Tennis racket theorem: Difference between revisions

Revision as of 22:27, 27 April 2023

Revision as of 22:43, 27 April 2023

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* The energy ellipsoid first intersects the momentum ellipsoid when <math>2E = L^2/I_3</math>, at the points <math>(0, 0, \pm L/I_3)</math>. This is when the body rotates around its axis with the largest moment of inertia.

* They intersect at two cycles around the points <math>(0, 0, \pm L/I_3)</math>. Since each cycle contains no point at which <math>\dot\omega=0</math>, the motion of <math>\omega(t)</math> must be a periodic motion around each cycle.

* They intersect at two "diagonal" curves that intersects at the points <math>(0, 0, \pm L/I_3I_2, 0)</math>, when <math>2E = L^2/I_2</math>. If <math>\omega(t)</math> starts anywhere on the diagonal curves, it would approach one of the points, distance exponentially decreasing, but never actually reach the point. In other words, we have 4 [[Heteroclinic orbit|heteroclinic orbits]] between the two saddle points.

* They intersect at two cycles around the points <math>(\pm L / I_1, 0, 0)</math>. Since each cycle contains no point at which <math>\dot\omega=0</math>, the motion of <math>\omega(t)</math> must be a periodic motion around each cycle.

* The energy ellipsoid last intersects the momentum ellipsoid when <math>2E = L^2/I_1</math>, at the points <math>(\pm L / I_1, 0, 0)</math>. This is when the body rotates around its axis with the smallest moment of inertia.

The tennis racket effect occurs when <math>\omega(0)</math> is very close to a saddle point. The body would linger near the saddle point, then rapidly move to the other saddle point, repeatingnear periodically<math>\omega(T/2)</math>, linger again for a long time, and so on. The motion repeats with period <math>T</math>.

The above analysis is all done in the perspective of an observer which is rotating with the body. An observer watching the body's motion in free space would see its angular momentum vector <math>\vec L = I\vec \omega</math> conserved, while both its angular velocity vector <math>\vec \omega(t)</math> and its moment of inertia <math>I(t)</math> undergoing complicated motions in space. At the beginning, the observer would see both <math>\vec \omega(0), \vec L</math> mostly aligned with the second major axis of <math>I(0)</math>. After a while, the body performs a complicated motion and ends up with <math>I(T/2), \vec \omega(T/2)</math>, and again both <math>\vec L, \vec \omega(T/2)</math> are mostly aligned with the second major axis of <math>I(T/2)</math>.

Consequently, there are two possibilities: either the rigid body's second major axis is in the same direction, or it has reversed direction. If it is still in the same direction, then <math>\vec\omega(0), \vec\omega(T/2)</math> viewed in the rigid body's reference frame are also mostly in the same direction. However, we have just seen that <math>\omega(0)</math> and <math>\omega(T/2)</math> are near opposite saddle points <math>(0, \pm L/I_2, 0)</math>. Contradiction.

Qualitatively, then, this is what an observer watching in free space would observe:

* The body rotates around its second major axis for a while.

* The body rapidly undergoes a complicated motion, until its second major axis has reversed direction.

* The body rotates around its second major axis again for a while. Repeat.

This can be easily seen in the video demonstration in microgravity.

== See also ==