
symplectic manifold

vibhedak jyaamiti mein, ganit kaa ack vishay, ack symplectic manifold ack chikna manifold, sootra _ 1 haye, joe ack bund gair-janitra vibhedak 2-roop sootra _ 2 seey lais haye, jise symplectic roop kahaa jaataa haye. symplectic manifolds key adhyayan koo symplectic jyaamiti yaa symplectic topology kahaa jaataa haye. shaastriya yantriki aur vishleshanatmak yantriki key amoort sutrikaran mein swabhavik roop seey samroopata vaale manifold swabhavik roop seey manifold key samroop bandalon key roop mein utpann hotey hain. udaaharan key liye, shaastriya yantriki key hamiltonian sutrikaran mein, joe kshetra key liye pramukh preranaon mein seey ack pradaan kartaa haye, ack pranali key sabhi sambhaavit vinyason key sett koo ack manifold key roop mein model kiya jaataa haye, aur yeh manifold kaa coatangent bundle pranali key charan sthaan kaa varnan kartaa haye.
prerana.
shaastriya yantriki seey symplectic manifold utpann hotey hain; vishesh roop seey, way ack bund pranali key charan sthaan kaa ack saamaanyikaran hain. usee tarah hamilton sameekaran ack pranali key samay vikas koo vibhedak samikaranon key ack samooh seey praapt karane kii anumati dete hain, symplectic roop koo ack vector kshetra praapt karane kii anumati deni chaahiye joe ack hamiltonian function "aich" key vibhedak "dh" seey pranali key pravaah kaa varnan kartaa haye. isaliye hamein sparshrekha manifold "tm" seey coatangent manifold "t" ≤ "m", yaa samkaksh, ack tatva key liye ack raikhik maanchitra kii aavashyakta hoti haye. ω" koo ack khand koo darshaane key liye, yeh aavashyakta kii "ω" gair-apakshayi hoe, yeh sunishchit kartaa haye kii pratyek antar "dh" key liye ack advitiya sambandhit vector kshetra "vh" haye joe kii haye. choonki koee chaahata haye kii hamiltonian pravaah rekhaaon key saath sthir rahey, isaliye ack honah chaahiye, jiska arth haye kii "ω" baari-baari seey haye aur isaliye ack 2-roop haye. anta mein, ack aavashyakta yeh haye kii "ω" pravaah rekhaaon key tahat naheen badalana chaahiye, i.ee. kii "ω" kaa jhooth vyutpann "vh" key saath gayab hoe jaataa haye. carton key sootra koo lagu karate hue, yeh raashi (yahaan sootra _ 3 aantarik utpaad haye): 
taaki, vibhinn chikane karyon key liye iss tark koo doharaane parr sootra _ 5 iss tarah seey kii sambandhit sootra _ 6 pratyek bindu parr sparshrekha sthaan koo failata haye, humm dekhate hain kii sootra _ 6 key pravaah key saath vikshepan jhooth vyutpann kii aavashyakta manmaane dhang seey chikane sootra _ 5 key anuroop haye, joe iss aavashyakta key barabar haye kii "ω" koo bund kiya jaanaa chaahiye.
paribhasha.
ack chikna manifold sootra _ 1 parr ack symplectic roop ack bund gair-apakshayi antar 2-roop sootra _ 2 haye. yahaan, gair-apakshayi kaa arth haye kii pratyek bindu sootra _ 11 key liye, sootra _ 2 dwara paribhaashit sparshrekha sthaan sootra _ 12 parr tirchha-samamit jodee gair-apakshayi haye. iska matlab haye, yadi koee sootra 14 maujood haye jaise kii sabhi sootra 16 key liye sootra 15, tou sootra 17. kyonki visham aayamon mein, tirchha-samamit metrices hamesha ekvachan hotey hain, isaliye sootra 2 key gair-janitra honey kii aavashyakta kaa tatparya haye kii sootra 1 kaa ack summ ayam haye. bund sthiti kaa arth haye kii sootra _ 2 kaa baahari vyutpann gayab hoe jaataa haye. ack symplectic manifold ack jodee sootra _ 21 haye jahan sootra _ 1 ack chikna manifold haye aur sootra _ 2 ack symplectic roop haye. sootra _ 1 koo ack symplectic roop dene koo sootra _ 1 koo ack symplectic sanrachana dene key roop mein sandarbhit kiya jaataa haye.
udaaharan.
symplectic vector space.
sootra _ 26 koo sootra _ 27 key liye ack aadhar honey dein humm iss aadhar parr apane symplectic roop "ω" koo nimnanusar paribhaashit karate hainh
iss mamle mein symplectic roop ack saral dwighaat roop mein cumm hoe jaataa haye. yadi "inn" "mein" aine "×" aine "pehchaan matricks koo darshaata haye tou iss vargaatmak roop kaa matricks, ω, block matricks dwara diyaa jaataa haih
cuttangant bundles.
