
aakshan (physics)

bhauthikashaasthrathil, oru bhauthika samvidaanam kaalakramena engane maariyenn vivarikkunna oru scaler alavaanu pravartthanam (athinte "chalanathmakatha"). sisttathinte chalana samavaakyangal nischalamaaya pravartthana thathwathiloode uruthirinjedukkaannaal pravartthanam pradhaanamaan.
oru kanika sthiramaaya vegathayil (ekeekritha rekheeya chalanam) neengunnathinte lalithamaaya saahacharyathil, pravartthanam ennathu kanikayude vegathayaanu, athu neengunna dooratthinte eratti, athinte pathayil kootticherkkunnu; thulyamaayi, pravartthanam kanikayude gathikorjjathinte eratti, athinu aaa alavilulla oorjjam. kooduthal sankeernamaaya samvidaanangalkkaayi, atharam alavukalellaam samyojippichirikkunnu.
kooduthal aupachaarikamaayi, sisttathinte paathaye (paatha allengil charithram ennum vilikkunnu) athinte vaadamaayi edukkukayum athinte falamaayi oru yadhaartha samkhya undaakukayum cheiyunna oru ganithashaasthra pravarthanamaanu pravartthanam. saadhaaranayaayi, pravartthanam vyathyastha paathakalkk vyathyastha moolyangal edukkunnu. pravarthanatthinu oorjjam × samayam allengil momentam × neelam ennivayude alavukalundu, athinte sii yoonit juul-second aanu (plank sthiraangam "echu" pole).
aamukham.
eathoru" bhauthika samvidaanathinteyum chalanatthinte deferential samavaakyangal thulyamaaya oru avibhaajya samavaakyamaayi punarnirmikkaan kazhiyumennu haamilttan thathwam prasthaavikkunnu. athinaal, chalanathmaka maathrkakal roopappeduthunnathinu rund vyathyastha sameepanangalund.
ithu orotta kanikayude classical mechanicsinu mathramalla, vaidyuthakaanthika, guruthwaakarshana mekhalakal polulla classical feeldukalkkum baadhakamaan. haamilttante thathwam kwaandam mekkaaniksilekkum kwaandam feeld sidhaanthathilekkum vyaapippichittund-prathyekichu kwaandam mekkaaniksinte paathu integral formulation eee ashayam upayogikkunnu-avide oru physical sisttam kramarahithamayi saadhyamaaya paathakalilonn pinthudarunnu, oro paathayudeyum probability aamplityoodinte ghattam paathayude pravarthanatthaal nirnnayikkappedunnu.
deferential samavaakyathinte pariharam.
anubhavaparamaya niyamangal palappozhum deferential samavakyangalaayi prakadippikkunnu, ithu sthaanam, momentam thudangiya bhauthika alavukal samayam, sthalam allengil athinte saamaanyavalkkaranam ennivaykkoppam thudarchayaayi engane maarunnuvenn vivarikkunnu. saahacharyathinte prarambha, athirthi vyavasthakal kanakkiledukkumbol, eee anubhava samavaakyangalude "pariharam" sisttathinte swabhaavathe vivarikkunna onno athiladhikamo functionukalaanu, avaye "chalana samavaakyangal" ennu vilikkunnu.
aakshan integralinte minimisation.
"atharam chalana samavaakyangal kandethunnathinulla oru badal sameepanathinte bhagamaanu "pravartthanam". oru bhauthika samvidaanam yadhaarthathil pinthudarunna paatha "pravartthanam minimise cheythirikkunnu" allengil kooduthal pothuve, nischalamaanennu classical mechanics anumaanikkunnu. mattoru vidhathil paranjaal, pravartthanam oru vyathiyana thathwathe thripthippeduthunnu nischalamaaya pravartthanatthinte thathwam (chuvadeyum kaanuka). pravartthanatthe nirvachikkunnath oru avibhaajyaghadakamaanu, koodaathe oru sisttathinte chalanatthinte classical samavaakyangal aaa avibhaajyaghadakathinte moolyam kurachukondu uruthirinjedukkaam.
eee lalithamaaya thathwam bhauthikashaasthrathekkurishu aazhathilulla ulkkaazhcha nalkunnu, ithu aadhunika saidhaanthika bhauthikashaasthrathile oru pradhana aashayamaanu.
charithram.
