Mechancs Physcs 151 Lecture 22 Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j P, Q, P j, P Q, P Necessary and suffcent P j P j for Canoncal Transf. = = j Q, Q, P j Q, Q, P Infntesmal CT u v u v Posson Bracket [ uv, ] Canoncal nvarant Fundamental PB [, ] = [, ] = [, ] = [, ] = δ j j u ICT exressed by δ u = ε[ u, G] + δt t Infntesmal tme transf. generated by Hamltonan Hamlton s euatons j j j 1
Two Ponts of Vew Canoncal Transformaton allows one system to be descrbed by multle sets of coordnates/momenta Same hyscal system s exressed n dfferent hase saces P Q Ths s the statc vew The system tself s unaffected Is there a dynamc vew? Dynamc Vew of CT A system evolves wth tme t ( ), t ( ) t (), t () At any moment, and satsfy Hamlton s euatons The tme-evoluton must be a Canoncal Transformaton! Ths movement s a CT Statc Vew = Coordnate system s changng Dynamc Vew = Physcal system s movng 2
Infntesmal Tme CT Infntesmal CT t (), t () t ( + dt), t ( + dt) We know that the generator = Hamltonan u du = dt[ u, H ] + dt = [, H] = [ H, ] t Hamltonan s s the generator of of the system s moton wth tme Integratng t wth tme should gve us the fnte CT that turns the ntal condtons (t ), (t ) nto the confguraton (t), (t) of the system at arbtrary tme That s a new defnton of solvng the roblem Statc vs. Dynamc Two ways of lookng at the same thng System s movng n a fxed hase sace Hamlton s euatons Integrate to get (t), (t) System s fxed and the hase sace s transformng ICT gven by the PB Integrate to get CT for fnte t Euatons are dentcal You ll fnd yourself ntegratng exactly the same euatons Dd we gan anythng? 3
Conservaton Consder an ICT generated by G Suose G s conserved and has no exlct t-deendence u δ u = ε[ u, G] + δt t [ GH, ] = How s H (wthout t-deendence) changed by the ICT? H δh = ε[ H, G] + δt = t If an ICT does not affect Hamltonan, ts generator s conserved A transformaton that does not affect H Symmetry of the system Generator of the transformaton s conserved Momentum Conservaton Smlest examle: What s the ICT generated by momentum? δ = ε[, ] = εδ δ = ε[, ] = j j j j j That s a shft n by ε satal translaton If Hamltonan s unchanged by such shft, then Momentum s conserved Ths s not restrcted to lnear momentum [ H, ] = Hamltonan s unchanged by a shft of a coordnate The generator of the ICT s the conjugate momentum [ H, ] = s conserved 4
Angular Momentum Let s consder a secfc case: Angular momentum Pck x-y-z system wth z beng the axs of rotaton n artcles ostons gven by ( x, y, z) Rotate all artcles CCW around z axs by dθ x = x ydθ y = y + xdθ Momenta are rotated as well ( x, y ) dθ x = x ydθ y = y + xdθ ( x, y) Generator s G = xy yx G G dθ[ x, G] = dθ = yd θ dθ[ x, G] = dθ = ydθ x x etc. Angular Momentum The generator G = xy yx s obvously L = ( r ).e. the z-comonent of the total momentum Generator for rotaton about an axs gven by a unt vector n should be G = Ln We now know generators of 3 mortant ICTs Hamltonan generates dslacement n tme Lnear momentum generates dslacement n sace Angular momentum generates rotaton n sace z z 5
Integratng ICT I sad we can ntegrate ICT to get fnte CT How do we ntegrate δ u = ε[ u, G]? du Frst, let s rewrte t as du = dα[ u, G] [ ug, ] dα = We want the soluton u(α) as a functon of α, wth the ntal condton u() = u Taylor exand u(α) from α = α α u( α) = u + α + + + α α α 2 2 3 3 du d u d u 2 3 d 2! d 3! d Ths s [u,g] What can I do wth these? Integratng ICT du Snce [ ug, ] s true for any u, we can say dα = d [, G dα = ] Now aly ths oerator reeatedly 2 j du d [ 2 ug, ] [[ ug, ], G ] dα = dα = du = [ [[ ug, ], G ],, G ] j dα Gong back to the Taylor exanson, α α u( α) = u + α + + + α α α 2 2 3 3 du d u d u 2 3 d 2! d 3! d 2 3 α α = u + α[ u, G] + [[ u, G], G] + [[[ u, G], G], G] + 2! 3! Now we have a formal soluton But does t work? 6
Rotaton CT Let s ntegrate the ICT for rotaton around z Let me forget the artcle ndex Parameter α s θ n ths case Let s see how x changes wth θ 2 3 θ θ x( θ) = x + θ[ xg, ] + [[ xg, ], G] + [[[ xg, ], G], G] + 2! 3! Evaluate the Posson Brackets [ x, G] = y [[ x, G], G] = x [[[ x, G], G], G] = y [[[[ x, G], G], G], G] Where does ths lead us? G = x y = x Reeats after ths y x Rotaton CT 2 3 θ θ x( θ) = x + θ[ xg, ] + [[ xg, ], G] + [[[ xg, ], G], G] + 2! 3! 2 3 4 θ θ θ = x θ y x + y + x 2! 3! 4! Smlarly 2 4 3 5 θ θ θ θ y θ = x 1 + + 2! 4! 3! 5! = x cosθ y snθ 2 3 θ θ y( θ) = y + θ[ yg, ] + [[ yg, ], G] + [[[ yg, ], G], G] + 2! 3! = y cosθ + x snθ 7