sootra _ 30 koo ayam sootra _ 31 kaa ack chikna manifold honey dein. phir coatangent bundle sootra _ 32 key kull sthaan mein ack praakritik symplectic roop hotaa haye, jise pointcare too-form yaa canonical symplectic roop kahaa jaataa haye.
yahaan sootra 34 sootra 30 parr koee bhee sthaaniya nirdeshank hain aur sootra 36 cottangent vactor sootra 37 key sambandh mein fibruz nirdeshank hain. oopari aur nichale soochkaank koo alag karane kaa bindu ack metrick tensor vaale manifold key mamle seey sanchaalit hotaa haye, jaisa kii remanian manifold key mamle mein hotaa haye. oopari aur nichale soochkaank samanvaya frame key parivartan key tahat vipareet aur saha-parivartansheel roop seey parivartit hotey hain. vakyansh "cottangent vactor key sambandh mein fibruz nirdeshank" kaa arth yeh haye kii moment sootra _ 38 wegg sootra _ 39 key liye "souldard" hain. souldering iss vichaar kii ack abhivyakti haye kii wegg aur gati collinear hain, jisamen donon ack hii disha mein chalate hain, aur ack paimaane key kaarak seey bhinn hotey hain.
kaee gunaa badh jaataa haye.
ack kahler manifold ack susangat samakalan jatil sanrachana seey lais ack symplectic manifold haye. way jatil manifolds kaa ack vishesh varg banaate hain. udaharanon kaa ack badaa varg jatil beejganitiya jyaamiti seey aataa haye. kissy bhee chikani jatil prakshepya vividhata sootra _ 40 mein ack symplectic roop hotaa haye joe prakshepya antariksh sootra _ 41 parr fubini-adhyayan roop kaa pratibandh haye.
lagbhag jatil kaee gunaa.
ack sootra _ 42-sangat lagbhag jatil sanrachana key saath remanian manifold koo lagbhag-jatil manifold kahaa jaataa haye. way saamaanyikrit karate hain kii way ekikrit honey kii aavashyakta naheen haye. yaani, way aavashyak roop seey ack jatil sanrachana seey utpann naheen hotey hain.
lagrangian aur anya upmaanivak.
ack samroop bahugunit sootra _ 21 key upa-aavrit kii kaee praakritik jyaamitiya dhaaranaaen hainh
ack pramukh udaaharan yeh haye kii utpaad mein ack symplectomorphism kaa graph lagrangian haye. unake pratichhedan chikane manifolds dwara dhaaran naheen kiye gaye kathorta gunon koo pradarshit karate hain; arnold anuman chikane mamle mein euler visheshta key bajaay ack chikane lagrangian upmanifold key swa pratichedanon kii sankhya key liye cumm seema key roop mein upmanifold kii betty sankhyaon kaa yog detaa haye.
udaaharan.
maan leejiye kii sootra _ 53 mein vaishvik nirdeshank hain jinhen sootra _ 54 label kiya gayaa haye. phir, humm sootra _ 55 koo canonical symplectic roop seey lais curr sakate hain. 
sootra _ 57 dwara diyaa gayaa ack maanak lagrangian upa-manank haye. formula _ 42 formula _ 59 parr gayab hoe jaataa haye kyonki sparshrekha vactor formula _ 60 kii kissy bhee jodee koo dekhate hue hamaare paas vaha formula _ 61 haye jise spasht karane key liye, mamle key formule _ 62 parr vichaar karein. phir, formula _ 63 aur formula _ 64. dhyaan dein kii jaba humm ise vistaarit karate hain.
donon shabdon mein hamaare paas ack sootra _ 66 kaarak haye, joe paribhasha key anusaar 0 haye.
udaaharan key liyeh cottengent bundle.
ack manifold kaa coatangent bundle sthaaniya roop seey pahale udaaharan key samaan sthaan parr model kiya gayaa haye. yeh dikhaaya jaa sakata haye kii humm inn affine symplectic roopon koo chipka sakate hain isaliye yeh bundle ack symplectic manifold banaata haye. lagrangian submanifold kaa ack cumm tuchchh udaaharan ack manifold key coatangent bundle kaa shoonya khand haye. udaaharan key liye, chalo 
taba, humm sootra _ 68 prastut curr sakate hain
jahan humm sanket sootra _ 70 koo sootra _ 71 key nirdeshank key roop mein maan rahey hain. humm usa upasamuchchay parr vichaar curr sakate hain jahan nirdeshank sootra _ 72 aur sootra _ 73 hamein shoonya khand dete hain. iss udaaharan koo smooth function formula _ 74 aur unake differential formula _ 75 key gayab honey vaale sthaan dwara paribhaashit kissy bhee manifold key liye dohraya jaa sakata haye.
udaaharan key liyeh parametric submanifold.
