- #1

Kashmir

- 439

- 73

Goldstein 3rd Ed pg 161.

Im not able to understand this paragraph about the ambiguity in the sense of rotation axis given the rotation matrix A, and how we ameliorate it.

Any help please.

"The prescriptions for the direction of the rotation axis and for the rotation angle are not unambiguous. Clearly if ##\mathbf{R}## is an eigenvector, so is ##-\mathbf{R}##; hence the sense of the direction of the rotation axis is not specified. Further, ##-\Phi## satisfies Eq. (4.61) if ##\Phi## does. Indeed, it is clear that the eigenvalue solution does not uniquely fix the orthogonal transformation matrix A. From the determinantal secular equation (4.52), it follows that the inverse matrix ##\mathrm{A}^{-1}=\tilde{\mathrm{A}}## has the same eigenvalues and eigenvectors as A. However, the ambiguities can at least be ameliorated by assigning ##\Phi## to ##A## and ##-\Phi## to ##A^{-1}##, and fixing the sense of the axes of rotation by the right-hand screw rule"

Im not able to understand this paragraph about the ambiguity in the sense of rotation axis given the rotation matrix A, and how we ameliorate it.

Any help please.

"The prescriptions for the direction of the rotation axis and for the rotation angle are not unambiguous. Clearly if ##\mathbf{R}## is an eigenvector, so is ##-\mathbf{R}##; hence the sense of the direction of the rotation axis is not specified. Further, ##-\Phi## satisfies Eq. (4.61) if ##\Phi## does. Indeed, it is clear that the eigenvalue solution does not uniquely fix the orthogonal transformation matrix A. From the determinantal secular equation (4.52), it follows that the inverse matrix ##\mathrm{A}^{-1}=\tilde{\mathrm{A}}## has the same eigenvalues and eigenvectors as A. However, the ambiguities can at least be ameliorated by assigning ##\Phi## to ##A## and ##-\Phi## to ##A^{-1}##, and fixing the sense of the axes of rotation by the right-hand screw rule"

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