Solving the Symmetric Top - In this problem you will both qualitatively (analyt- ically) and quantitatively...

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Solving the Symmetric Top - In this problem you will both qualitatively (analyt- ically) and quantitatively (numerically) solve for the motion of a symmetric top that is fixed at one end and in a gravitational field. Recall that the Lagrangian in this case was: L = (MR² + 1₁) (0² + 6² sin² 0)+1(+cos 0)² - MgR cos, where R = |R| is the distance from the fixed tip of the top and its center of mass and I₁ and I3 are the principle moments of inertia. Rederive the expressions given in lecture. That is, show that the equations of motion for the Euler angles are: lz-l3 cos 0 I sin²0 2 E - l3 v= 13 (l₂-l3 cos 0)² 21 sin²0 8 -cos-l3 cos I sin²0 lz 213 MgR cos e where I = MR² + 1₁. Focusing on the last of these, the equation of motion for 0, show that upon setting u = cos this can be simplified to find i² = (1-²)(a-Bu)-(b-au)², for appropriate constants a, 3, a and b. That is, find the constants! For hints see Classical Mechanics by Goldstein. Solving the Symmetric Top - In this problem you will both qualitatively (analyt- ically) and quantitatively (numerically) solve for the motion of a symmetric top that is fixed at one end and in a gravitational field. Recall that the Lagrangian in this case was: L = (MR² + 1₁) (0² + 6² sin² 0)+1(+cos 0)² - MgR cos, where R = |R| is the distance from the fixed tip of the top and its center of mass and I₁ and I3 are the principle moments of inertia. Rederive the expressions given in lecture. That is, show that the equations of motion for the Euler angles are: lz-l3 cos 0 I sin²0 2 E - l3 v= 13 (l₂-l3 cos 0)² 21 sin²0 8 -cos-l3 cos I sin²0 lz 213 MgR cos e where I = MR² + 1₁. Focusing on the last of these, the equation of motion for 0, show that upon setting u = cos this can be simplified to find i² = (1-²)(a-Bu)-(b-au)², for appropriate constants a, 3, a and b. That is, find the constants! For hints see Classical Mechanics by Goldstein.