Mechanics Physics 151

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1 Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9)

2 Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ p ( ) δ p ( ) 0 No srcly needed, bu adds flexbly o he defnon of he acon negral Ths connecs o: Canoncal Transformaons Prncple of Leas Acon ( ) p qd 0 Go no hs a b

3 Canoncal Transformaon Goal: To fnd ransformaons ( q,, qn, p,, pn, ) P P( q,, qn, p,, pn, ) ha sasfy Hamlon s equaon of moon q H dp p H dq K s he ransformed Hamlonan Hamlon s prncple requres ( ) δ pq H q p d K K dp P K d K(, P, ) (,, ) 0 and δ ( P (,, )) K P d 0

4 General Transformaon ( ) (,, ) 0 and δ ( P (,, )) K P d 0 δ pq H q p d Two ypes of ransformaons are possble P K λ( p q ) P K + pq d Boh sasfy Hamlon s prncple Combned, we fnd P K + λ( pq ) d Scale ransformaon Canoncal ransformaon Exended Canoncal ransformaon

5 Scale Transformaon We can always change he scale of (or un we use o measure) coordnaes and momena P ν p µ q To sasfy Hamlon s prncple, we can defne K( P,, ) µν H( p, q, ) Ths s rval P K µν ( p q ) Scale ransformaon We now concenrae on Canoncal ransformaons

6 Canoncal Transformaon P K + pq d Hamlon s prncple ( ) δ P K d δ pq d δ[ F] d 0 Sasfed f δp δq δp δ 0 a and F can be any funcon of p, q, P, and I defnes a canoncal ransformaon Call he generang funcon of he ransformaon or generaor

7 Smple Example [] F qp P Canoncal ransformaon generaed by F s Try a generang funcon: P K + K + ( q ) P + Pq pq d K q P p H OK, ha was oo smple Ideny ransformaon Le s push hs one sep furher P K + pq d

8 Smple Example [] Le s ry hs one: F f ( q,, q, ) P P n f are arbrary funcons of q q n and f f P K + K + ( f ) P + P q j + P pq d q f ( q,, q, ) p n f q j P j f K H + P We can do all wha we could do before P K + pq d j All pon ransformaons of generalzed coordnaes are covered Mus nver hese n equaons o ge P

9 Arbrary Generang funcon F a canoncal ransformaon Oppose mappng s no unque There are many possble Fs for each ransformaon e.g. add an arbrary funcon of me g() o F () P dg K + P K + + Does no affec d d d K K + dg() d F s arbrary up o any funcon of me only So s he Hamlonan he acon negral Jus modfes he Hamlonan whou affecng physcs

10 Fndng he Generaor Le s look for a generang funcon Suppose KP (,,) Hqp (,,) for smplcy p q P d Eases way o sasfy hs would be F F( q, ) p P q Trval example: p P Fq (, ) q P K + pq d q In he Hamlonan formalsm, you can freely swap he coordnaes and he momena

11 Type- Generaor F F( q, ) s no very general I does no allow -dependen ransformaon Fx hs by exendng o F F( q,, ) p (,, ) q q P Call Type- (,, ) q Ths affecs he Hamlonan F F F q + + pq P + K d q K H + F P K + pq d

12 Harmonc Oscllaor Consder a -dmensonal harmonc oscllaor p kq H ( q, p) + p + m q m m Sum of squares Can we make hem sne and cosne? Suppose Trck s o fnd f(p) so ha he ransformaon s canoncal How? p f( P)cos { f( P) } ( ω ) q f( P) sn mω k ω m K H s cyclc P s consan m

13 Harmonc Oscllaor Le s ry a Type- generaor F( q,, ) p P q Express p as a funcon of q and f( P) p f( P)cos q sn p mω qco mω m ω qq Inegrae wh q F co P mω q sn We are geng somewhere

14 Harmonc Oscllaor p mω qco q We need o urn H(q, p) no K(, P) Solve he above equaons for q and p P q sn p Pmω cos mω Now work ou he Hamlonan ( K H p + m ω q ) ωp m P mω q sn Thngs don ge much smpler han hs

15 Harmonc Oscllaor K ωp E Solvng he problem s rval E P cons ω Fnally K P ω ω + α p Pmω cos mecos( ω+ α) P E q sn sn( ω α) mω mω +

16 Phase Space Oscllaor moves n he p-q and P- phase spaces me p q E mω π π E One cycle draws he same area n boh spaces ω The area swep by a cyclc sysem n he phase space s nvaran P E ω Wll come back o hs n Lecure 3

17 Oher Types of Generaors Type- generaor F F q s sll no so general q P p (,, ) Jus ry o fnd a generaor for We need generang funcons of dfferen se of ndependen varables In fac, we may have 4 basc ypes of hem F( q,, ) F ( q, P, ) F ( p,, ) F ( p, P, ) 3 4 We can derve hem usng he now-famlar rule.e. we can add any /d nsde he acon negral

18 Type- Generaor In he las lecure, I used F q p o conver ( ) δ pq H q p d Swch he defnon of canoncal ransformaons P K + pq P K + pq d d pq P K H d + + To sasfy hs F F ( q, P, ) (,, ) 0 q P p ( ) δ pq H q p d K (,, ) 0 F H +

19 Type- Generaor If we go back o he orgnal defnon of generang P K + pq d (,, ) F F q P P p q P funcon Trval case: p F qp P q Ideny ransformaon We push he same dea o defne he oher ypes K H + F

20 Four Basc Generaors Generaor F( q,, ) F (,, ) q P P F (,, ) 3 p + qp F (,, ) 4 p P + qp P Dervaves Trval Case p P F q q P p q q q P 3 3 P p F qp 4 4 p P F F 3 p pp 4 P P P p q q p q p p q

21 Four Basc Generaors The 4 ypes of generaors are almos equvalen I may look as f F s specal, bu sn P K + pq d P K + pq d 3 P K + pq d 4 P K + pq d There s no reason o consder any of hese 4 defnons o be more fundamenal han he ohers We arbrarly chose he frs form (whch happens o be he Lagrangan form) o wre he generang funcons n he able

22 Four Basc Generaors Some canoncal ransformaons canno be generaed by all 4 ypes e.g. deny ransf. s generaed only by F or F 3 Ths does no presen a fundamenal problem One can always swap coordnae and momenum p P q One can always change sgn by scale ransformaon q P ± p ± These ransformaons make he 4 ypes praccally equvalen

23 One More Example -dm sysem wh Try P pq Le s use Type- F F ( q, P, ) H p + q p q P E V q Sep : Express p wh q and P Sep : Inegrae wh q o ge p P q F Plog q assumng q > 0 Sep 3: Dfferenae o ge Now we have a canoncal ransformaon log q q e

24 One More Example P F Plog q q e p Pe q Now rewre he Hamlonan p P + H + e E q Equaon of moon: P ( P + ) e E P E+ C consan P + C + q e E + C+ E E

25 Summary P K + pq Hamlonan formalsm s d nvaran under canoncal + scale ransformaons Generang funcons defne canoncal ransformaons Four basc ypes of generang funcons Canoncal ransformaons F( q,, ) F ( q, P, ) F ( p,, ) F ( p, P, ) 3 4 They are all praccally equvalen Used o smplfy a harmonc oscllaor Invarance of phase space area

Mechanics Physics 151

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