Question 12.5: Demonstrate that an n-fold rotational symmetry axis is a pri...
Demonstrate that an n-fold rotational symmetry axis is a principal axis of inertia, and that in the case n ≥ 3, the two other principal axes can be freely chosen in the plane perpendicular to the first axis.
If a body has an n-fold symmetry axis, then the tensor of inertia must be equal in two coordinate systems rotated from each other by \varphi = 2π/n:
\widehat{\Theta} = \widehat{\Theta }^{′} =\widehat{A} \widehat{\Theta } \widehat{A}^{-1} .
If we select the z-axis as a rotation axis, the rotation matrix reads
\widehat{A}=\begin{pmatrix} \cos \varphi & \sin \varphi & 0 \\ −\sin \varphi & \cos \varphi& 0\\ 0 & 0 & 1 \end{pmatrix}Multiplying the matrices out, one obtains the components Θ^{′}_{ij} of the new tensor of inertia which shall coincide with Θ_{ij} .
Θ^{′}_{11}= Θ_{11} = Θ_{11} cos^{2} \varphi +Θ_{22} \sin^{2} \varphi +2 Θ_{12} \sin \varphi \cos \varphi,
Θ^{′}_{22}= Θ_{22} = Θ_{11} \sin^{2} \varphi + Θ_{22} \cos^{2} \varphi −2 Θ_{12} \sin \varphi \cos \varphi,
Θ^{′}_{12}= Θ_{12} =− Θ_{11} \cos \varphi \sin \varphi + Θ_{22} \cos \varphi \sin \varphi+ Θ_{12}(1 −2 \sin^{2} \varphi),
Θ^{′}_{13}= Θ_{13} =+ Θ_{13} \cos \varphi + Θ_{23} \sin \varphi,
Θ^{′}_{23}= Θ_{23} =− Θ_{13} \sin \varphi+ Θ_{23} \cos \varphi.
The determinant of the system of the last two equations,
\begin{vmatrix} \cos \varphi −1 & \sin \varphi \\ -\sin \varphi & \cos \varphi −1 \end{vmatrix}= 2(1− \cos \varphi),vanishes only for \varphi = 0, 2π, . . .. If there is symmetry (n ≥ 2), then we must have Θ_{13} = Θ_{23} = 0, i.e., the z-axis must be a principal axis.
Two of the remaining three equations are identical, and there remains the system of equations
(Θ_{22} −Θ_{11}) \sin^{2} \varphi +2Θ_{12} \sin \varphi \cos \varphi = 0,(Θ_{22} −Θ_{11}) \cos \varphi \sin \varphi −2Θ_{12} \sin^{2} \varphi = 0.
The determinant of coefficients has the value
D =−2 \sin^{4} \varphi −2 \sin^{2} \varphi \cos^{2} \varphi =−2 \sin^{2} \varphi.There holds D = 0 for \varphi = 0,π, 2π, . . . . Hence, Θ_{11} = Θ_{22} and Θ_{12} = 0, if n>2. If the axis of rotational symmetry z is at least 3-fold, the tensor of inertia is diagonal for each orthogonal pair of axes in the x,y-plane.