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→Geometric analysis: 2 𝐸 = 𝐿 2 / 𝐼 3 |
→Geometric analysis: intersection curves |
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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:
[[File:Contour plot of all solutions to Euler's equations.png|thumb|All intersection curves of the angular momentum ellipsoid with energy ellipsoid (not shown).]]
* 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.
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