"aashayathinte vikasana samayat "aakshan" ippol kaalaharanapetta niravadhi vazhikalil nirvachikkapettu.
ganithashaasthraparamaaya nirvachanam.
vyathiyaanangalude calculus upayogichu ganithashaasthra bhashayil prakadippicha oru bhauthika samvidaanathinte parinamam (i.iii.oru samsthaanathu ninnu mattonnilekku sisttam yadhaarthathil engane purogamikkunnu ennathu pravartthanatthinte oru nischala pooyintumaayi (saadhaaranayaayi, kuranjathu) poruthappedunnu.
pravartthanam" ennathinte niravadhi vyathyastha nirvachanangal bhauthikashaasthrathil saadhaarana upayogathilundu. pravartthanam saadhaaranayaayi kaalakramena oru avibhaajya gadakamaanu. ennirunnaalum, pravartthanam feeldukalumaayi bandhappettirikkumbol, athu speshyal veriyabilukalilum samyojippikkaam. chila sandarbhangalil, bhauthika samvidaanam pinthudarunna paathayiloode pravartthanam samyojippikkappedunnu.
sisttathinte vikasanathinte prarambha samayathinum avasana samayathinum idayil sisttathinte paathayiloode edukkunna pravartthanatthe saadhaaranayaayi kaalakramena oru avibhaajya gadakamaayi prathinidheekarikkunnu
soothravaakyam _ 1
ivide "l" enna integraantine lagranchian ennu vilikkunnu. aakshan integral nannayi nirvachikkanamenkil, paatha samayathilum sthalathum parimithappedutthanam.
pravarthanatthinu [oorjjam] × [samayam] enna alavukalundu, athinte sii yoonit juul-secondaanu, ithu koneeya momentatthinte yoonittin samaanamaan.
classical physixil pravartthanam.
classical physixil, "aakshan" enna padatthinu niravadhi ardhangalundu.
pravartthanam (pravartthanaparam).
ettavum saadhaaranayaayi, eee padam oru functional formula _ 2-naayi upayogikkunnu, ithu samayathinte oru function (feeldukalkku) sthalavum inputtaayi edukkukayum oru scaler nalkukayum cheiyunnu. classical mechanicsil, inputt function ennathu "t" 1 num "t" 2 num edayilulla sisttathinte parinaamamaan, ivide q saamaanyavalkkaricha kordinettukale prathinidheekarikkunnu. rund thavanakalkkidayilulla oru inputt parinaamathinu lagranchian "l"-nte avibhaajyaghadakamaayi aakshan formula _ 3 nirvachikkappedunnu
formula _ 4
parinaamatthinte anthima bindukkal sthiramaayi nirvachikkappedukayum soothravaakyam 5, soothravaakyam 6 enningane nirvachikkappedukayum cheiyunna haamilttan thathwamanusarichu, yadhaartha parinamam qtrue ("t") ennathu 3 enna pravartthana soothravaakyam nischalamaaya (oru minimum, maximum allengil oru saddle pooyint) oru parinaamamaan. eee thathwam lagranchian mechanicsile chalana samavakyangalil kalaashikkunnu.
hrasvamaaya pravartthanam (pravartthanaparam).
churukkia pravartthanavum oru pravarthanamaanu. ithu saadhaaranayaayi formula _ 8 aayi soochippikkunnu. ivide inputt function ennathu bhauthika samvidaanathe pinthudarunna "paathu" aanu, samayakramam kanakkiledukkathe. udaaharanathinu, oru graha bhramanapathatthinte paatha deerghavrithaakrithiyilullathana, oru ekeekritha guruthwaakarshana mandalatthile oru kanikayude paatha oru parabolayan; rund saahacharyangalilum, kanika ethra vegathil kadannupokunnu ennathine aashrayichalla paatha. saamaanyavalkkaricha kordinettukalile oru paathayiludaneelamulla saamaanyavalkkaricha momentayude avibhaajyaghadakamaayi churukkia aakshan formula _ 8 nirvachikkappedunnu
soothravaakyam _ 10
ithu krithyamaayi ucharikkappedunnu,
soothravaakyam _ 11
moupertyoosinte thathwamanusarichu, yadhaartha paatha ennathu _ 8 enna hrasvamaaya pravartthana soothravaakyam nischalamaaya oru paathayaan.
