
finsler anek putt

ganitat, visheshtah vibhedak bhoomitimadhye, finsler manifold haah eka vibhedak manifold aahe jithe pratyek sparshikechya jagevar (bahudha asamamiteey) minkowski karyatmak pradan kele jaate, jyamule konatyahi gulgulit vakrareshechi lambi ashee paribhashit karata yete
finsler manifold heey remanion manifoldspeksha adhik samanya asatat kaaran sparshika nikashanna aateel utpadananni prerit karanyachi aavashyakta nasate.
pratyek finsler manifold eka aantarik ardh-parimani jaaga banate jevha done bindunmadhil antar tyanna jodana vakranchi anant lambi mhanoon paribhashit kele jaate.
 paul finslerchya navavarun finsler manifolds asey naav denyat aale, jyane tyachya prabandhat yaa bhoomiticha abhyas keela.
vyakhya.
finsler manifold heey finsler metricks ektritpane eka bhinn karnyayogya manifold aahe, jeey sparshika bandalvar paribhashit kelele eka satatche nakaratmak nasalele karya aahe jenekarun pratyek bindusathi,
dusya shabdant, pratyek sparshikechya jagevaril eka asamamiteey manak aahe. finsler metrik dekhil adhik tantotantpane gulgulit asane aavashyak aaheh
tyanantar upadataratmakata svayamsiddhichi jaaga khaleel tivra bahirmukh sthitine ghetali jaau shakateh
yethe etcha hessian haah samamiteey dwillingi prakar aahe.
etcha moolbhut tensor mhanoonahi olakhala jaato. jarr asel tarr, majboot bahirmukhata mhanje kadak asamanatesah upaductivity. jarr teey jordar bahirmukh asel tarr pratyek sparshikechya jagevar teey minkowskiche manak aahe.
finsler metrik ulat karata yenyajoge aahe, jarr, yavyatirikt,
prativarti finsler metrik pratyek sparshikechya jagevar eka manak (nehamichya arthane) paribhashit karate.
udaharane.
randers anek patinni vadhatat.
sootra _ 3 heey remanian manifold aanee 'bii' heey 'emm' varr eka vibhedak swaroop asoo dya.
jethe sootra 5 heey sootra 6 chey inverse matricks aahe aanee einstein notation vaparale jaate. mug
m" varr randers metrik paribhashit karate aanee sootra _ 8 heey randers manifold aahe, jeey naun-reversible finsler manifoldche eka vishesh prakaran aahe.
gulgulit ardh-parimani jaaga.
("m", "d") heey eka ardhamapi asoo dya jenekarun "m" dekhil eka bhinn karnyayogya anekpat asel aanee "d" khaleel arthane "m" chya bhinn sanrachaneshi susangat aselah
mug ekhadi vyakti finsler karya "f": "tm" → [0, φ] dware paribhashit karoo shakate.
jethe 'γ' mhanje 'm' madheel konatihi vakrata jyamadhye 'γ' (0) = 'x' aanee 'γ' (0) = v aahe. asha prakare midavalele finsler karya 'f' heey 'm' chya pratyek sparshikechya jagevaril asamamiteey (samanyatah naun-minkowski) manakanparyant maryadit asate. mool ardh-parimaniche prerit aantarik metrik yethoon punarprapt kele jaau shakate
aanee kharey tarr konatehi finsler karya "f": t "m" → [0, φ] yaa sutradvare "m" varr aantarik ardhamapi "d" l "paribhashit karate.
bhooganiti.
f" lambichya eksandhatemule
bhinn karnyayogya vakr "γ": ["a", "b"] "m" madheel "m" sakaratmak dishene punarparivartan antargat aparivartaniy aahe. sthir gatee vakr "γ" haah finslerchya anek patancha bhooganitiy asato, jarr tyache purese lahan vibhag "γ" asateel.".["c," "d" "] hii" "m" "madhye" "γ" "(" "c" ") teey" "γ" "(" "d" ") paryantchi lambi-qamee karnari aahet". samaan ritine, jarr teey urja karyatmakatesathi sthir asel tarr 'γ' haah bhooganitiy aahe.
yaa arthane qee tyache karyatmak vyutpann nishchit antim bindunsah bhinn karnyayogya vakranmadhye nahise hote aanee.
finslerchya anek patanvar pramanit favarani rachana.
urja karyatmak "e" ["γ"] sathee yular-lagrange samikaran sthanik nirdeshankat ("x" 1,..., "x" n ", v" 1,..., v" n) chaa t "m" mhanoon
jethe "k" = 1,..., aine" aanee "jee" ij heey moolbhut tensorche samanvay pratinidhitva aahe, jyachi vyakhya ashee aaheh
v" ωt "x" m "chya sandarbhat" f "2 (" x "," v ") chee majboot bahirmukhata gruheet dharalyas, matricks" g "" ij "(" x "," v ") ulat aahe aanee tyacha ulat" g "" ij "(" x "," v ") dware darshavila jaato. jarr aanee keval jarr tyachi sparshika vakrata hii tii "emm" aanee setminswar chikat vector kshetra "etch" chee avibhajya vakrata asel; {0} sthanik patalivar paribhashit kelee asel tarr ("emm", "eff") chee bhooganitiy aahe.
jethe sthanik favarani gunank "g" i dware dile jaatat
t "m" aanee setminsvaril vector kshetra "h"; {0} heey "jh" = "v" aanee ["v", "h"] = "h" chey samadhan karate, jethe "j" aanee "v" heey canonical endomorphism aahet aanee t "m" aanee setminsvaril canonical vector kshetra aahe; {0}. mhanoon, vyakhyenusar, 'etch' hii 'emm' varr favarani aahe. favarani "etch" ubhya prakshepanadvare tantunchya bandalvaril araikhik jodani paribhashit karate.
remanian prakaranashi sadrushya sadhatana, eka aavrutti aahe
hehresmann vakrata aanee nonlinear covarient vyutpattichya drushtine samanya favarani sanrachanesathi ("m", "h") jacoby samikaran.
bhaugolikshastrache vaishishtya aanee gunadharmanche kamitkamikaran.
hopph-rino prameyanusar ("emm", "eff") varr nehmich lambi qamee karnare vakr (kiman puresha chhotya parisarat) astitvat asatat. lambi qamee karananya vakranna nehmich sakaratmak reparametries kele jaau shakate aanee konatyahi geodesicne "eae" ["γ"] sathee euler-lagrange samikaran purn kele pahije. f" 2 chee majboot bahirmukhata gruheet dharalyas avibhajya vakranchya advitiyatenusar konatyahi ("" x "," v ") ωt" m "aanee setminsasathi" γ "(0) = v saha eka adviteey kamal bhooganitiy" γ "astitvat aahe.
jarr "f" 2 jordar bahirmukh asel, tarr bhooganiti "γ": [0 ", b"] "m" heey jawalchya vakranmadhye lambi-qamee karnare aahet, joou pahila bindu "γ" ("s") paryant "γ" sobat "γ" (0) shhee sanyugmit hoto aanee "t"> "s" sathee remanian prakaranapramane "γ" chya jawal "γ" (0) teey "γ" ("t") paryant nehmich lahan vakr asatat.