Moment of inertia

• Moment of inertia is a symmetric tensor. For a continuous distribution

  $$ I_{ij}=I_{ji}=\int\text dM(r^2\delta_{ij}-r_ir_j) $$

  where $\text dM=\rho \text dV$or $\sigma \text dA$ or $\mu \text d \mathcal l$. For a discrete distribution we get

  $$ I_{ij}=\sum^N_{k=1}m_k\left ( |\vec r^{(k)}|^2\delta_{ij}-r_i^{(k)}r_j^{(k)}\right ) $$

Axis theorems

• Parallel axis theorem: $I'{ij}=I{ij}+M(|\vec R|^2-R_iR_j)$ where $M$ is total mass $I_{ij}$ is the moment of inertia about the center of mass and $\vec R$ is the position vector relative to the center of mass
• Perpendicular axis theorem: for a lamina $I_{33}=I_{11}+I_{22}$

Rotational angular momentum and rotational kinetic energy

$$ \begin{aligned} \begin{pmatrix} L_1 \\ L_2 \\ L_3 \end{pmatrix}&=L_i=I_{ij}\omega_{j}=\begin{bmatrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{bmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \end{pmatrix} \\ T&=\frac 12 \omega_i I_{ij} \omega_j = \frac 12 \begin{pmatrix} \omega_1 & \omega_2 & \omega_3 \end{pmatrix}\begin{bmatrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{bmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \end{pmatrix} \end{aligned} $$

Diagonalization

• Symmetric square matrix $P=(\vec E^{(1)},\cdots,\vec E^{(n)})$ where $A\vec E^{(k)}=\lambda^{(k)}\vec E^{(k)}$ defines the eigenvectors and eigenvalues of $A$ with $P^{-1}AP=D=\text{diag}(\lambda^{(1)},\cdots,\lambda^{(n)})$ this can be chosen to an orthogonal diagonalization with $\vec E^{(k)}\cdot \vec E^{(m)}=\delta^{km}$ and $P^{-1}=P^{T}$. If the eigenvalues are degenerate then one has to be careful in the choice of eigenvectors in the degenerate surface
• Since the moment of inertia matrix is symmetric it can be diagonalized and the eigenvectors define the principal axes of the rigid body. The three principal moments of inertia $I_1,I_2$ and $I_3$ are the eigenvalues of the moment of inertia matrix

Euler’s equation

$$ \vec M=\dot {\vec L}|S=\dot {\vec L}|{S'}+\vec\omega\times\vec L=I\dot\omega+\vec\omega\times(I\vec\omega) \\ \begin{aligned} M_1&=I_1\dot \omega_1+(I_3-I_2)\omega_2\omega_3 \\ M_2&=I_2\dot \omega_2+(I_1-I_3)\omega_3\omega_1 \\ M_3&=I_3\dot \omega_3+(I_2-I_1)\omega_1\omega_2 \end{aligned} $$

Symmetric free top

• symmetric free body $\vec M=0$ with spin frequency $\omega_t$ and $I_3>I$ will precess with frequency

  $$ \Omega=\left ( \frac{I_3-I}{I} \right )\omega_t $$

• Intermediate axis theorem: If $I_1<I_2<I_3$ then rotations are unstable about the $2^{\text{nd}}$-axis

• Heavy spinning tops, or gyroscopes can exhibit fast/slow precession and nutation