haamilttante pradhana pravartthanam.
haamilttaninte pradhana function formula _ 13 ennathu prarambha samaya formula _ 15 umm prarambha end pooyint formula _ 16 umm nischayikkunnathiloode aakshan functional formula _ 2 ll ninnu labikkum, athesamayam uyarnna samayaparidhi formula _ 17 umm randaamathe end pooyint formula _ 18 umm vyathyaasappedaan anuvadikkunnu. classical mekkaaniksinte roopeekaranamaaya haamilttan-jacoby samavaakyathe haamilttaninte pradhana pravartthanam thripthippeduthunnu. shroadinger samavakyavumaayi saamyamullathinaal, haamilttan-jacoby samavaakyam kwaandam mechanicsumaayi ettavum nerittulla bandham nalkunnu.
haamilttante savisheshamaaya pravartthanam.
motham oorjjam "e" samrakshikkappedumbol, veriyabilukalude additteev verthirikkal upayogichu haamilttan-jacoby samavaakyam pariharikkan kazhiyum
formula _ 19
ivide samaya-swatantra function "w" ("q" 1 ", q" 2,) aanu..., qn") "" haamilttaninte swabhaavaparamaaya pravartthanam "" ennu vilikkunnu ". eee functionte bhauthika praadhaanyam athinte motham samaya derivative edukkunnathiloode manassilakkunnu.
soothravaakyam _ 20
ithu nalkaan samyojippikkaam
soothravaakyam _ 21
ithu churungiya pravartthanam maathramaan.
haamilttan-jacoby samavaakyangalude mattu pariharangal.
haamilttan-jacoby samavaakyangal palappozhum additteev separability upayogichaanu pariharikkappedunnathu; chila sandarbhangalil, parihaaratthinte vyakthigatha padangal, udha.jii.ska" ("qk") enna padatthe "aakshan" ennum vilikkunnu.
oru saamaanyavalkkaricha ekopanathinte pravartthanam.
karangunnatho aandolanamo aaya chalanathinu anusrithamaayi ghatta idathile oru adacha paathaykk chuttum orotta saamaanyavalkkaricha aakkam samyojippichu nirvachikkunna aakshan angle kordinettukalile orotta variable "jeke" aanith
soothravaakyam _ 22
jeke" enna variable saamaanyavalkkaricha coordinate "kyuke" yude "aakshan" ennu vilikkunnu; aakshan-angle kordinettukalkku keezhil kooduthal poornnamaayum vivarichirikkunna kaaranangalal "jeke" yumayi bandhapetta canonical variable athinte "angle" "dablyuke" aanu. samyojanam orotta variable "qk" nuu mukalilaanu, athinaal, mukalilulla samkshiptha aakshan integralile integrated dott prodakttil ninnu vyathyasthamaan. qk" adachitta paathaykk chuttum vyathyasappettirikkunnathu "" kk "variable" "sk" ("qk") leeii maattathinu thulyamaanu. thaalpparyamulla niravadhi bhauthika samvidaanangalkku, jeke onnukil oru sthiraankamaanu allengil valare pathukke vyathyaasappedunnu; athinaal, perterbation kanakkukoottalukalilum adiabatic inevarantukal nirnnayikkunnathilum "jeke" enna variable palappozhum upayogikkunnu.
oru hamiltonian ozhukkinte pravartthanam.
tatological vann phoom kaanuka.
euler-lagranch samavaakyangal.
lagranchian mechanicsil, cheria aswasthathakalkku keezhil aakshan integral nischalamaayirikkendathindi aavasyakatha vyathiyaanangalude calculus upayogichu nedaavunna oru koottam deferential samavaakyangalkk (euler-lagranch samavaakyangal ennu vilikkunnu) thulyamaanu.
