Physics'601'Homework'11222Due'Friday'Dec.'3''!1. In!class!we!algebraically!derived!the!expression!for!€ ωzbody!in!terms!of!Euler!angles!and!there!time!derivatives.!Using!the!same!methods!derive!the!expression!for!€ ωxbody!and!!€ ωybody.!2. Consider!a!symmetric!top!where!the!two!identical!moments!of!inertia!a re!given !b y!I0!and!the!other!moment!is!by!I0/2!.!The!(conserved)!angular!momentum!about!the!body!fixed!zDaxis!is!given!by€ I0ω0!.!!!Suppose!that!€ mgl =ω02I0!.!a. Suppose!!that!the!system!precesses!without!nutating!with!a!precession!frequency!of!2€ ω0!!Find!the!angle!θ.!b. Suppose!that!at!t=0!the!angle!θ!is!slightly!perturbed!away!from!the!equilibrium!angle!found!in!a.!by!an!amount!Δθ.!!Find!the!nutation!frequency.!3. Consider!the!application!of!the!formal!short!time!expansion!developed!in!class!to!a!simple!Harmonic!oscillator:!€ L =12m˙ x 2−12λ2mω02˙ x 2!!with!€ x = x0+λx1+λ2x2+ ...!.!!!Suppose!the!initial!conditions!are!fixed!by!€ x0(0) = x(0);˙ x 0(0) = 0;x1(0) = 0,˙ x 1(0) =˙ x (0)!with!€ xn(0) = 0,˙ x n(0) = 0for!n>1.!!λ is!taken!to!unity!at!the!end!of!the!problem.!a. Solve!for!x(t)!to!5th!order.!b. Verify!that!the!solution!is!identical!to!the!exact!solution!Taylor!expended!in!time!up!to!t5.!4. In!class!we!formally!developed!the!shortDtime!!expansion!for!a!system!with!one!degree!of!freedom.!!In!this!problem!I!would!like!you!to!use!the!same!methods!to!derive!it!for!2!degrees!of!freedom!:!€ L =12m(˙ x 2+˙ y 2) −λ2V (x, y)!!with!€ x = x0+λx1+λ2x2+ ... ;y = y0+λy1+λ2y2+ ....!!Choosing!sensible!boundary!conditions!find!an!expression!x(t),y(t)!up!to!4th!order.!!The!result!should!be!given!in!terms!of!the!initial!position,!the!initial!velocity,!and!partial!!derivatives!with!respect!to!V+evaluated!at!the!initial!position.!!5. Apart!from!short!time!expansion!there!is!a!context!in!which!one!can!build!an!expansion!in!which!the!full!potential!is!treated!perturbatively:!if!one!is!working!in!a!regime!in!which!potential!energy!difference!are!always!much!less!then!the!kinetic!energy!the!approximation!can!be!justified!up!to!comparative!long!times.!!!Consider!the!case!of!a!!physical!pendulum!whose!Lagrangian!is!given!by!€ L =12I˙ θ 2+ V0cos(θ)where!I!is!the!moment!of!inertia!and!V0!is!the!maximum!value!of!the!potential!energy!(i.e.!m+g+L)!where!L!is!length!from!the!pivot!point!to!the!center!of!mass.!!!The!problem!under!consideration!is!this:!suppose!that!at!t=0!the!system!is!at!the!minimum!of!the!potential!!(θ!=0)!with!an!initial!angular!velocity!€ ˙ θ =ω0.!!!!To!develop!the!expansion!insert!powers!of!λ:!€ L =12I˙ θ 2+λV0cos(θ)with€ θ=ω0t +λθ1+λ2θ2+ ...!and!impose!boundary!conditions!€ θi(0) = 0,˙ θ i(0) = 0!for!i.!!These!boundary!conditions!are!designed!to!ensure!that!the!boundary!conditions!to!our!problem!is!solved!!! Find!explicitly!the!form!for!€ θ1!and!€ θ2.!!6. !This!is!a!continuation!of!the!problem!discussed!above.!!While!the!solution!found!there!is!formally!correct!based!on!the!expansion,!as!with!naïve!perturbation!theory!for!an!anharmonic!oscillator!it!has!a!problem!with!periodicity!and!a!reorganized!series!can!give!much!more!accurate!answers..!a. Show!on!general!grounds!that!for!this!problem!that!in!the!regime!where!!system!has!sufficient!energy!to!go!“over!the!top”!that!θ(t) is!of!!the!form!!€ θ(t) =ωt + cnsin(nωt)∑!,!where!ω!is!in!general!not!€ ω0.!b. Using!the!power!counting!scheme!€ ω=ω0+λω1+λ2ω2+ ...,!€ cn=λn(cn(0)+λcn(1)+λ2cn(2)...)!compute!€ θ(t)!up!to!2nd!order.!!7. This!is!a!continuation!of!problems!4!and!5.!!Consider!initial!conditions!w here!the!total!energy!E!is!!€