Finding Euler angles from rotation about arbitrary axis

[TheSource007]

Homework Statement

An object is rotated 45 degrees about an axis whose + direction is that of (i-k). Find zxz Euler angles (that is, Euler angles as introduced by Goldstein) for a set of three active rotations that gives the same net motion of the object.

Homework Equations

Goldstein chapter 4, eq 4.46. Not sure how to input it here.

The Attempt at a Solution

What I tried to do was to find the rotation matrix corresponding to the rotation about the given axis using the Rodriguez rotation formula http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle, and then compared to eq. 4.46 in Goldstein and get the Euler angles that way. But the problem specifies to find rotation of the object, and eq 4.46 describes the rotation of the axes. How can I find the Euler angles that describes the rotation of the object, not the axes? Was my approach the indicated/correct or is there another way?
Also, Can someone explain to me was is the meaning of eq 4.47 (the transpose of that matrix)
Thanks.

 

[mark726]

Thank you for your question. It seems like you have a good understanding of the Rodriguez rotation formula and how to use it to find the rotation matrix for the given axis. However, as you mentioned, this only gives the rotation of the axes, not the object. To find the rotation of the object, we need to consider the order in which the rotations are performed.

In this case, we are given a rotation of 45 degrees about the axis (i-k). This is equivalent to a rotation of 45 degrees about the z-axis followed by a rotation of 45 degrees about the x-axis. Therefore, the set of three active rotations that gives the same net motion of the object would be a rotation of 45 degrees about the z-axis, followed by a rotation of 45 degrees about the x-axis, and finally another rotation of 45 degrees about the z-axis.

To find the Euler angles corresponding to this set of rotations, we can use the same approach as before, but this time we will use the inverse rotation matrix. This means that instead of rotating the axes, we will rotate the object in the opposite direction. The resulting Euler angles will give the rotation of the object, not the axes.

As for the meaning of eq 4.47, it is simply the transpose of the rotation matrix. This is because the transpose of a rotation matrix is equal to its inverse, and in this case, we are using the inverse rotation matrix to find the rotation of the object.

I hope this helps clarify your doubts. Let me know if you have any further questions.