Mechanics of particles and systems Definition

Nutation

Builds on Rigid body Euler angles Precession

Definition

Let a rigid body turn about a fixed point $O$ . Let $e_3$ be a unit vector fixed in the body and $u$ a fixed unit vector in space. Describe the orientation of the body by Euler angles $(\varphi, \theta, \psi)$ with respect to $u$ . Here $\theta \in [0, \pi]$ is the angle between $u$ and $e_3$ . The nutation of the body is the change of the angle $\theta$ in time. It is the motion of the axis $e_3$ towards or away from $u$ . Its rate is $\dot\theta$ .

Remarks

Nutation and precession together describe the motion of the axis $e_3$ , which is the point of the unit sphere with polar angle $\theta$ and azimuth $\varphi$ .

Example: for the heavy symmetric top, $\cos\theta$ oscillates periodically between two roots of a cubic polynomial, so the axis nutates between two cones about the vertical. Depending on the initial conditions, the tip of the axis then traces a wavy curve, a curve with loops, or a curve with cusps on the sphere.

Non-example: in steady precession $\theta$ is constant, so there is no nutation, although $\varphi$ changes.

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Euler angles concept
Precession concept
Rigid body concept
Nutation definition

Tag	`6611H0ZA8F`
URI	`m2://6611H0ZA8F`
Kind	definition
Writing	W0 stub
MSC	70E05, 70E17
Defines	Nutation
Uses	Rigid body, Euler angles, Precession
Changed	3 November 2026

Nutation

Definition

Remarks

Reading path

Further reading

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