
molecular hamiltonian

paramaanu, paramaanu, optical bhouthika shaastram mariyu quantum chemistrylo, molecular hamiltonian anedi oka anuvulooni electronlu mariyu kendrakaala saktini soochinche hamiltonian operator. eee operator mariyu anubandha shroadinger sameekaranam ushna vaahakata, nirdishta vaedi, vidyut vaahakata, optical mariyu ayaskanta lakshanaalu mariyu praticharya vanti anuvula mariyu anuvula samuuhaala yokka lakshanaalanu ganinchadaaniki ganana rasayana shaastram mariyu bhouthika saastramlo pradhana paatra poeshistaayi.
anuvu yokka praathamika bhagalu nucleus, vaati paramaanu sankhyalu, "z", mariyu electronlu, ivi pratikuula praathamika chaarj, − "e" kaligi untaayi. vaati paraspara charya "z" + "q" yokka anhu aavesam istundi, ikkada, "n" thoo electronla sankhyaku samanam. electronlu mariyu kendrakaalu chaalaa manchi ujjaayimpu, point chaarjeelu mariyu point maas. molecular hamiltonian anedi aneka padaala mottamha dheeni pradhana padaalu electronla yokka kinetic energylu mariyu rendu rakaala chaarj cheyabadina kanaala madhya coolumb (electrostatic) paraspara charyalu. electronlu mariyu kendrakaala yokka gathi saktulanu matrame kaligi unna hamiltonian, mariyu vaati madhya coolumb paraspara charyalanu coolumb hamiltonian anie pilustaaru. daani nundi aneka chinna padaalu laevu, veetilo ekkuva bhaagam electronic mariyu nuclear spin kaaranamgaa unnaayi.
saadhaaranamgaa coolumb hamiltonianku sambandhinchina samaya-swatantra shroadinger sameekaranam yokka parishkaram anuvu yokka chaalaa lakshanaalanu anchanaa vestundani bhaavinchinappatikii, daani aakaram (trimiteeya nirmaanam) thoo sahaa, puurti coolumb hamiltonian aadhaaramgaa lekkalu chaalaa arudu. pradhana kaaranam daani shroadinger sameekaranaanni parishkarinchadam chaalaa kashtam. anuvartanaalu hydrogen anuvu vanti chinna vyavasthalaku parimitam cheyabaddaayi.
paramaanu taranga kriyala yokka daadaapu anni gananalu janminchina mariyu oppenheimer roopondinchina coolumb hamiltonian yokka vibhajanapai aadharapadi untaayi. nuclear kinetic energy padaalu coolumb hamiltonian nundi tolaginchabaddaayi mariyu migilina hamiltonian electronla hamiltonianga matrame pariganinchabadutundi. sthira kendrakaalu samasyaloki electronlu quantum yaantrika maargamlo kadile vidyut sambhaavyata yokka generatorluga matrame pravesistaayi. eee chatramlo molecular hamiltonian clamped nucleus hamiltonian anie pilavabadedigaa saralikrutam cheyabadindi, dheenini electronic hamiltonian anie kooda pilustaaru, idhi electronic coordinatela vidhulapai matrame panichestundi.
clamped nucleus hamiltonian yokka shroadinger sameekaranam nucleus yokka tagina sankhyalo nakshatraraashula kosam parishkarinchabadina tarvaata, tagina aigen viluvanu (saadhaaranamgaa atyalpam) anhu akshaamsaala yokka vidhiga chudavachu, idhi sambhaavya shakti uparitalaaniki daariteestundi. aacharanaatmaka gananalalo uparitalam saadhaaranamgaa konni vislaeshanhaatmaka vidhula paramgaa amarchabadutundi. puttukatho vachhe-oppenheimer ujjaayimpu yokka rendava dasalo electronlapy aadhaarapade puurti coolumb hamiltonian yokka bhaagam sambhaavya shakti uparitalam dwara bharti cheyabadutundi. idhi mottam paramaanu hamiltonian nuu anhu akshaamsaalapai matrame panichese maroka hamiltonian gaa maarustundi. vividha electronic sthithula saktulu daggaragaa unnappudu sambhavinche janma-oppenheimer ujjaayimpu vichchinnam vishayamlo-porugu sambhaavya shakti uparitalaalu avasaram, dheeni gurinchi marinta vivaraala kosam eee kathanaanni chudandi.
