Editing Nutation (section)

==In a rigid body==
{{further|Rigid body dynamics}}
If a [[Spinning top|top]] is set at a tilt on a horizontal surface and spun rapidly, its rotational axis starts precessing about the vertical. After a short interval, the top settles into a motion in which each point on its rotation axis follows a circular path. The vertical force of gravity produces a horizontal torque {{math|'''&tau;'''}} about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity {{math|'''&Omega;'''}} such that at any moment
:<math> \boldsymbol{\tau} = \mathbf{\Omega} \times \mathbf{L},</math>       (vector [[cross product]])

where {{math|'''L'''}} is the instantaneous angular momentum of the top.<ref name=Feynman>{{harvnb|Feynman|Leighton|Sands|2011|pp=20–7{{clarify|date=December 2013}}}}</ref>

Initially, however, there is no precession, and the upper part of the top falls sideways and downward, thereby tilting. This gives rise to an imbalance in torques that starts the precession. In falling, the top overshoots the amount of tilt at which it would precess steadily and then oscillates about this level. This oscillation is called ''nutation''. If the motion is damped, the oscillations will die down until the motion is a steady precession.<ref name=Feynman/><ref name=Goldstein220>{{harvnb|Goldstein|1980|p=220}}</ref>

The physics of nutation in tops and [[gyroscope]]s can be explored using the model of a ''heavy [[symmetrical top]]'' with its tip fixed. (A symmetrical top is one with rotational symmetry, or more generally one in which two of the three principal moments of inertia are equal.) Initially, the effect of friction is ignored. The motion of the top can be described by three [[Euler angles]]: the tilt angle {{math|''&theta;''}} between the symmetry axis of the top and the vertical (second Euler angle); the [[azimuth]] {{math|''&phi;''}} of the top about the vertical (first Euler angle); and the rotation angle {{math|''&psi;''}} of the top about its own axis (third Euler angle). Thus, precession is the change in {{math|''&phi;''}} and nutation is the change in {{math|''&theta;''}}.<ref name=Goldstein217>{{harvnb|Goldstein|1980|p=217}}</ref>

If the top has mass {{math|''M''}} and its [[center of mass]] is at a distance {{math|''l''}} from the pivot point, its [[gravitational potential]] relative to the plane of the support is
:<math>V = Mgl\cos(\theta).</math>

In a coordinate system where the {{math|''z''}} axis is the axis of symmetry, the top has [[angular velocity|angular velocities]] {{math|''&omega;''<sub>1</sub>, ''&omega;''<sub>2</sub>, ''&omega;''<sub>3</sub>}} and [[moments of inertia]] {{math|''I''<sub>1</sub>, ''I''<sub>2</sub>, ''I''<sub>3</sub>}} about the {{math|''x'', ''y''}}, and {{math|''z''}} axes. Since we are taking a symmetric top, we have {{math|''I''<sub>1</sub>}}={{math|''I''<sub>2</sub>}}. The [[kinetic energy]] is

:<math>E_\text{r} = \frac{1}{2}I_1\left(\omega_1^2 + \omega_2^2\right) + \frac{1}{2}I_3\omega_3^2.</math>

In terms of the Euler angles, this is
:<math>E_\text{r} = \frac{1}{2}I_1\left(\dot{\theta}^2 + \dot{\phi}^2\sin^2(\theta)\right) + \frac{1}{2}I_3\left(\dot{\psi} + \dot{\phi}\cos(\theta)\right)^2.</math>

If the [[Lagrangian mechanics|Euler–Lagrange equations]] are solved for this system, it is found that the motion depends on two constants {{math|''a''}} and {{math|''b''}} (each related to a [[constant of motion]]). The rate of precession is related to the tilt by
:<math>\dot{\phi} = \frac{b - a\cos(\theta)}{\sin^2(\theta)}.</math>

The tilt is determined by a differential equation for {{math|''u'' {{=}} cos(''&theta;'')}} of the form
:<math>\dot{u}^2 = f(u)</math>

where {{math|''f''}} is a [[cubic function|cubic polynomial]] that depends on parameters {{math|''a''}} and {{math|''b''}} as well as constants that are related to the energy and the gravitational torque. The roots of {{math|''f''}} are [[cosine]]s of the angles at which the [[time derivative|rate of change]] of {{math|''&theta;''}} is zero. One of these is not related to a physical angle; the other two determine the upper and lower bounds on the tilt angle, between which the gyroscope oscillates.<ref>{{harvnb|Goldstein|1980|pp=213–217}}</ref>