2.2.70)
dt
Dropping the bar notation and performing the indicated summations over the range 1,2,3 we find the
Euler equations have the form

dω 1 dω 2 dω 3
I 11 + I 21 + I 31 +(I 13 ω 1 + I 23 ω 2 + I 33 ω 3 ) ω 2 − (I 12 ω 1 + I 22 ω 2 + I 32 ω 3 ) ω 3 = M 1
dt dt dt
dω 1 dω 2 dω 3
I 12 + I 22 + I 32 +(I 11 ω 1 + I 21 ω 2 + I 31 ω 3 ) ω 3 − (I 13 ω 1 + I 23 ω 2 + I 33 ω 3 ) ω 1 = M 2 (2.2.71)
dt dt dt
dω 1 dω 2 dω 3
I 13 + I 23 + I 33 +(I 12 ω 1 + I 22 ω 2 + I 32 ω 3 ) ω 1 − (I 11 ω 1 + I 21 ω 2 + I 31 ω 3 ) ω 2 = M 3 .
dt dt dt
In the special case where the barred axes are principal axes, then I ij =0 for i 6= j and the Euler’s
equations reduces to the system of nonlinear differential equations
dω 1
I 11 +(I 33 − I 22 )ω 2 ω 3 = M 1
dt
dω 2
I 22 +(I 11 − I 33 )ω 3 ω 1 = M 2 (2.2.72)
dt
dω 3
I 33 +(I 22 − I 11 )ω 1 ω 2 = M 3 .
dt
In the case of constant coefficients and constant moments the solutions of the above differential equations
can be expressed in terms of Jacobi elliptic functions.