Rigid Body Dynamics: Advanced Topics

For a rigid body rotating freely (no external torques), the angular momentum in the body frame satisfies:

$I_1\dot{\omega}_1 - (I_2 - I_3)\omega_2\omega_3 = 0$

$I_2\dot{\omega}_2 - (I_3 - I_1)\omega_3\omega_1 = 0$

$I_3\dot{\omega}_3 - (I_1 - I_2)\omega_1\omega_2 = 0$

Where $I_1, I_2, I_3$ are the principal moments of inertia and $\omega_1, \omega_2, \omega_3$ are the angular velocity components in the body frame.

First integral: The kinetic energy $T = \frac{1}{2}(I_1\omega_1^2 + I_2\omega_2^2 + I_3\omega_3^2)$ and the angular momentum magnitude $L^2 = I_1^2\omega_1^2 + I_2^2\omega_2^2 + I_3^2\omega_3^2$ are both conserved.

Geometric interpretation. The trajectory on the angular velocity ellipsoid $I_1\omega_1^2 + I_2\omega_2^2 + I_3\omega_3^2 = 2T$ intersects the angular momentum sphere $I_1^2\omega_1^2 + I_2^2\omega_2^2 + I_3^2\omega_3^2 = L^2$ to give the polhode curve.

For an axisymmetric body ($I_1 = I_2 \neq I_3$):

• Rotation about the symmetry axis ($\omega_3$): The body is stable if $I_3$ is either the largest or smallest moment. This explains why a spinning top is stable but rotation about the intermediate axis is not.

• Tennis racket theorem (Dzhanibekov effect): Rotation about the intermediate axis ($I_1 < I_2 < I_3$, spinning about the $I_2$ axis) is unstable. Small perturbations cause the body to flip periodically.

Proof of instability for intermediate axis. Linearise Euler’s equations about $\boldsymbol{\omega} = (0, \Omega, 0)$:

$I_1\dot{\omega}_1 = (I_2 - I_3)\Omega\,\omega_3$

$I_3\dot{\omega}_3 = (I_1 - I_2)\Omega\,\omega_1$

Combining: $\ddot{\omega}_1 = \frac{(I_2 - I_3)(I_1 - I_2)}{I_1 I_3}\Omega^2\,\omega_1$. Since $I_1 < I_2 < I_3$, both factors in the numerator are negative, giving a positive coefficient: $\omega_1$ grows exponentially. The motion is unstable. $\blacksquare$

Physical examples:

1. Book toss: Throw a book spinning about each of its three axes. Rotation about the shortest and longest axes is stable; rotation about the intermediate axis causes flipping.
2. Satellite attitude: Gravity-gradient stabilisation exploits the fact that rotation about the axis of minimum moment of inertia is stable in a gravitational field.

A symmetric top ($I_1 = I_2$) with one point fixed, under gravity, is described by three Euler angles $(\phi, \theta, \psi)$.

The Lagrangian:

$L = \frac{1}{2}I_1(\dot{\theta}^2 + \dot{\phi}^2\sin^2\theta) + \frac{1}{2}I_3(\dot{\psi} + \dot{\phi}\cos\theta)^2 - Mgd\cos\theta$

Conserved quantities: $p_\phi$ (angular momentum about the vertical) and $p_\psi$ (angular momentum about the symmetry axis) are cyclic:

$p_\phi = I_1\dot{\phi}\sin^2\theta + I_3(\dot{\psi} + \dot{\phi}\cos\theta)\cos\theta = \text{const}$

$p_\psi = I_3(\dot{\psi} + \dot{\phi}\cos\theta) = \text{const}$

Steady precession. For $\dot{\theta} = 0$ (constant inclination $\theta_0$):

$\dot{\phi} = \frac{Mgd}{I_3\omega_3} \quad \text{(regular precession)}$

Nutation. When $\theta$ varies, the tip of the symmetry axis traces a nutation path. The effective potential:

$V_{\text{eff}}(\theta) = \frac{(p_\phi - p_\psi\cos\theta)^2}{2I_1\sin^2\theta} + Mgd\cos\theta$

has a minimum at $\theta_0$ for stable regular precession. Oscillation about $\theta_0$ gives nutation with frequency:

