Torque-free precession of a symmetric body

Let the first two principal moments equal A and the third equal C. The third Euler equation immediately shows that the third angular-velocity component is constant without torque. The remaining two equations become a pair of first-order linear equations. Combining them into a complex transverse component turns the pair into one phase-rotation equation, whose signed rate depends on the inertia difference.

Define a positive body circulation as motion from the first toward the second body axis. If C exceeds A and the third angular velocity is positive, the transverse angular-velocity components circulate in that positive sense. If C is smaller, the circulation reverses. The transverse amplitude stays constant. Its period uses the absolute value of the signed rate; when the rate vanishes, the component vector does not circulate.

Choose the fixed space z direction along the conserved angular momentum, whose magnitude is L. The body symmetry axis maintains a constant angle theta to that direction because its projection on angular momentum is C times the constant third angular velocity. This produces a cone in space. For nonzero tilt, its azimuth increases at L divided by A in the right-handed convention about positive angular momentum.

This inertial rate is positive by the choice of positive space axis. The final Euler spin-coordinate rate is instead negative Omega_b in the declared active Z-X-Z convention. Neither should be mislabeled as the physical third component of angular velocity: that component also includes the projection of azimuthal precession. At exact alignment the cone collapses, and an azimuth rate is a redundant coordinate description rather than an observable sweep.

Torque-free precession does not require friction, energy input, or a force continually pushing the symmetry axis sideways. The geometry follows from the relationship between angular momentum and angular velocity in an anisotropic body. The ideal model keeps both the rotational energy and angular-momentum magnitude constant. Internal dissipation would invalidate that energy conservation while possibly leaving total angular momentum approximately fixed.

The vector lab can test the geometry of a projection: predict that increasing an angle from thirty to sixty degrees decreases its axial cosine projection while preserving the chosen magnitude, then inspect the components. Connect that observation to the tilt formula on paper. This single planar vector does not animate either precession cone, integrate Euler’s equations, or demonstrate the different body and space rates.