Geodezi chna li niya kriva na gladkomu mnogovidi golovna normal yakoyi ortogonalna do mnogovidu Geodezichna liniya ye uzagalnennyam ponyattya pryamoyi na vikrivleni neevklidovi prostori taka liniya dlya dvoh blizko roztashovanih tochok bude najkorotshoyu Geodezichnij trikutnik na sferi Geodezichni liniyi ce dugi velikih kil Zokrema geodezichnimi liniyami budut na ploshini pryama na sferi velike kolo na elipsoyidi v zagalnomu vipadku ce nezamknena kriva sho zadayetsya cherez eliptichni dovgotu i shirotu 1 u prostori Minkovskogo geodezichnoyu ye svitova liniya vilnoyi materialnoyi tochki U metrichnih prostorah ponyattya geodezichnoyi liniyi uzagalnyuyetsya ponyattyam kvazigeodezichnoyi liniyi V zagalnij teoriyi vidnosnosti Ajnshtajna po geodezichnih liniyah vikrivlenogo prostoru chasu ruhayutsya probni chastinki Rivnyannya geodezichnoyi v comu vipadku ye rivnyannyam ruhu chastinki Geodezichna na splyusnutomu elipsoyidiZmist 1 Vlastivosti krivoyi na mnogovidi 2 Funkcional dovzhini krivoyi 3 Persha variaciya 4 Druga variaciya 5 Obgovorennya formul variacij geodezichnoyi liniyi 6 Rivnyannya geodezichnoyi dlya dovilnogo parametra 7 Geodezichna liniya dvomirnoyi poverhni 8 Div takozh 9 Primitki 10 Literatura Vlastivosti krivoyi na mnogovidi Redaguvati V ohoplyuyuchomu mnogovid evklidovomu prostori rivnyannya krivoyi zadayetsya funkciyeyu radius vektora r displaystyle mathbf r tochki krivoyi vid parametra t displaystyle t krivoyi r r t displaystyle mathbf r mathbf r t Oskilki cya kriva takozh lezhit na n displaystyle n vimirnomu mnogovidi yakij zadayetsya rivnyannyam r r u 1 u 2 u n displaystyle mathbf r mathbf r u 1 u 2 u n to rivnyannya krivoyi dayetsya funkciyami koordinat mnogovidu vid parametra krivoyi t displaystyle t 1 u i u i t i 1 2 n displaystyle 1 qquad u i u i t qquad i 1 2 n Vektor krivini krivoyi ye drugoyu pohidnoyu vid radius vektora po naturalnomu parametru s displaystyle s krivoyi k d 2 r d s 2 displaystyle qquad mathbf k d 2 mathbf r over ds 2 Odinichnij vektor n displaystyle n vzdovzh vektora krivini k displaystyle mathbf k ye golovnoyu normallyu krivoyi Vektor krivini k displaystyle mathbf k mozhna rozklasti na dvi chastini paralelnu do mnogovidu j ortogonalnu do nogo k k k displaystyle qquad mathbf k mathbf k mathbf k bot Paralelna chastina krivini k displaystyle mathbf k nazivayetsya geodezichnoyu krivinoyu krivoyi Zgidno z oznachennyam dlya geodezichnoyi liniyi vona dorivnyuye nulyu Obchislimo geodezichnu krivinu 2 k d 2 r u t d s 2 d d s r i d u i d s r i d 2 u i d s 2 r i j d u i d s d u j d s d 2 u i d s 2 G j k i d u j d s d u k d s r i b i j d u i d s d u j d s displaystyle 2 qquad mathbf k d 2 mathbf r u t over ds 2 d over ds left mathbf r i du i over ds right mathbf r i d 2 u i over ds 2 mathbf r ij du i over ds du j over ds d 2 u i over ds 2 Gamma jk i du j over ds du k over ds mathbf r i mathbf b ij du i over ds du j over ds Otzhe kontravariantni koordinati geodezichnoyi krivini dorivnyuyut 3 k i d 2 u i d s 2 G j k i d u j d s d u k d s displaystyle 3 qquad k i d 2 u i