Physics'601'H omework'4333Due ' Friday'Octob e r'1'!!Goldstein:!9.7,!9.25!!In!addition:!!1. This!problem!uses!the!result!in!problem!2!of!homework!3!in!the!context!of!a!6‐dimension!phase!space!associated!with!a!single!particle!in!3!dimensions.!!!!The!canonical!variables!are!!€ x, y,z, px, py, pz!.!!!In!this!problem!various!transformations!with!clear!physical!meanings!are!proposed.!!!State!the!physical!meaning,!show!that!they!are!canonical!and!find!the!generator!of!the!transformation!!(as!defined!in!in!problem!2!of!homework!3)!and!show!it!satisfies!the!relation! € ∂ ξ ( η ;ε)∂ε= [ ξ ,g]PB!given!in!problem!2!of!homework!3.!In!all!of!these!ε!is!a!constant.!!€ X = x +εY = yZ = zPx= pxPy= pyPz= pza.€ X = x cos(ε) + y sin(ε)Y = −x sin(ε) + y cos(ε)Z = zPx= pxcos(ε) + pysin(ε)Py= −pxsin(ε) + pycos(ε)Pz= pzb.€ X = xY = yZ = zPx= px+εPy= pyPz= pzc.€ X = (1+ε)xY = (1+ε)yZ = (1+ε)zPx= (1+ε)−1pxPy= (1+ε)−1pyPz= (1+ε)−1pzd!!!!!2. Show!that!if!a!one‐parameter!family!of!time‐independent!canonical!transformations!leaves!the!Hamiltonian!invariant!and!its!generator!has!no!explicit!time!dependence!then!its!generator!is!conserved.!!That!is!given!a!canonical!transformation! €  ξ ( η ;ε)!with! € H( ξ ( η ;ε)) = H( η )!and!with!g!as!a!generator‐‐‐! € ∂ ξ ( η ;ε)∂ε= [ ξ ,g]PB‐‐‐‐then!g!is!conserved.!!3. In!problem!1!of!HW!2!we!showed!an!explicit!example!of!a!conserved!quantity!which!was!not!associated!with!a!point!transformation.!!!Use!the!result!shown!above!in!problem!2!to!show!that!the!conserved!quantity!Δ!is!associated!with!a!family!of!!canonical!transformations!by!explicitly!constructing!the!family!of!transformations.!4. Consider!the!canonical!transformation!for!a!system!with!one!degree!of!freedom!generated!by:! .!a. Find!the!explicit!form!for!the!canonical!transform.!b. Verify!that!it!satisfies!the!canonical!Poisson!bracket!relations.!!For!the!remainder!of!this!problem!consider!particle!moving!in!one!dimension!in!a!constant!gravitational!field!€ L(q,˙ q ) =12m˙ q 2− mgq.!c. Use!the!fact!that!for!any!A,$€ dAdt= [A,H]PB+∂A∂tto!show!that!! !and!.!!!d. Part!c.!implies!that!K!the!Hamiltonian!associated!with!Q$,$P$must!be!zero!up!to!a!possibly!time!dependent!constant!independent!of!Q$,$P.!!Show!from!the!explicit!form!of! !that!this!is!in!fact!the!case.!e. What!is!the!physical!interpretation!of!Q$,$P.!f. !Show!that!€ F2(q,P,t)!satisfies!a!Hamilton‐Jacobi!equation:!€ H q,∂F2∂q      +∂F2∂t= 0.!g. Introduce!the!function!!€ f (Q,P,t) = F2(q(Q,P,t),P ,t).!!Show!that!€ ∂f (Q,P,t)∂t= L q(Q,P,t),˙ q (Q,P,t)( ).!!5. In!class!we!derived!the!Hamilton!Jacobi!equation! € H q , ∇ S( q , P )( )+∂S( q , P )∂t= 0!where! €  P !is!a!constant!of!the!motion.!!!Derive!an!!analogus!expression! €  Q € H − ∇ p˜ S ( Q , p ), p )( )+∂˜ S ( Q , p )∂t= 0!where!! €  Q !is!a!constant!of!the!motion!and!the!tilde!is!on!€ ˜ S !to!distinguish!it!from!S.!!! F2(q,P,t) = (q +12gt2)(P " m g t) "P2t2m! dQdt= 0! dPdt= 0!