Editing Precession (section)

==Torque-free or torque neglected==
Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the [[angular momentum]] is a constant, but the [[angular velocity]] vector changes orientation with time. What makes this possible is a time-varying [[moment of inertia]], or more precisely, a time-varying [[Moment of inertia#The inertia tensor|inertia matrix]]. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate [[Basis (linear algebra)|coordinate axes]] (e.g. {{math|''x''}}, {{math|''y''}}, {{math|''z''}}). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the [[Vector component#Decomposition|component]] of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia.

The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:<ref>{{Citation|author1-link=Hanspeter Schaub| last =Schaub| first =Hanspeter| year =2003| title =Analytical Mechanics of Space Systems| publisher =AIAA| isbn =9781600860270| pages =149–150| url =https://books.google.com/books?id=qXvESNWrfpUC}}</ref>
<math display="block">\boldsymbol\omega_\mathrm{p} = \frac{\boldsymbol I_\mathrm{s} \boldsymbol\omega_\mathrm{s} } {\boldsymbol I_\mathrm{p} \cos(\boldsymbol \alpha)}</math>
where {{math|'''''ω'''''<sub>p</sub>}} is the precession rate, {{math|'''''ω'''''<sub>s</sub>}} is the spin rate about the axis of symmetry, {{math|'''''I'''''<sub>s</sub>}} is the moment of inertia about the axis of symmetry, {{math|'''''I'''''<sub>p</sub>}} is moment of inertia about either of the other two equal perpendicular principal axes, and {{mvar|'''α'''}} is the angle between the moment of inertia direction and the symmetry axis.<ref>{{cite web| url = https://www.sfu.ca/~boal/211lecs/211lec26.pdf| title = Lecture 26 – Torque-free rotation – body-fixed axes| first = David| last = Boal| year = 2001 | access-date = 2008-09-17}}</ref>

When an object is not perfectly [[Rigid body dynamics|rigid]], inelastic dissipation will tend to damp torque-free precession,<ref>{{cite journal |doi=10.1111/j.1365-2966.2005.08864.x |title=Nutational damping times in solids of revolution |journal=Monthly Notices of the Royal Astronomical Society |volume=359 |issue=1 |page=79 |year=2005 |last1=Sharma |first1=Ishan |last2=Burns |first2=Joseph A. |last3=Hui |first3=C.-H. |doi-access=free |bibcode=2005MNRAS.359...79S }}</ref> and the rotation axis will align itself with one of the inertia axes of the body.

For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix {{mvar|'''R'''}} that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal [[moment of inertia tensor]] {{math|'''''I'''''<sub>0</sub>}} and fixed external angular momentum {{mvar|'''L'''}}, the instantaneous angular velocity is
<math display="block">\boldsymbol\omega\left(\boldsymbol R\right) = \boldsymbol R \boldsymbol I_0^{-1} \boldsymbol R ^T \boldsymbol L</math>
Precession occurs by repeatedly recalculating {{mvar|'''ω'''}} and applying a small [[Rotation representation (mathematics)#Euler axis and angle (rotation vector)|rotation vector]] {{math|'''''ω''' dt''}} for the short time {{math|''dt''}}; e.g.:
<math display="block">\boldsymbol R_\text{new} = \exp\left(\left[\boldsymbol\omega\left(\boldsymbol R_\text{old}\right)\right]_{\times} dt\right) \boldsymbol R_\text{old}</math>
for the [[Cross product#Conversion to matrix multiplication|skew-symmetric matrix]] {{math|['''''ω''''']<sub>×</sub>}}. The errors induced by finite time steps tend to increase the rotational kinetic energy:
<math display="block">E\left(\boldsymbol R\right) = \boldsymbol \omega\left(\boldsymbol R\right) \cdot \frac{\boldsymbol L }{ 2}</math>
this unphysical tendency can be counteracted by repeatedly applying a small rotation vector {{mvar|'''v'''}} perpendicular to both {{mvar|'''ω'''}} and {{mvar|'''L'''}}, noting that
<math display="block">E\left(\exp\left(\left[\boldsymbol v\right]_{\times}\right) \boldsymbol R\right) \approx E\left(\boldsymbol R\right) + \left(\boldsymbol \omega\left(\boldsymbol R\right) \times \boldsymbol L\right) \cdot \boldsymbol v</math>