nuclear motion shroadinger sameekaranaanni space-fixed (prayogasaala) framelo parishkarinchavacchu, conny appudu anuvaada mariyu bhramana (bahya) saktulu lekkinchabadavu. (antargata) anhu kampanaalu matrame samasyaloki pravesistaayi. inca, triotomic kante pedda anuvula kosam, harmonic ujjaayimpunu pravesapettadam chaalaa saadhaaranam, idhi sambhaavya shakti uparitalaanni anhu sthaanabhramsham yokka chaturasraakaara vidhiga anchanaa vestundi. idhi harmonic nuclear motion hamiltonian nuu istundi. harmonic ujjaayimpugaa, manam hamiltonian nuu jodinchani oka dimensional harmonic ausilator hamiltonianla mottamgaa maarchavacchu. shroadinger sameekaranam yokka khachitamaina parishkaranni anumatinche konni vyavasthalalo oka dimensional harmonic ausilator okati.
pratyamnaayamgaa, anhu kadalika (rovibrational) shroadinger sameekaranaanni anuvutho tirige mariyu anuvadinche pratyeka chatramlo (eccart chatram) parishkarinchavacchu. eee shareera-sthira chatraniki sambandhinchi roopondinchabadina hamiltonian kendrakaala bhramanam, anuvaadam mariyu kampanaaniki kaaranamavutundi. 1968loe watson eee hamiltonianku oka mukhyamaina saralikarananu pravesapettinappati nundi, dheenini tarachugaa watson yokka nuclear motion hamiltonian anie pilustaaru, ayithe dheenini eccart hamiltonian anie kooda pilustaaru.
coolumb hamiltonian.
aneka pariseelaneeya vastuvula beejaganita roopam-i.ii.gamaninchadagina parimaanaalanu soochinche hermician operatorlu-eee krindi parimaana niyamaala dwara pondutaruh
saastriiyamgaa anuvulooni electronlu mariyu kendrakaalu "p" 2/(2 "m") roopamlo gathi saktini kaligi untaayi mariyu
coolumb paraspara charyala dwara sankarshana chendutaayi, ivi dhooram "r" ij kii vilomaanupaatamlo untaayi."
i" mariyu "j" kanaala madhya.
suutram _ 3
eee vyakteekaranalo r "i" anedi edainaa kanaaniki (electron ledhaa nucleus) samanvaya vahakanni suuchistumdi, conny ikkada nundi manam anhu samanvayaanni suuchimchadaaniki pedda aksharam r nii mariyu vyavastha yokka electronlaku chinna aksharam r nii reserve chestaamu. coordinatelanu antarikshamlo ekkadainaa kendrikrutamai unna edainaa cartician frameku sambandhinchi vyakteekarinchavachani teesukovachhu, endukante dhooram, antargata utpatti kaavadamtho, frame yokka bhramanamlo asthiramgaa untundi mariyu, difference vector yokka pramaanam kaavadamtho, frame yokka anuvaadam kinda dhooram kooda asthiramgaa untundi.
hamilton roopamlo shaastreeya saktini lekkinchadam dwara okaru molecular hamilton operatornu pondutaaru, dheenini tarachugaa coolumb hamiltonian anie pilustaaru. eee hamiltonian aidu padaala mottam. vaaru
ikkada "m" i anedi kendrakam dravyarashi "i", "z" i "anedi kendrakam yokka paramaanu sankhya" i ", mariyu" m "e anedi electron dravyarashi. kana "i" yokka laplace operator suutram _ 9. kinetic energy operator oka antargata utpatti kabatti, cartician frame yokka bhramanamlo idhi asthiramgaa untundi, dheeniki sambandhinchi "x" i "," y "i" mariyu "z" i "vyakteekarinchabadataayi.
chinna padaalu.
