Precession

Precession describes how a spinning object's rotation axis slowly changes direction over time. When an object's spin axis itself rotates around a different axis, the object is precessing around that second axis. In a specific coordinate system, precession corresponds to a shift in the first Euler angle, while the third Euler angle represents the spin itself. A change in the second Euler angle is known as nutation. Physicists distinguish between two kinds of precession: one that occurs without any external torque, and one that is caused by an applied torque.

In astronomy, precession refers to gradual shifts in a celestial body's rotation or orbit. A well-known case is the slow drift of Earth's spin axis, called the precession of the equinoxes.

When no external torque acts on a spinning body, its angular momentum stays constant, but the direction of its angular velocity changes over time. This happens because the body's moment of inertia—or more precisely, its inertia matrix—varies with time. The inertia matrix contains the moments of inertia measured along different coordinate axes. If an object is not symmetric around its main spin axis, the moment of inertia along each axis changes as it rotates, even though angular momentum remains fixed. As a result, the angular velocity component around each axis shifts inversely with that axis's moment of inertia.

For a symmetric object like a disk spinning around an axis that is not its symmetry axis, the torque-free precession rate is given by ωₚ = (Iₛ ωₛ) / (Iₚ cos α), where ωₚ is the precession rate, ωₛ is the spin rate around the symmetry axis, Iₛ is the moment of inertia about that symmetry axis, Iₚ is the moment of inertia about either of the two equal perpendicular axes, and α is the angle between the symmetry axis and the moment of inertia direction.

If the object is not perfectly rigid, internal energy loss will gradually damp out torque-free precession, causing the rotation axis to line up with one of the body's principal inertia axes.

For a solid object with no symmetry axis, its orientation—represented by a rotation matrix R that maps internal coordinates to external ones—can be simulated numerically. Given the fixed internal inertia tensor I₀ and constant external angular momentum L, the instantaneous angular velocity is ω(R) = R I₀⁻¹ Rᵀ L. Precession is simulated by repeatedly recalculating ω and applying a small rotation ω dt over a short time step dt, using R_new = exp([ω(R_old)]ₓ dt) R_old, where [ω]ₓ is the skew-symmetric matrix. Finite time steps introduce errors that tend to increase rotational kinetic energy, expressed as E(R) = ω(R) · L / 2.