over ds 2 Gamma jk i du j over ds du k over ds Funkcional dovzhini krivoyi Redaguvati Najkorotshoyu liniyeyu na mnogovidi sho spoluchaye dvi tochki mnogovida ye vidrizok geodezichnoyi liniyi Rozglyanemo variaciyu funkcionala dovzhini krivoyi parametr krivoyi t displaystyle t probigaye znachennya vid a displaystyle a do b displaystyle b 4 S a b r d t displaystyle 4 qquad S int a b dot mathbf r dt a b r 2 d t displaystyle int a b sqrt dot mathbf r 2 dt Persha variaciya Redaguvati U tochci lokalnogo ekstremumu persha variaciya dorivnyuye nulyu dlya sproshennya zapisu v nastupnih peretvorennyah ne budemo pisati mezhi integruvannya 5 d S d r 2 d t r r d r d t d d t r r d r d t d 2 r d s 2 d r d s k d r d s displaystyle 5 qquad delta S int delta sqrt dot mathbf r 2 dt int dot mathbf r over dot mathbf r cdot delta dot mathbf r dt int d over dt dot mathbf r over dot mathbf r cdot delta mathbf r dt int d 2 mathbf r over ds 2 cdot delta mathbf r ds int mathbf k cdot delta mathbf r ds V ostannij formuli variaciya tochok krivoyi d r displaystyle delta mathbf r lezhit u dotichnomu do mnogovida afinnomu prostori i mi mozhemo zapisati d r r i d u i displaystyle delta mathbf r mathbf r i delta u i Oskilki variaciyi d u i displaystyle delta u i dovilni hocha mali to dlya rivnosti nulyu ostannogo integrala v formuli 5 treba shob vektor krivini krivoyi 2 buv ortogonalnim do mnogovidu tobto geodezichna krivina 3 dorivnyuvala nulyu 6 d 2 u i d s 2 G j k i d u j d s d u k d s 0 displaystyle 6 qquad d 2 u i over ds 2 Gamma jk i du j over ds du k over ds 0 Formula 6 ye rivnyannyam geodezichnoyi liniyi diferencijnim rivnyannyam vidnosno nevidomih funkcij u i u i s displaystyle u i u i s pri zadanij metrici na mnogovidi a otzhe i zadanih simvolah Kristofelya G j k i displaystyle Gamma jk i Druga variaciya Redaguvati Povtorimo obchislennya variaciyi dovzhini krivoyi 4 ale teper budemo vrahovuvati odnochasno dodanki pershogo j drugogo poryadkiv Dlya obchislen nam znadobitsya rozklad u ryad Tejlora do chleniv drugogo poryadku vklyuchno funkciyi kvadratnogo korenya f x x displaystyle f x sqrt x f x D x x D x x 1 2 x D x 1 8 x 3 2 D x 2 displaystyle f x Delta x sqrt x Delta x approx sqrt x 1 over 2 sqrt x Delta x 1 over 8x 3 over 2 Delta x 2 Pidintegralnij viraz formuli 4 dlya provarijovanoyi krivoyi dorivnyuye r d r 2 r 2 2 r d r d r 2 displaystyle qquad sqrt dot mathbf r delta dot mathbf r 2 sqrt dot mathbf r 2 2 dot mathbf r cdot delta dot mathbf r delta dot mathbf r 2 abo rozkladayuchi v ryad z tochnistyu do chleniv drugogo poryadku 7 r d r 2 r 1 2 r 2 r d r d r 2 1 8 r 3 2 r d r 2 d r 2 r r r d r 1 2 r 3 r 2 d r 2 r d r 2 displaystyle 7 qquad sqrt dot mathbf r delta dot mathbf r 2 approx dot mathbf r 1 over 2 dot mathbf r 2 dot mathbf r cdot delta dot mathbf r delta dot mathbf r 2 1 over 8 dot mathbf r 3 2 dot mathbf r cdot delta dot mathbf r 2 delta dot mathbf r 2 approx dot mathbf r dot mathbf r over dot mathbf r delta dot mathbf r 1 over 2 dot mathbf r 3 dot mathbf r 2 delta dot mathbf r 2 dot mathbf r cdot delta dot mathbf