Questions: Euler's Equations for Rigid Body Rotation

Question 1

A rigid body with three distinct principal moments of inertia I₁ < I₂ < I₃ spins freely in space with no external torque. About which axis will its rotation be unstable?

• A. The axis of minimum moment of inertia (I₁)
• B. The axis of intermediate moment of inertia (I₂)
• C. The axis of maximum moment of inertia (I₃)
• D. All three principal axes are unstable under torque-free rotation

Answer: (B)

Rotation about the intermediate principal axis (I₂) is unstable — this is the tennis racket theorem (intermediate axis theorem). Small perturbations grow, causing the body to tumble. Rotation about the minimum (I₁) and maximum (I₃) axes is stable: perturbations lead to bounded wobbling (polhode motion) but not tumbling. This asymmetry follows directly from the Euler equations: the coupling terms (I₂−I₃)ω₂ω₃ change sign depending on whether I₂ is between or outside the other two moments.

Question 2

Euler's equations are written in the body-fixed frame rather than an inertial frame. What is the key advantage?

• A. In the body frame, the angular momentum L is always zero, simplifying computation
• B. In the body frame, the inertia tensor I is constant, even as the body rotates
• C. In the body frame, all torques vanish, reducing to torque-free dynamics
• D. The body frame rotates with the body, eliminating all cross-product coupling terms

Answer: (B)

The inertia tensor I is attached to the body's mass distribution. In an inertial frame, I changes continuously as the body rotates, making the equations intractable. In the body-fixed frame, the body does not move relative to itself, so I remains constant (diagonal along principal axes). The price is a coupling term ω × Iω that arises from the rotating frame's kinematics — but this is far more manageable than a time-varying inertia tensor.

Question 3

Torque-free rotation of a rigid body about its axis of maximum moment of inertia is stable under small perturbations.

• T. True
• F. False

Answer: True

True. The tennis racket theorem (intermediate axis theorem) states that of the three principal rotation axes, the maximum and minimum inertia axes support stable rotation, while the intermediate axis is unstable. For the maximum inertia axis, small perturbations produce bounded wobbling (polhode motion) — the body oscillates around the spin axis but does not tumble. This is observable by spinning a book or phone: it spins stably about its thinnest or thickest axis.

Question 4

If no external torque acts on a rigid body, its angular velocity vector ω remains constant in both direction and magnitude.

• T. True
• F. False

Answer: False

False. Conservation of angular momentum L means L = Iω is constant in the inertial frame when there is no torque. But ω itself can change direction because I is a tensor, not a scalar — as the body rotates, the relationship between L and ω changes. The coupling terms in Euler's equations (e.g., (I₂−I₃)ω₂ω₃) show that the components of ω interact and evolve even in torque-free motion. This is the source of precession and the intermediate-axis instability.

Question 5

Why does switching to the body-fixed frame simplify Euler's equations, and what new term does this switch introduce into the rotational equations of motion?

Model answer: In the body-fixed frame, the inertia tensor I is constant (diagonal along principal axes), eliminating the need to track its time-varying entries. The price is that the frame itself rotates, so the transport theorem must be applied when taking time derivatives: dL/dt (inertial) = (dL/dt)_body + ω × L. This introduces the coupling term ω × Iω into the equations of motion. The full Euler equations are τ = I(dω/dt) + ω × (Iω), where the second term is the gyroscopic coupling responsible for precession and the intermediate-axis instability.

This is a standard classical mechanics trade-off: you exchange a complicated time-varying coordinate system (changing I in the inertial frame) for a simpler one (constant I in the body frame) by accepting an extra pseudo-force-like term. The coupling term ω × Iω is not a complication — it is physically meaningful. It encodes gyroscopic effects: even without external torques, the cross-product term drives the angular velocity components to interact, producing the rich behavior of spinning tops, satellites, and tumbling asteroids.