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matrix and geometric analysis |
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== Theory ==
[[File:Intersecting ellipsoids.gif|thumb|upright=1.5|A visualization of the instability of the intermediate axis. The magnitude of the angular momentum and the kinetic energy of a spinning object are both conserved. As a result, the angular velocity vector remains on the intersection of two ellipsoids. ]]▼
[[File:Dzhanibekov effect.ogv|thumb|upright=1.5|Dzhanibekov effect demonstration in [[microgravity]], [[NASA]].]]
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Note that <math>\omega_1</math> is ''not'' opposed (and therefore will grow) and so rotation around the second axis is ''unstable''. Therefore, even a small disturbance along other axes causes the object to 'flip'.
=== Matrix analysis ===
If we the object is mostly rotating along its third axis, so <math>|\omega_3 | \gg |\omega_1 |, |\omega_2 | </math>, we can assume <math>\omega_3</math> does not vary much, and write the equations of motion as a matrix equation:<math display="block">\frac{d}{dt}\begin{bmatrix}
\omega_1\\
\omega_2
\end{bmatrix} =
\begin{bmatrix}
0 & \omega_3(I_3-I_2)/I_1 \\
\omega_3(I_1 - I_3)/I_2 & 0
\end{bmatrix} \begin{bmatrix}
\omega_1\\
\omega_2
\end{bmatrix}</math>which has [[Stability theory#Stability of fixed points in 2D|zero trace and positive determinant]], implying the motion of <math>(\omega_1, \omega_2)</math> is a stable rotation around the origin -- a neutral equilibrium point. Similarly, the point <math>(\omega_1, 0,0)</math> is a neutral equilibrium point, but <math>(0, \omega_2, 0)</math> is a saddle point.
== Geometric analysis ==
▲[[File:Intersecting ellipsoids.gif|thumb|upright=1.5|A visualization of the instability of the intermediate axis. The magnitude of the angular momentum and the kinetic energy of a spinning object are both conserved. As a result, the angular velocity vector remains on the intersection of two ellipsoids. ]]During motion, both the energy and angular momentum-squared are conserved, thus we have two conserved quantities:<math display="block">\begin{cases}
2E = \sum_i I_i \omega_i^2\\
L^2 = \sum_i I_i^2 \omega_i^2
\end{cases}</math>and so for any initial condition <math>\omega(0)</math>, the trajectory of <math>\omega(t)</math> must stay on the intersection curve between two ellipsoids defined by <math display="block">\begin{cases}
\sum_i I_i \omega_i^2 = \sum_i I_i \omega_i(0)^2\\
\sum_i I_i^2 \omega_i^2 = \sum_i I_i^2 \omega_i(0)^2
\end{cases}</math>This is shown on the animation to the left.
By inspecting Euler's equations, we see that <math>\omega(t) = 0</math> implies that two components of <math>\omega(t)</math> are zero -- that is, the object is exactly spinning around one of the principal axes. In all other situations, <math>\omega(t)</math> must remain in motion.
By Euler's equations, if <math>\omega(t)</math> is a solution, then so is <math>c \omega(ct)</math> for any constant <math>c > 0</math>. Consequently we can analyze the geometry of motion with a fixed value of <math>L^2</math>, and vary <math>\omega(0)</math> on the fixed ellipsoid of constant squared angular momentum. As <math>\omega(0)</math> varies, the value of <math>2E</math> also varies -- thus giving us a varying ellipsoid of constant energy. This is shown in the animation as a fixed orange ellipsoid and increasing blue ellipsoid.
For concreteness, consider <math>I_1 = 1, I_2 = 2, I_3 = 3</math>, then the angular momentum ellipsoid's major axes are in ratios of <math>1 : 1/2 : 1/3</math>, and the energy ellipsoid's major axes are in ratios of <math>1 : 1/\sqrt 2 : 1/\sqrt 3</math>. Thus the angular momentum ellipsoid is both flatter and sharper, as visible in the animation. In general, the angular momentum ellipsoid is always more "exaggerated" than the energy ellipsoid.
Now inscribe on a fixed ellipsoid of <math>L^2</math> its intersection curves with the ellipsoid of <math>2E</math>, as <math>2E</math> increases from zero to infinity. We can see that the curves evolve as follows:
* For small energy, there is no intersection, since we need a minimum of energy to stay on the angular momentum ellipsoid.
* The energy ellipsoid first intersects the momentum ellipsoid 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_3)</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 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.
== See also ==
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