1920laloe chaalaa spectroscopic aadharalu coolumb hamiltonian konni padaalanu kolpoyindani spashtam chesaayi. mukhyamgaa bhaarii anuvulanu kaligi unna anuvulaku, eee padaalu, gathi mariyu coolumb saktula kante chaalaa chinnavi ayinappatikee, avi lekkinchalenivi. eee spectroscopic pariseelanalu electronlu mariyu kendrakaalaku kotta sthaayi swechhanu pravesapettadaaniki daariteesaayi, avi spin. eee anubhavika bhavanaku paal dirac oka-kanam schroedinger sameekaranam yokka saapekshamgaa saraina (lorentz covarient) roopaanni pravesapettinappudu saiddhaantika aadharam icchaaru. oka kanam yokka spin mariyu praadaesika kadalika spin-arbit coupling dwara sankarshana chendutayani diarock sameekaranam anchanaa vestundi. saaruupyataloo spin-maroka-kakshya kalayika pravesapettabadindi. kanaala spin ayaskanta dvidhruvam yokka konni lakshanaalanu kaligi undatam anedi spin-spin couplingku daariteesindi. shaastreeya pratiroopam laeni tadupari padaalu fermi-contact padam (nucleusto parimita parimaana kendrakampai electronic saandrata yokka paraspara charya), mariyu nuclear kwadrupol coupling (electronla kaaranamgaa vidyut kshetram yokka pravanathatho nuclear kwadrupol yokka paraspara charya). chivaraku praamaanika namuunaa anchanaa vesina samaanatvaanni ullanghinche padaanni tappaka prastaavinchaali. idhi chaalaa chinna paraspara charya ayinappatikee, idhi kairal anuvulalo ananciomerlaku vibhina saktulanu istundi kabatti idhi shaastreeya saahityamlo taginanta drushtini aakarshinchindi.
eee vyaasam yokka migilina bhaagam spin nibandhanalanu vismaristundi mariyu coolumb hamiltonian yokka aigen value (time-independent shroadinger) sameekaranam yokka parishkaranni pariseelistundi.
coolumb hamiltonian yokka shroadinger sameekaranam.
coolumb hamiltonian anedi sajatiya pradaesamloe anuvu yokka dravyarashi (com) kadalika kendram kaaranamgaa nirantara varnapataanni kaligi untundi. shaastreeya yantrasaastramlo binduvula dravyarashi vyavastha yokka com kadalikanu vaeru cheyadam sulabham. saastriiyamgaa com yokka kadalika itara kadalikala nundi vidadeeyabadadu. com oke vidhamgaa kadulutundi (i.ii.sthiramaina vegamtho) anni kanaala dravyaraasula mottam "m" kuu samaanamaina dravyarashi kaligina binduvu kanamgaa unnatlugaa antariksham gundaa.
quantum mechanicslo oka swechhaa kanaaniki sthithi vidhiga samatala taranga vidhi untundi, idhi baga nirvachinchabadina momentum yokka chadarapu-samagrata laeni vidhi. gathi shakti
eee kanaaniki edainaa saanukuula viluvanu teesukovachhu. heisenberg anischiti suutraaniki anugunamgaa, com yokka sthaanam pratichota oke vidhamgaa saadhyapadutundi.
