On the Euler Angles

The two tables here show the relation between the body axes and the inertial frame for each step

of the Euler angle rotation process. Table 1 gives the location of the body axes in the inertial

frame at each step, and Table 2 gives the apparent position of the inertial frame at each step. The

subscripts in each case denote the number of steps completed. The rotations and their order are

those shown in Goldstein’s Figure 4-7, which I have reproduced here.

Table 1: The Euler Rotations: Body Axes in the Inertial Frame

!

i

!

i 0

!

i

!

j

!

j 0

!

j

!

k

!

k 0

!

k

!

i

!

i 1

!

cos"i + sin"j

!

j

!

j 1

!

"sin#i + cos#j

!

k

!

k 1

!

k

!

i

!

i 2

!

cos"i + sin"j

!

j

!

j 2

!

"cos# sin$i + cos# cos$j+ sin#k

!

k

!

k 2

!

sin" sin#i $ sin" cos#j+ cos"k

!

i

!

i 3

!

cos" cos# $ cos% sin& sin"( )i + cos" sin# + cos% cos& sin"( )j+ sin% sin"k

!

j

!

j 3

!

"sin# cos$ " cos% sin& cos#( )i + "sin# sin$ + cos% cos& cos#( )j+ sin% cos#k

!

k

!

k 3

!

sin" sin#i $ sin" cos#j+ cos"k

Table 2. The Euler Rotations: Inertial Axes in the Body Frame

!

i

!

i0

!

i

!

j

!

j0

!

j

!

k

!

k0

!

k

!

i

!

i1

!

cos"i # sin"j

!

j

!

j1

!

sin"i + cos"j

!

k

!

k1

!

k

!

i

!

i2

!

cos"i # sin"j

!

j

!

j2

!

cos" sin#i + cos" cos#j $ sin"k

!

k

!

k2

!

sin" sin#i + sin" cos#j + cos"k

!

i

!

i3

!

cos" cos# $ cos% sin& sin"( )i + $cos" sin# $ cos% cos& sin"( )j + sin% sin"k

!

j

!

j3

!

sin" cos# + cos$ cos" sin#( )i + %sin# sin& + cos$ cos" cos#( )j % sin$ cos"k

!

k

!

k3

!

sin" sin#i + sin" cos#j + cos"k