pravartthana thathwam.
classical feeldukal.
vaidyuthakaanthikakshethram allengil guruthwaakarshanakshethram polulla feeldukalude chalana samavaakyangal labhikkunnathin pravartthana thathwam vipuleekarikkaam.
ainsteen samavaakyam "ainsteen-hilbert pravartthanam" oru vyathiyana tatwatal parimithappedutthiyirikku.
oru guruthwaakarshana mandalatthile oru vasthuvinte paatha (bahirakasha samayathile paatha) pravartthana thathwam upayogichu kandethaam. swathanthramaayi veezhunna oru pindatthinu, eee paatha oru geodesic aanu.
samrakshana niyamangal.
oru bhauthika saahacharyathile samamithikalude prathyaaghaathangal pravartthana thathwathil ninnu uruthirinja euler-lagranch samavakyangalkkoppam pravartthana thathwathil kandethanakum. oru udaaharanam notharinte sidhaanthamaanu, athu oru bhauthika saahacharyathile oro thudarchayaaya samamitikkum oru samrakshana niyamavumaayi (neremarichu) poruthappedunnuvennu prasthaavikkunnu. eee aazhathilulla bandhathinu pravartthana thathwam anumaanikkendathundu.
kwaandam mechanics kwaandam feeld sidhaanthavum.
kwaandam mechanicsil, sisttam nischalamaaya oru paatha pinthudarunnilla, pakshe sisttathinte swabhaavam anuvadaneeyamaaya ellah paathakaleyum avayude pravartthanatthinte moolyatheyum aasrayichirikkunnu. vividha paathakalumayi bandhapetta pravartthanam paathu integral kanakkakkaan upayogikkunnu, ithu vividha phalangalude probability aamplityoodukal nalkunnu.
classical mechanicsil nyoottante niyamangalumayi thulyamaanenkilum, aakshan thathwam saamaanyavalkkaranathin anuyojyamaanu, koodaathe aadhunika bhauthikashaasthrathil oru pradhana punk vahikkunnu. vaasthavathil, eee thathwam bhauthikashaasthrathile mahathaaya saamaanyavalkkaranangalilonu. kwaandam mechanicsil, prathyekichu richard fainmaante paathu integral formulationil, kwaandam aamplityoodukalude vinashakaramaaya edapedalil ninnu ithu uyarnnuvarunnu.
maiks vellinte samavaakyangal nischalamaaya pravartthanatthinte vyavasthakalaayum uruthirinjekkaam.
eaka aapekshika kanika.
aapekshika phalangal praadhaanyamarhikkumbol, sariyaaya samaya soothravaakyam _ 23 anusarichu paaraameettar cheytha oru lokarekha "sii" sanjarikkunna pindamulla "emm" enna oru pooyint kanikayude pravartthanam
formula _ 24
pakaram, kanikayude coordinate samayam "t" kondu kanika paaraameettar cheyyappedukayum coordinate samayam "t" 1 muthal "t" 2 vare aayirikkukayum cheythaal, pravartthanam maarunnu.
soothravaakyam _ 25
lagranchian evideyaanu
soothravaakyam _ 26
aadhunika vipuleekaranangal.
pravartthana thathwam kooduthal saamaanyavalkkarikkam. udaaharanathinu, pravartthanam oru avibhaajyaghadakamaayirikkanani, kaaranam praadeshikamallaatha pravartthanangal saadhyamaan. noonn commutative jyamiti polulla chila savisheshathakal kanakkiledukkumbol konfigareshan space oru functional space polum aayirikkanamennilla. ennirunnaalum, eee ganitha vipuleekaranangalkk oru bhauthika adisthaanam pareekshanaathmakamaayi sthaapikkappedendathundu.
uravidangalum kooduthal vayana.
oru vyaakhyaana grandhasoochikaykku edwin eph kaanuka. mattu karyangalkkoppam thaazhepparayunna pusthakangal pattikappedutthunna tailor

