Analytical mechanics

Rigid-body rotation, the gyroscope, Euler angles

The orientation of a free rigid body has three rotational degrees of freedom. Euler angles provide generalized coordinates for orientation; angular momentum in body and space frames generally point in different directions.

Rotation about a principal axis is simple, but general motion requires tracking the body’s orientation. The ZXZ Euler convention decomposes it into three successive rotations.

T = \frac12(I_1\omega_1^2+I_2\omega_2^2+I_3\omega_3^2), \qquad \mathbf L_{body}=\mathrm{diag}(I_1,I_2,I_3)\boldsymbol\omega_{body}

Definition: Euler angles and principal axes

Under the ZXZ convention, angles $(φ,θ,ψ)$ rotate successively about the space z axis, the intermediate x′ axis, and the body-fixed z″ axis. The angles are coordinates, not components of angular velocity. Rotational energy is simplest in principal axes, with moments $I_1,I_2,I_3$ and body-frame angular-velocity components.

A symmetric top has a tilted body axis; the trace shows that axis precessing around the vertical. Drag to rotate the view and vary spin and tilt.

Gyroscopes and precession

For a rapidly spinning symmetric top, angular momentum is nearly aligned with its symmetry axis. The gravitational torque changes its direction, producing precession; in the slow-precession approximation $Ω_p≈τ/L$ when torque is perpendicular to L. This does not capture all nutation.

Example: Example

A top has gravitational torque τ=0.020 N·m and angular momentum L=0.40 kg·m²/s. Estimate its slow-precession rate.

Solution

Using $Ω_p≈τ/L$ , $Ω_p≈0.020/0.40=0.050$ rad/s. This is the slow-precession approximation, with the axis near vertical and angular momentum dominated by spin; it does not resolve nutation.

Quick check

In body-fixed principal axes, Euler’s equations include $I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3=\tau_1$ , with two cyclically permuted equations. With zero external torque, steady rotation about the largest- or smallest-inertia principal axis is stable to small disturbances, whereas rotation about the intermediate axis is unstable. A rapidly spinning gyroscope under a transverse torque changes the direction of its angular momentum slowly, producing precession. Euler angles still have coordinate singularities for certain orientations; these are limitations of the angle chart, not physical singularities of the rigid body.

For steady rotation about principal axis 3, $\omega_3$ is constant and the two transverse equations govern small disturbances. Stability depends on the sign of $(I_3-I_1)(I_3-I_2)$ : the largest- and smallest-inertia axes give bounded oscillations, while the intermediate axis permits growing disturbances. This is the basis of the tennis-racket flip.

For a rapidly spinning symmetric top, a torque τ perpendicular to L gives an approximate precession rate of:

For an autonomous system, which quantity is conserved along Hamiltonian motion?

References

Herbert Goldstein, Charles Poole, John Safko (2002). Classical Mechanics
L. D. Landau, E. M. Lifshitz (1976). Mechanics