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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>. In particular, the motion of the body in free space (obtained by integrating <math>c\omega(ct) dt</math>) is ''exactly the same'', just completed faster by a ratio of <math>c</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.
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