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matrix and geometric analysis |
→Geometric analysis: 2 𝐸 = 𝐿 2 / 𝐼 3 |
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* 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 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_3)</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.
== See also ==
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