dravyarashi kendram yokka coordinate vector x nii vyavastha yokka degrees off freedamlo muudugaa parichayam cheyadam dwara mariyu oka (ekapaksha) kanaala coordinate vektarnu tolaginchadam dwara, tadwara degrees off friedam sankhya alaage untundi, oka leanier transformation dwara oka kotta coordinates samitini pondavacchu. eee akshaamsaalu "anni" kanaala (kendrakaalu "mariyu" electronlu ") paatha akshaamsaala sarala kalayikalu. golusu niyamanni vartimpajeyadam dwara okaru choopinchavachhu
suutram _ 10
suutram _ 11 yokka modati padam com kadalika yokka gathi shakti, dheenini suutram _ 12 x paie aadharapadi lenanduna vidigaa chikitsa cheyavachu. ippude cheppinatlugaa, daani igenstatelu plain taramgaalu. sambhaavya "v" (t) kotta akshaamsaalalo vyakteekarinchabadina coolumb padaalanu kaligi untundi. suutram _ 12 yokka modati padam kinetic energy operator yokka saadhaarana roopaanni kaligi untundi. rendava padaanni maas polarization padam antaaru. anuvaadaparamgaa asthiramaina hamiltonian suutram _ 12 sweeya-anubandhamgaa mariyu diguva nundi sarihaddugaa choopabadutundi. antey, daani atyalpa aigen viluva vaastavamainadi mariyu parimitamainadi. suutram _ 12 anedi okela unde kanaala kramaparivartanallo tappanisarigaa asthiramgaa unnappatikee (suutram _ 11 mariyu kaam gatisakti asthiramgaa unnanduna), daani asthirata spashtamgaa kanipinchadu.
suutram _ 12 yokka aneka vaastava paramaanu anuvartanaalu laevu; ayithe, prarambha anuvartanam kosam hydrogen anuvupai praathamika panini chudandi. paramaanu taranga kriyala gananalalo ekkuva bhaagam electronic
puttukatho vachhe oppenheimer anchanaa yokka modati dasalo talette clamped nucleus hamiltoniantho samasya parishkarinchabadutundi.
reifer chudandi. coolumb hamiltonian yokka ganita lakshanaala gurinchi samagra charcha kosam. coolumb hamiltonian lakshanaala nundi matrame anuvu yokka bhavana (baga nirvachinchabadina jyaamititoe electronlu mariyu kendrakaala sthiramaina vyavasthagaa) "praadhaanyata" pondagalara anie kooda eee kaagitamlo charchinchabadindi.
clamped nucleus hamiltonian.
clamped nucleus hamiltonian nucleus yokka sthiravidyuth kshetramloni electronla saktini vivaristundi, ikkada nucleus oka jadatva chatraniki sambandhinchi sthiramgaa unnatlu bhaavistaaru.
electronic hamiltonian yokka roopam
suutram _ 18
electronlu mariyu kendrakaala akshaamsaalu kendrakaalato kadile frameku sambandhinchi vyakteekarinchabadataayi, tadwara kendrakaalu eee frameku sambandhinchi visraantigaa untaayi. frame space-fixed frameku samaantaramgaa untundi. idhi oka jadatva chatram endukante kendrakaalu bahya saktulu ledhaa tarkla dwara vegavantam cheyabadavani bhaavistaaru. frame yokka muulam ekapakshamgaa untundi, idhi saadhaaranamgaa kendra kendrakampai ledhaa dravyarashi yokka anhu kendramlo unchabadutundi. konnisaarlu kendrakaalu "space-fixed framelo visraantigaa" unnayani perkonabadindi. eee prakatana kendrakaalanu shaastreeya kanaalugaa chuustaarani suuchistumdi, endukante quantum yaantrika kanaalu visraantigaa undalevu. (dheeni ardham idhi ekakaalamlo sunna momentum mariyu baga nirvachinchabadina sthaanaanni kaligi undhi, idhi heisenberg yokka anischiti suutraaniki viruddhamgaa undhi).
