Mechanics Physics 151

Transcription

1 Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( ) = δt ( ) = δ pt ( ) = δ pt ( ) = 0 Not strctly needed, but adds flexblty to the defnton of the acton ntegral Ths connects to: Canoncal Transformatons Prncple of Least Acton t t t t ( ) p = 0 Got nto ths a bt

2 Canoncal Transformaton Goal: To fnd transformatons = (,, n, p,, pn, t) P = P(,, n, p,, pn, t) that satsfy Hamlton s euaton of moton H = dp H p = d K = dp K P = d K s the transformed Hamltonan K = K(, P, t) Hamlton s prncple reures t t ( p H(, p, t) ) 0 and ( ) δ = t t δ P K(, P, t) = 0 General Transformaton ( ) ( ) t t δ p H(, p, t) = 0 and δ P t K(, P, t) = 0 t Two types of transformatons are possble P K = λ( p ) Scale transformaton P K + = p Canoncal transformaton Both satsfy Hamlton s prncple Combned, we fnd P K + = λ( p Extended Canoncal ) transformaton

3 Scale Transformaton We can always change the scale of (or unt we use to measure) coordnates and momenta P ν p = µ = To satsfy Hamlton s prncple, we can defne K( P,,) t = µν H( p,,) t P K = µν ( p ) Scale transformaton Ths s trval We now concentrate on Canoncal transformatons Canoncal Transformaton P K + = p Hamlton s prncple t t t δ ( P K ) = δ p H δ[ F] 0 t = = t t Satsfed f δp = δ = δp = δ = 0 at t and t F can be any functon of p,, P, and t It defnes a canoncal transformaton Call t the generatng functon of the transformaton or generator 3

4 Smple Example [] Try a generatng functon: F = P P Canoncal transformaton generated by F s P K + = K + ( ) P + P = p = P p K = H OK, that was too smple = Identty transformaton Let s push ths one step further P K + = p Smple Example [] Let s try ths one: F = f (,,, t) P P n f are arbtrary functons of n and t f f P K + = K + ( f ) P + P j + P = p t = f (,,, t) n f j p = Pj f K = H + P t We can do all what we could do before P K + = p j All pont transformatons of generalzed coordnates are covered Must nvert these n euatons to get P 4

5 Arbtrarty Generatng functon F a canoncal transformaton Opposte mappng s not unue There are many possble Fs for each transformaton e.g. add an arbtrary functon of tme g(t) to F () P dg t K + P K + + Does not affect the acton ntegral dg() t K K + F s arbtrary up to any functon of tme only So s the Hamltonan Just modfes the Hamltonan wthout affectng physcs Fndng the Generator P K + = p Let s look for a generatng functon Suppose KPt (,,) = Hpt (,,) for smplcty p P = Easest way to satsfy ths would be F = F(, ) = p = P Trval example: p = F (, ) = P = In the Hamltonan formalsm, you can freely swap the coordnates and the momenta 5

6 Type- Generator F = F(, ) s not very general It does not allow t-dependent transformaton Fx ths by extendng to F F(,, t) (,, ) t p = = Call t Type- (,, ) t P = Ths affects the Hamltonan = + + = p P + K t K = H + t F P K + = p Harmonc Oscllator Consder a -dmensonal harmonc oscllator p k k Hp (, ) = + = ( p+ ) ω m m m Sum of suares Can we make them sne and cosne? Suppose Trck s to fnd f(p) so that the transformaton s canoncal How? p = f( P)cos { f( P) } = f( P) sn K = H = s cyclc P s constant m 6

7 Harmonc Oscllator Let s try a Type- generator F(,, t) p = P = Express p as a functon of and f( P) p = f( P)cos = sn p = cot m ω Integrate wth F = cot P = = sn We are gettng somewhere Harmonc Oscllator p = = cot = = We need to turn H(, p) nto K(, P) Solve the above euatons for and p P = sn p = P cos Now work out the Hamltonan K = H = ( p + m ω ) = ωp m P sn Thngs don t get much smpler than ths 7

8 Harmonc Oscllator K = ωp= E Solvng the problem s trval E P = const = ω Fnally K = = P ω = ω t + α p= P cos= mecos( ωt+ α) P E = sn sn( ωt α) = + Phase Space Oscllator moves n the p- and P- phase spaces p P me E One cycle draws the same area n both spaces ω The area swept by a cyclc system n the phase space s nvarant E ω π E Wll come back to ths n Lecture 3 π 8

9 Other Types of Generators Type- generator F = F(,, t) s stll not so general Just try to fnd a generator for = P = p We need generatng functons of dfferent set of ndependent varables In fact, we may have 4 basc types of them F(,, t) F (, P, t) F ( p,, t) F ( p, P, t) 3 4 We can derve them usng the now-famlar rule.e. we can add any / nsde the acton ntegral Type- Generator In the last lecture, I used t t ( ) δ p H(, p, t) = 0 to convert Swtch the defnton of canoncal transformatons P K + = p P K + = p p P K H = + + To satsfy ths F F (, P, t) = p F = p = P t t ( ) δ p H(, p, t) = 0 = K = H + t F 9

10 Type- Generator If we go back to the orgnal defnton of generatng functon P K + = p F = F (,, ) P t P Trval case: F = P = p P We push the same dea to defne the other types = K = H + t p = P = Identty transformaton F Four Basc Generators Generator F(,, t) F (,, ) P t P F (,, ) 3 p t + p F (,, ) 4 p P t + p P Dervatves Trval Case = p p = P = F = P = p = P = F = P 3 3 = P = p 4 4 = = p P F F = p 3 = pp 4 P = p = P = p = = p P = 0

11 Four Basc Generators The 4 types of generators are almost euvalent It may look as f F s specal, but t sn t P K + = p P K + = p 3 P K + = p 4 P K + = p There s no reason to consder any of these 4 defntons to be more fundamental than the others We arbtrarly chose the frst form (whch happens to be the Lagrangan form) to wrte the generatng functons n the table Four Basc Generators Some canoncal transformatons cannot be generated by all 4 types e.g. dentty transf. s generated only by F or F 3 Ths does not present a fundamental problem One can always swap coordnate and momentum = p P = One can always change sgn by scale transformaton P =± p =± These transformatons make the 4 types practcally euvalent

12 One More Example -dm system wth Try P = p Let s use Type- F F (, P, t) p H = + = = p Step : Express p wth and P Step : Integrate wth to get F = Plog Step 3: Dfferentate to get = log Now we have a canoncal transformaton = P assumng > 0 P p = E = e V = One More Example P F = Plog = e p = = Pe Now rewrte the Hamltonan p P + H = + = e = E constant Euaton of moton: P = ( P + ) e = E P = Et+ C P + C + = e = = Et + Ct+ E E

13 Summary Canoncal transformatons P K + = p Hamltonan formalsm s nvarant under canoncal + scale transformatons Generatng functons defne canoncal transformatons Four basc types of generatng functons F(,, t) F (, P, t) F ( p,, t) F ( p, P, t) 3 4 They are all practcally euvalent Used t to smplfy a harmonc oscllator Invarance of phase space area 3