Rigid Body Dynamics: Advanced Topics

9.1 Euler”s Equations in the Body Frame

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.

9.2 Stability of Free Rotation

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.

9.3 The Symmetric Top with One Point Fixed

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:

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

9.4 Gyroscopic Precession

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 $\omega_3$.

9.5 Coupled Rigid Bodies

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.

Common Pitfalls

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.

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).