anhu sthaanaalu sthiraamkaalu kabatti, electronic kinetic energy operator edainaa anhu vector meeda anuvaadam kinda asthiramgaa untundi. coolumb sambhaavyata, difference vectorsnu batti, alaage asthiramgaa untundi. anhu kakshyala vivaranalo mariyu anhu kakshyalapai samakalanam yokka gananalo eee asthirata anuvulooni anni anuvulanu antariksha-sthira frameku samaantaramgaa vaati swantha sthaanikeekarinchina framelatho sannaddham cheyadam dwara upayoginchabadutundi.
puttukatho-oppenheimer ujjaayimpupai vyaasamlo vivarinchinatlugaa, suutram _ 19 yokka shroadinger sameekaranam yokka tagina sankhyalo parishkaralu sambhaavya shakti uparitalam (pays) suutram _ 20kii daariteestaayi. daani akshaamsaalapai "v" yokka kriyaatmaka aadharapadatam anedi
suutram _ 21
kosam
suutram _ 22
ikkada t mariyu s ekapaksha vahakalu mariyu δφ anedi anantamaina konam,
δφ "δφ2. pespai eee asthirata paristhiti saadhaaranamgaa jarige rii yokka thedaalu mariyu koonaala paramgaa pespai vyakteekarinchabadinappudu svayamchaalakamgaa neraverutundi.
harmonic nuclear motion hamiltonian.
eee vyaasam yokka migilina bhaagamlo anuvu paakshikamgaa kathinamgaa undani manamu bhaavistaamu. boow ujjaayimpu yokka rendava dasalo anhu gathi shakti "t" n tirigi pravesapettabadindi mariyu hamiltoniantho shroadinger sameekaranam
suutram _ 23
pariganinchabadutundi. daani draavanamloo gurtinchaalanukuntunnadi dravyarashi yokka anhu kendram yokka kadalika (3 degreela swechcha), anuvu yokka mottam bhramanam (3 degreela swechcha) mariyu anhu kampanaalu. saadhaaranamgaa, icchina anhu gathi saktitoe idhi saadhyam kaadu, endukante idhi 3 "n" − 6 antargata degreela swechcha nundi 6 bahya degreela swechhanu (mottam anuvaadam mariyu bhramanam) spashtamgaa vaeru cheyadu. vaastavaaniki, ikkada kinetic energy operator space-fixed (sf) frameku sambandhinchi nirvachinchabadindi. manam sf frame yokka muulaanni dravyarashi yokka anhu kendraaniki taraliste, appudu, golusu niyamanni vartimpajeyadam dwara, anhu dravyarashi dhruveekarana nibandhanalu kanipistaayi. eee nibandhanalanu puurtigaa vismarinchadam aacharam mariyu memu eee aacharaanni anusaristaamu.
vibajananu saadhinchadaaniki manam antargata mariyu bahya akshaamsaalanu vaeru cheyali, dheeniki eccart akshaamsaala dwara santrupti chendadaaniki paristhitulanu pravesapettaadu. maas-waited cartician coordinatelaloo harmonic vislaeshanha nundi eee paristhitulu sahajamgaa elaa talettutayo chuupistaamu.
kinetic energy vyakteekarananu saralikrutam cheyadaaniki memu maas-waited displacement coordinatelanu parichayam chestaamu
suutram _ 24.
appati nundi
suutram _ 25
kinetic energy operator avutundi,
suutram _ 26
manam samathoulya jyaamiti chuttoo "v" yokka taylor vistarana chesthe,
suutram _ 27
mariyu moodu padaala tarvaata kattirinchandi (harmonic ujjaayimpu anie pilavabadedi), manam "v" nii moodava padamtho matrame vivarinchavacchu. v" 0 anae padaanni saktilo grahinchavacchu (shakti yokka kotta sunnani istundi). samathoulya paristhiti kaaranamgaa rendava padam adrushyam avtondi. migilina padam "v" yokka hessian maatrika f nuu kaligi untundi, idhi samaruupata kaligi untundi mariyu sthiramaina muulakaalatoe orthogonal 3 "n" × 3 "n" maatrikatoe vikarnam kaavachuh
suutram _ 28
f yokka aaru aigen vektarlu (q yokka chivari aaru varusalu) aigen value sunna (sunna-frequency modlu) kaligi unnayani rotation mariyu translation kinda "v" yokka invarians nundi chuupavachchu. avi "bahya sthalam" nii vistarinchi untaayi. q yokka modati varusalu-vaati ground state looni anuvula kosam-sunna kaani aigen value unna aigen vectors; avi antargata coordinatelu mariyu a (3 "n"-6)-dimensional subspace kosam orthonormal besisnu yerparustaayi.
