Mechancs hyscs 151 Lecture Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j, Q, j, Q, Necessary and suffcent j j for Canoncal Transf. = = j Q, Q, j Q, Q, Infntesmal CT u v u v osson Bracket [ uv, ] Canoncal nvarant Fundamental B [, ] = [, ] = [, ] = [, ] = 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 Two onts 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 Q Ths s the statc vew The system tself s unaffected Is there a dynamc vew? 1
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 = hyscal system s movng 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 B Integrate to get CT for fnte t Euatons are dentcal You ll fnd yourself ntegratng exactly the same euatons Dd we gan anythng?
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 Angular Momentum Let s consder a secfc case: Angular momentum ck x-y- system wth beng the axs of rotaton n artcles ostons gven by ( x, y, ) Rotate all artcles CCW around axs by d x = x yd y = y + xd Momenta are rotated as well x = x yd y = y + xd ( x, y ) d ( x, y) Generator s G = xy yx G G d[ x, G] = d = yd d[ x, G] = d = yd x x etc. 3
Angular Momentum The generator G = xy yx s obvously L = ( r ).e. the -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 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 = du α d u α d u u( α) = u + α + + + α α α 3 d! 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 j du d du = [ ug, ] = [[ ug, ], G ] = [ [[ ug, ], G],, G] j dα dα dα Gong back to the Taylor exanson, 3 du α d u α d u d! d 3! d u( α) = u + α + + + α α α α α = u + α[ u, G] + [[ u, G], G] + [[[ u, G], G], G] + Now we have a formal soluton But does t work? 4
Rotaton CT Let s ntegrate the ICT for rotaton around Let me forget the artcle ndex G = xy yx arameter α s n ths case Let s see how x changes wth x( ) = x + [ xg, ] + [[ xg, ], G] + [[[ xg, ], G], G] + Evaluate the osson Brackets [ x, G] = y [[ x, G], G] = x [[[ x, G], G], G] = y [[[[ x, G], G], G], G] Where does ths lead us? = x Reeats after ths Rotaton CT x( ) = x + [ xg, ] + [[ xg, ], G] + [[[ xg, ], G], G] + 4 = x y x + y + x 4! Smlarly 4 3 5 y = x 1 + +! 4! 3! 5! = x cos y sn y( ) = y + [ yg, ] + [[ yg, ], G] + [[[ yg, ], G], G] + = y cos + x sn Free Fall An object s fallng under gravty Hamltonan s H = + mg m Integrate the tme ICT t t t () = + th [, ] + [[, H], H] + [[[, H], H], H] + [, H] = [[ H, ], H] = g [[[, H], H], H ] = m g t () t t m = + It s easer than t looked 5
Infntesmal Rotaton ICT for rotaton s generated by G = Ln We ve studed nfntesmal rotaton n Lecture 8 Infntesmal rotaton of d about n moves a vector r as dr = nd r Comare the two exressons dr = d[, r L n] = nd 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 Scalar roducts [, 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- comonents are [ LLn, ] = n L [ L, L ] = [ L, L ] = L [ L, L ] = L x x x y x y [ L, L ] = L [ L, L ] = [ L, L ] = L y x y y y x [ L, L ] = L [ L, L ] = L [ L, L ] = x y y x These relatonshs are well-known n QM They tell us two rather nterestng thngs [ L, L ] = ε L j jk k 6
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 osson bracket of two conserved uanttes s conserved Now consder [ L, L ] = ε L j jk k If comonents of L are conserved, the 3 rd comonent must Total vector L s conserved Angular Momentum Remember the Fundamental osson Brackets? [, ] = [, ] = [, ] = [, ] = δ j j Now we know [ L, L ] = ε L j j j B of two canoncal momenta s j jk k osson brackets between L x, L y, L are non-ero Only 1 of the 3 comonents of the angular momentum can be a canoncal momentum On the other hand, [ L, L ] =, so L may be a canoncal momentum QM: You may measure L and, e.g., L smultaneously, but not L x and L y, etc. hase 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. d How does the area change? d Q 7
hase Volume Easy to calculate the Jacoban for 1-dmenson Q Q dqd = M dd where M = Q Q M = = [ Q, ] = 1 dqd = dd.e., volume n 1-dm. hase sace s nvarant Ths s true for n-dmensons Goldsten roves t usng smlectc aroach Volume n hase Sace s a Canoncal Invarant Harmonc Oscllator We ve seen t n the oscllator examle (Lecture 1) me E mω E ω One cycle draws the same area That s statc vew π E ω π 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 8
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 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 hyscs 181 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 α α u( α) = u + α[ u, G] + [[ u, G], G] + [[[ u, G], G], G] + Dscussed nfntesmal rotaton [, rln ] = n r Angular momentum QM Invarance of the hase volume Louvlle s theorem Stat. Mech. 9