Problem $3$: An asymmetric top, with $EL2E{I}_1{L}^{2}2E{I}_3{L}^2=2E{I}_{2}hat(e{)}_1$hat$(e{)}_3{}_2=0t=0{}_{}=\frac{2E}{L},$ and $,=\frac{1}{{}_{}}\sqrt[\phantom{2}]{\frac{I_1{I}_3}{(I_3-I_2)(I_2-I_1)}}.{}_1(t)={}_{}\sqrt[\phantom{2}]{\frac{I_2(I_3-I_2)}{I_1(I_3-I_1)}}sech(\frac{t}{})$

${}_2(t)={}_{}tanh(\frac{t}{})$

${}_3(t)={}_{}\sqrt[\phantom{2}]{\frac{I_2(I_2-I_1)}{I_3(I_3-I_1)}}sech(\frac{t}{}).{}^2$hat$()I_1,$ executes torque$-$free motion with energy

$E$ and angular momentum $L.(a)$ Verify that the $(squared)$ angular momentum $is$ confined

$to$ the range $2E{I}_1{L}^{2}2E{I}_3.(b)$ Assume $L^2=2E{I}_2,$ and that $$ initially lies $in$ the

plane formed $by$ hat$(e{)}_1$ and hat$(e{)}_3(i.e,$ that ${}_2=0att=0).$ Define the quantities

${}_{}=\frac{2E}{L},$ and $,=\frac{1}{{}_{}}\sqrt[\phantom{2}]{\frac{I_1{I}_3}{(I_3-I_2)(I_2-I_1)}}.$

Integrate Euler's equations $to$ obtain the solutions

${}_1(t)={}_{}\sqrt[\phantom{2}]{\frac{I_2(I_3-I_2)}{I_1(I_3-I_1)}}sech(\frac{t}{})$

${}_2(t)={}_{}tanh(\frac{t}{})$

${}_3(t)={}_{}\sqrt[\phantom{2}]{\frac{I_2(I_2-I_1)}{I_3(I_3-I_1)}}sech(\frac{t}{}).$

$(c)$ Discuss the time dependence $of{}^2,$ and sketch the motion $of$ the unit vector hat$()as$ seen

from the body frame. $(If$ you're feeling creative, try $to$ create a visualization $of$ using

Desmos, Mathematica, $or$ something similar.$)$