nuclear configuration space r3 "n", "antargata space". sunna-frequency igenvectorlu sunna kaani frequency yokka igenvectorlaku artogonalga untaayi. eee orthogonalitylu vaastavaaniki eccart paristhitulu anie chuupavachchu. antargata akshaamsaalalo vyakteekarinchabadina gathi shakti antargata (kampana) gathi shakti.
saadhaarana akshaamsaala parichayamtho
suutram _ 29
anhu kadalika kosam hamiltonian yokka kampana (antargata) bhaagam "harmonic ujjaayimpu" loe maarutundi."
suutram _ 30
sambandhita shroadinger sameekaranam sulabhamgaa parishkarinchabadutundi, idhi oka dimensional harmonic ausilatorla kosam 3 "n"-6 sameekaranalugaa karakamga maarutundi. anhu chalana shroadinger sameekaranam yokka eee sumaru parishkaaramlo pradhana prayatnam "v" yokka hessian f yokka ganana mariyu daani vikarnata.
hessian f yokka khachitamaina gananala kosam algorithamlu andubaatuloki vacchina rojula nundi (1980luu-1990luu), 3 "n" maas-waited cartician coordinatelaloo vivarinchina anhu chalana samasyaku eee ujjaayimpu, quantum chemistrylo pramaanikamgaa maarindi. harmonic ujjaayimpu kakunda, anuvu yokka bahya (bhramana mariyu anuvaada) kadalikalanu parigananaloki teesukoni marinta lopam undhi. avi konnisaarlu "watsons hamiltonian" anie piluvabade rovibrational hamiltonianlo lekkinchabadataayi.
watson yokka anhu chalana hamiltonian.
antargata (kampana) kadalikalato jatacheyabadina bahya (anuvaadam mariyu bhramanam) kadalikala kosam hamiltonian pondadaaniki, eee samayamlo shaastreeya yantrasaastraaniki tirigi ravadam mariyu kendrakaala yokka eee kadalikalaku anugunamgaa shaastreeya gathi saktini roopondinchadam sarvasaadhaaranam. saastriiyamgaa, anuvaada-dravyarashi-kadalika kendraanni itara kadalikala nundi vaeru cheyadam sulabham. ayithe, kampana kadalika nundi bhramanaanni vaeru cheyadam marinta kashtam mariyu puurtigaa saadhyam kaadu. eee roow-vibrational vibajananu 1935loe eccart modatagaa ippudu eccart paristhitulu anie piluvabade vaatini vidhinchadam dwara saadhinchaadu. anuvutho tirige frame (eccart frame) loe samasya vivarinchabadindi, anduvalla idhi jadatvarahita frame, kalpita saktulatoe anubandhinchabadina saktuluh kendrapagami mariyu coriolis shakti gathi saktilo kanipistaayi.
saadhaaranamgaa, shaastreeya gatisakti "t" anedi vakrarekha akshaamsaalatoo anubandhinchabadina metrik tensor g = ("g" ij) nii s = ("s" i) dwara nirvachistundi.
suutram _ 31
quantisation dasha anedi eee shaastreeya kinetic energini quantum mechanical operatorga marchadam. shaastreeya roopaaniki upayoginchina adhe (saadhaaraneekarinchina, vakrarekha) akshaamsaalalo laplace-beltramy operator nuu vrayadam dwara podolskini anusarinchadam saadhaaranam. eee operator yokka sameekaranaaniki metrik tensor g mariyu daani determinant yokka inverse avasaram. laplace-beltramy operator suutram _ 32 dwara guninchadam valana avasaramaina quantum mechanical kinetic energy operator labhistundi. eee vantakaanni unit metrik kaligina cartician coordinatelaku vartimpajesinappudu, quantization niyamaala anuvartanam dwara adhe kinetic energy pondabadutundi.
