Physics'601'H omework'3333Due ' Friday'Septem ber'24'!!Goldstein!problems!!8.1,!!8.6,!8.9,!8.26,!8.23!!In!addition:!!1. For!a!particle!in!3!dimensions!the!angular!momentum!operator!is!given!by! €  L = x × p !where! €  x !and! €  p !satisfy!are!canonical!(i.e.!they!satisfy!the!canonical!Poisson!bracket!relations.!a. Show!that!€ [Li,Lj]PB=εijkLk!where$i,j,k$!take!on!the!values!x,y,z.!!(Note!that!this!is!isomorphic!to!the!!commutators!for!the!angular!momentum!in!quantum!mechanics.!!For!those!with!mathematically!inclinations,!this!is!the!Lie!algebra!SO(3).)!!b. Show!that! € [ L ,L2]PB= 0!!(that!is!that€ [Li,L2]PB= 0!for!all!i)!where! € L2= L ⋅ L !.!c. Show!that! € [ L , f (r)]PB= 0!!where!f$is!an!arbitrary!function!and! € r = x ⋅ x .!Note!that!parts!b.!&c.!reflect!a!deeper!result! € [ L ,s]PB= 0!for!any!scalar!s.!!This!reflects!the!fact!that!the!angular!momentum!is!the!generator!of!rotations.!!!!!2. Label!our!canonical!variables!by!a!phase‐space!vector!!!!! €  η =q1q2...qnp1p2...pn1                        !which!satisfies!canonical!Poisson!brackets!€ [ηi,ηj]PB= Jij!!now!suppose!that!!there!is!a!one!parameter!continuous!family!of!!canonical!transformations!depending!on!one!parameter!ε :! €  ξ ( η ;ε)!!. a. !Show!that! €  ξ ( η ;ε)!!satisfies!€ [ξi,ξj]PB= Jij!if!it!satisfies!the!conditions!!€ dξidε= [ξi,g]PBwith €  ξ ( η ;0) = η !for!some!!g.!!Note!that!this!is!the!same!form!as!usual!Hamiltonian!time!!with!t!replaced!by!ε and!H!replaced!by!g.!!The!function!g!is!called!the!generator!of!the!transformation.!!!!!b. The!time!evolution!of!the!phase‐space!position!under!Hamiltonian!gives!a!transformation!from!an!initial!point!in!phase!space!to!a!subsequent!one.!!That!is! €  η (t)is!really!a!function!the!time!and!the!initial!conditions:! €  η ( η 0,t).!!Note!that!the!initial!conditions!are!a!set!of!phase!space!variable!which!are!canonical.!!Now!let!me!define!a!canonical!transformation! €  ξ ( η ,T )!which!has!the!same!functional!relation!as!! €  η ( η 0,t).!!That!is! €  ξ ( η ,T )corresponds!to!the!value!the!point!in!phase!space!that!any!point!in!phase!space!evolves!into!from! €  η !a!time!T!later.!!Using!part!a)!show!that!for!any!T!,! €  η ( ξ ,T )!!satisfies!the!canonical!Poission!bracket!relation!with$H!acting!as!the!generator!of!time!translations.!!!!!!