nirdeshank sootra _ 77 key saath vihit sthaan sootra _ 76 parr vichaar karein. sootra _ 76 kaa ack parametric upa-aavartak sootra _ 78 vaha haye jise nirdeshank sootra _ 80 dwara iss tarah seey maapdandit kiya jaataa haye kii 
yeh manifold ack lagrangian submanifold haye yadi lagrange bracket sootra _ 82 sabhi sootra _ 83 key liye gayab hoe jaataa haye. yaani, yeh lagrangian haye yadi 
sabhi sootra _ 83 key liye. ise vistaar karke dekhaa jaa sakata haye
ack lagrangian submanifold sootra _ 78 kii sthiti mein. yeh haye kii symplectic roop koo sparshrekha manifold sootra _ 88 parr gayab honah chaahiye; yaani, yeh sabhi sparshrekha vactors key liye gayab honah chaahieh
sabhi sootra key liye _ 83. sootra _ 76 parr canonical symplectic roop kaa upayog karke parinam koo saral banaaenh
aur baaki sub gayab hoe jaate hain.
jaisa kii ack symplectic manifold parr sthaaniya chart canonical roop mein lete hain, iss udaaharan seey pataa chalata haye kii lagrangian submanifold apekshaakrit aniyantrit hain. symplectic manifolds kaa vargeekaran flore homology key madhyam seey kiya jaataa haye-yeh lagrangian submanifolds key beach maanchitron key liye karyatmak kaarravaai key liye morse siddhant kaa ack anuprayog haye. bhautiki mein, kriya ack bhautik pranali key samay vikas kaa varnan karati haye; yahaan, ise brain kii gatishilata key vivaran key roop mein liya jaa sakata haye.
udaaharan key liyeh morse siddhant.
lagrangian upa-prakateekaran kaa ack aur upayogi varg morse siddhant mein hotaa haye. ack morse function sootra koo diye jaane parr aur ack chhote seey paryaapt sootra key liye koee bhee ack lagrangian upa-manifold kaa nirmaan curr sakata haye joe kii gayab hoe rahey locus sootra dwara diyaa gayaa haye.
vishesh lagrangian upa-manifolds.
kahler manifolds (yaa kailabi-yau manifolds) key mamle mein humm sootra _ 98 parr ack holmorphic aine-form key roop mein ack vikalp sootra _ 97 banaa sakate hain, jahaan sootra _ 99 vaastavik bhaag haye aur sootra _ 100 kaalpanik haye. ack lagrangian submanifold sootra _ 78 koo vishesh kahaa jaataa haye yadi uparokt lagrangian sthiti key alawa pratibandh sootra _ 100 seey sootra _ 78 gayab hoe rhaa haye. dusare shabdon mein, sootra 78 parr pratibandhit vaastavik bhaag sootra _ 99 sootra _ 78 parr aayatan roop kii or leey jaataa haye. nimnalikhit udaharanon koo vishesh lagrangian upa-manifold key roop mein jaanaa jaataa haye,
siz anuman darpan samroopata mein vishesh lagrangian upa-prakateekaran key adhyayan seey sambandhit haye; dekhein. 
thomas-yau anuman bhavishyavani kartaa haye kii lagrangian key hamiltonian isotopy vargon mein kailabi-yau manifold parr ack vishesh lagrangian upa-manifold kaa astitva manifold kii fukaaya shreni parr sthirata kii sthiti key sambandh mein sthirata key barabar haye.
lagrangian fiber.
ack symplectic manifold "aim" kaa lagrangian fiber ack fiber haye jahan sabhi fiber lagrangian submanifold hotey hain. choonki "m" summ-ayami haye, isaliye humm sthaaniya nirdeshank leey sakate hain aur darbox kii pramey dwara symplectic roop "ω" koo cumm seey cumm sthaaniya roop seey likhaa jaa sakata haye, jahaan d baahari vyutpann koo darshaata haye aur ω baahari utpaad koo darshaata haye. iss roop koo dou-roop yaa vihit dou-roop kahaa jaataa haye. iss sett-upp kaa upayog karke humm sthaaniya roop seey "m" koo coatangent bundle sootra _ 107 key roop mein soch sakate hain aur laghu tantu sootra _ 108 key roop mein lagrangian tantu yeh vihit chitra haye.
lagrangian maanchitran.
l" koo ack symplectic manifold ("k", ω) kaa ack lagrangian upa-manifold honey dein joe ack visarjan dwara diyaa jaataa haye ("i" koo lagrangian visarjan kahaa jaataa haye). key" kaa lagrangian fiber dein. yeh ack lagrangian maanchitran haye. π" δ "i" key mahatvapurna moolya samooh koo castic kahaa jaataa haye.
dou lagrangian maanchitra aur unhein lagrangian samkaksh kahaa jaataa haye yadi "σ", "τ" aur "ν" bhinn roop hain jaise kii rekhaachitra key donon taraf sahee awagaman parr diyaa gayaa haye, aur "τ" symplectic roop koo sanrakshit kartaa haye. prateekaatmak roop seh
jahan "τ" ± "ω" 2 "ω" 2 key peechhe kii or "τ" seey kheenchane koo darshaata haye.