Free Fall z An object s fallng under gravty 2 Hamltonan s H = + mgz 2m Integrate the tme ICT 2 3 t t zt ( ) = z + tzh [, ] + [[ zh, ], H] + [[[ zh, ], H], H] + 2! 3! [, zh] = [[ zh, ], H] = g [[[ zh, ], H], H ] = m g zt () z t t m 2 2 = + It s easer than t looked Infntesmal Rotaton ICT for rotaton s generated by G = Ln We ve studed nfntesmal rotaton n Lecture 8 Infntesmal rotaton of dθ about n moves a vector r as dr = ndθ r Comare the two exressons dr = dθ[, r L n] = ndθ r [, rln ] = n r Euaton [, rln ] = n rholds for any r that rotates together wth the system Several useful rules can be derved from ths 8
Scalar Products [, rln ] = n r Consder a scalar roduct ab of two vectors Try to rotate t [ abln, ] = a [ bln, ] + b [ aln, ] = a ( n b) + b ( n a) = a ( n b) + a ( b n) = Obvous: scalar roduct doesn t change by rotaton Also obvous: length of any vector s conserved Angular Momentum Try wth L tself x-y-z comonents are [ LLn, ] = n L [ L, L ] = [ L, L ] = L [ L, L ] = L x x x y z x z y [ L, L ] = L [ L, L ] = [ L, L ] = L y x z y y y z x [ L, L ] = L [ L, L ] = L [ L, L ] = z x y z y x z z [ L, L ] = ε L j jk k These relatonshs are well-known n QM They tell us two rather nterestng thngs 9
Angular Momentum Imagne two conserved uanttes A and B [ AH, ] = [ BH, ] = How does [A,B] change wth tme? [[ AB, ], H] = [[ BH, ], A] [[ H, A], B] = Jacob s dentty Posson bracket of two conserved uanttes s conserved Now consder [ L, L ] = ε L j jk k If 2 comonents of L are conserved, the 3 rd comonent must Total vector L s conserved Angular Momentum Remember the Fundamental Posson Brackets? [, ] = [, ] = [, ] = [, ] = δ j j Now we know [ L, L ] = ε L j j j PB of two canoncal momenta s j jk k Posson brackets between L x, L y, L z are non-zero Only 1 of the 3 comonents of the angular momentum can be a canoncal momentum 2 On the other hand, [ L, L ] =, so L may be a canoncal momentum QM: You may measure L and, e.g., L z smultaneously, but not L x and L y, etc. 1
Phase Volume Statc vew: CT moves a ont n one hase sace to a ont n another hase sace Dynamc vew: CT moves a ont n one hase sace to another ont n the same sace If you consder a set of onts, CT moves a volume to anther volume, e.g. P d How does the area change? d Q Phase Volume Easy to calculate the Jacoban for 1-dmenson Q Q dqdp = M dd where M = P P Q P P Q = = [ QP, ] = 1 =.e., volume n 1-dm. hase sace s nvarant M dqdp dd Ths s true for n-dmensons Goldsten roves t usng smlectc aroach Volume n Phase Sace s a Canoncal Invarant 11
Harmonc Oscllator We ve seen t n the oscllator examle (Lecture 21) P 2mE 2E 2 mω E ω One cycle draws the same area That s statc vew 2π E ω 2π n both saces Q Dynamc Vew Consder many artcles movng ndeendently e.g., deal gas molecules n a box They obey the same EoM ndeendently Can be reresented by multle onts n one hase sace They move wth tme CT Tme 12
Ideal Gas Dynamcs Imagne deal gas n a cylnder wth movable ston Each molecule has ts own oston and momentum They fll u a certan volume n the hase sace What haens when we comress t? Extra momenta Gas gets hotter! Comress slowly Louvlle s Theorem The hase volume occued by a grou of artcles (ensemble n stat. mech.) s conserved Thus the densty n hase sace remans constant wth tme Known as Louvlle s theorem Theoretcal bass of the 2 nd law of thermodynamcs Ths holds true when there are large enough number of artcles so that the dstrbuton may be consdered contnuous More about ths n Physcs 181 13
Summary Introduced dynamc vew of Canoncal Transf. Hamltonan s the generator of the moton wth tme Symmetry of the system Hamltonan unaffected by the generator Generator s conserved How to ntegrate nfntesmal transformatons 2 3 α α u( α) = u + α[ ug, ] + [[ ug, ], G] + [[[ ug, ], G], G] + 2! 3! Dscussed nfntesmal rotaton [, rln ] = n r Angular momentum QM Invarance of the hase volume Louvlle s theorem Stat. Mech. 14