1936loe willson mariyu howard anhu chalana hamiltonian nuu pondaaru, vaaru eee vidhananni anusarinchaaru, mariyu 1940loe darling mariyu dennison dwara marinta suddhi cheyabadindi. 1968 varaku idhi pramaanamgaa undipoyindi, appudu watson metrik tensor yokka nirnayakam ayina utpannaala dwara prayaaninchadam dwara daanini teevramgaa saralikrutam cheyagaligadu. watson pondina roow-vibrational hamiltonian nuu istaamu, dheenini tarachugaa watson hamiltonian anie pilustaaru. manam ilaa chese mundu manam prastaavinchaali
cartician roopamlo laplace operator nundi praarambhinchadam, samanvaya parivartanala anuvartanam mariyu golusu niyamanni upayoginchadam dwara kooda eee hamiltonian yokka utpannam saadhyamavutundi.
watson hamiltonian, "enn" kendrakaala yokka anni kadalikalanu vivaristuu,
suutram _ 33
modati padam maas turm yokka kendram
suutram _ 34
rendava padam anedi drudamaina rotter yokka gathi saktini pooli unde bhramana padam. ikkada
suutram _ 35 anedi boddy-fixed "rigid rotter angular momentum operator" yokka α bhaagam,
euler koonaala paramgaa daani vyakteekarana kosam eee kathanaanni chudandi. operator suutram _ 36 anedi telisina operator yokka oka bhaagam
kampana koneeya vaega operator" gaa (idhi koneeya vaega sanchaara sambandhaalanu "santrupti parachakapoyinappatiki"),
suutram _ 37
coriolis coupling constant" thoh
suutram _ 38
ikkada levi-poura chihnam undhi. suutram _ 35 looni chaturasra padaalu kendrapagami padaalu, suutram _ 35 mariyu suutram _ 41 looni dwipaarshwaalu coriolis padaalu. parimaanaalu "q" s, iγ anedi paina pravesapettina saadhaarana akshaamsaala bhagalu. pratyamnaayamgaa, willson yokka gf paddhatini upayoginchadam dwara saadhaarana akshaamsaalanu pondavacchu. 3x3 cemetric matricks formula _ 42 nuu "effective reciprocal inertia tensor" antaaru. anni "q" luu sunna (drudamaina anuvu) ayithe eccart frame pradhana akshaala frame thoo samanamga untundi (drudamaina rotter chudandi) mariyu suutram _ 42 vikarnamgaa untundi, vikarnampai sthirathvam yokka samathoulya paraspara kshanaalu untaayi. anni "q" luu sunna ayithe, anuvaadam mariyu drudamaina bhramana yokka gathi saktulu matrame manugada saagistaayi.
sambhaavya padam "yuu" anedi "watson padam":
suutram _ 44
samarthavanthamaina paraspara jadatva tensor yokka jadaku anulomaanupaatamlo untundi.
watson hamiltonianlo nalgava padam anedi saadhaarana akshaamsaalalo vyakteekarinchabadina anuvula (kendrakaalu) kampanaalatoo sambandham unna gathi shakti, idhi paina perkonna vidhamgaa, anhu sthaanabhramsham paramgaa ivvabadutundi.
suutram _ 45
chivaraku "v" anedi antargata akshaamsaalanu matrame batti nirvachanam prakaram vistarinchani sambhaavya shakti. harmonic ujjaayimpulo idhi roopaanni teesukuntundi
suutram _ 46