$\omega_{\text{nut}} = \frac{I_3\omega_3}{I_1} \quad \text{(for rapid spin)}$

A spinning wheel with angular momentum $\mathbf{L}$ subject to a torque $\boldsymbol{\tau}$ precesses:

$\boldsymbol{\Omega}_{\text{prec}} = \frac{\boldsymbol{\tau} \times \mathbf{L}}{L^2}$

Why the wheel doesn’t fall. Gravity produces a torque perpendicular to $\mathbf{L}$, causing $\mathbf{L}$ to rotate horizontally rather than the wheel falling. The precession rate is:

$\Omega_{\text{prec}} = \frac{Mgd}{I_3\omega_3}$

Gyroscopic inertia. A rapidly spinning top resists tilting because changing the direction of $\mathbf{L}$ requires a torque proportional to $\omega_3$.

When two rigid bodies are connected (e.g., a gyroscope on a gimbal), the system has additional degrees of freedom. The equations of motion couple through constraint forces at the joint.

Gimbal lock. In a three-gimbal system, when two gimbal axes align, one degree of freedom is lost. This is a kinematic singularity, not a dynamic one. Quaternion-based representations avoid gimbal lock entirely.

1. Using lab-frame equations in the body frame. Euler’s equations apply in the body frame where the inertia tensor is diagonal. In the lab frame, the inertia tensor is time-dependent.

2. Confusing $\omega_3$ (body frame) with $\dot{\phi}$ (lab frame). The angular velocity about the symmetry axis in the body frame includes contributions from both $\dot{\psi}$ and $\dot{\phi}$.

3. Ignoring the stability criterion. Free rotation about the intermediate axis is always unstable. Assuming stability without checking the moment ordering leads to incorrect predictions.

4. Neglecting nutation in fast-spinning tops. For $\omega_3 \gg \Omega_{\text{prec}}$, nutation is rapid and small-amplitude, but it is always present unless the initial conditions are precisely tuned to regular precession.

5. Assuming $I_1 = I_2$ (symmetric top) applies generally. The symmetric-top simplification $\dot{\phi} = I_3\omega_3/(I_1\omega_1)$ only holds when the two transverse moments of inertia are equal. For an asymmetric top ($I_1 \neq I_2$), the motion is far more complex and may exhibit chaotic tumbling.

6. Confusing precession with rotation. Precession is the slow rotation of the angular momentum vector $\mathbf{L}$ about the vertical; the top itself spins rapidly about its symmetry axis. These are distinct motions with very different timescales.

The effective potential for the $\theta$ motion:

$V_{\text{eff}(\theta) = \frac{(p_\phi - p_\psi\cos\theta)^2}{2I_1\sin^2\theta} + \frac{p_\psi^2}{2I_3} + Mgd\cos\theta}$

Nutation: The top nutates (oscillates in $\theta$) while precessing in $\phi$ and spinning in $\psi$. The type of nutation (looping, cusped, or smooth) depends on the initial conditions.

Fast top ($p_\psi \gg Mgd$): The precession rate is:

$\dot{\phi} \approx \frac{Mgd}{p_\psi} = \frac{Mgd}{I_3\omega_3}$

This is independent of $\theta$ to leading order (steady precession).

• Inertial navigation. Gyroscopes maintain a fixed orientation in space, providing angular rate measurements for aircraft and spacecraft. Ring laser gyroscopes measure rotation via the Sagnac effect with accuracy $\sim 0.001°$/hr.
• Satellite attitude control. Reaction wheels (spinning flywheels) exchange angular momentum with the spacecraft body to reorient without thrusters.
• Balance in animals. The inner ear contains fluid-filled semicircular canals that act as biological gyroscopes, detecting head rotation for balance. Damage to these canals causes vertigo and loss of spatial orientation.
• Tennis racket theorem (Dzhanibekov effect). A rigid body rotating about its intermediate axis spontaneously flips 180° periodically, visible in microgravity. This confirms the instability of intermediate-axis rotation predicted by Euler’s equations.

These relationships apply to free rigid bodies (no external torque) and tops (constant gravity). For coupled or driven systems, numerical integration of the full Euler equations is typically required.