r 2 Rozglyanemo detalnishe serednij dodanok v ostannomu virazi U nomu mi mayemo odinichnij dotichnij vektor t r r d r d s displaystyle boldsymbol tau dot mathbf r over dot mathbf r d mathbf r over ds a b t d r d t a b d t d t d r d t 0 S k d r d s displaystyle int a b boldsymbol tau cdot delta dot mathbf r dt int a b d boldsymbol tau over dt cdot delta mathbf r dt int 0 S mathbf k cdot delta mathbf r ds Variaciya d r displaystyle delta mathbf r za formuloyu Tejlora virazhayetsya cherez variaciyu d u i displaystyle delta u i koordinat na mnogovidi z tochnistyu do chleniv drugogo poryadku d r r i d u i 1 2 r i j d u i d u j displaystyle delta mathbf r approx mathbf r i delta u i 1 over 2 mathbf r ij delta u i delta u j Zbirayuchi vse dokupi znahodimo pershu j drugu variaciyi pri comu vvazhayuchi parametr krivoyi naturalnim 8 S u d u S u d S 1 2 d 2 S displaystyle 8 qquad S u delta u S u delta S 1 over 2 delta 2 S cdots 9 d S k i d u i d s displaystyle 9 qquad delta S int k i delta u i ds 10 d 2 S k G i j s r s b i j d u i d u j d t 2 t d t d s displaystyle 10 qquad delta 2 S int left mathbf k cdot Gamma ij s mathbf r s mathbf b ij delta u i delta u j delta boldsymbol tau 2 boldsymbol tau cdot delta boldsymbol tau right ds Drugu variaciyu mozhna povnistyu podati cherez variaciyi koordinat d u i d u i d t i displaystyle delta u i delta dot u i delta tau i d t d r i t i r i d t i r i j t i d u j d t i G j k i t k d u j r i b i j t i d u j displaystyle delta boldsymbol tau delta mathbf r i tau i mathbf r i delta tau i mathbf r ij tau i delta u j delta tau i Gamma jk i tau k delta u j mathbf r i mathbf b ij tau i delta u j Poznachimo D t i displaystyle qquad D tau i variaciyu odinichnogo dotichnogo vektora razom iz paralelnim perenosom na variaciyu zmishennya 11 D t i d t i G j k i t k d u j displaystyle 11 qquad D tau i delta tau i Gamma jk i tau k delta u j Todi obchislyuyemo vrahovuyuchi ortogonalnist vektoriv r i b j k displaystyle mathbf r i mathbf b jk d t 2 D t i r i b i j t i d u j 2 g i j D t i D t j b i j b k l t i t k d u j d u l displaystyle delta boldsymbol tau 2 D tau i mathbf r i b ij tau i delta u j 2 g ij D tau i D tau j mathbf b ij cdot mathbf b kl tau i tau k delta u j delta u l t d t g i j t i D t j displaystyle boldsymbol tau cdot delta boldsymbol tau g ij tau i D tau j I nareshti vrahovuyemo zv yazok tenzora Rimana cherez vektori povnoyi krivini R i j k l b i k b j l b i l b j k displaystyle R ijkl mathbf b ik cdot mathbf b jl mathbf b il cdot mathbf b jk Pidstavlyayemo obchisleni virazi v drugu variaciyu 12 d 2 S G i j s k s d u i d u j 1 2 g i j g k l t i D t k t j D t l 1 4 R i j k l t i d u j t k d u l d s displaystyle 12 qquad delta 2 S int left Gamma ij s k s delta u i delta u j 1 over 2 g ij g kl tau i wedge D tau k tau j wedge D tau l 1 over 4 R ijkl tau i wedge delta u j tau k wedge delta u l right ds De vvedeno poznachennya zovnishnogo dobutku vektoriv bivektora abo oriyentovanoyi ploshadki pobudovanoyi na dvoh vektorah a i b j a i b j a j b i displaystyle a i wedge b j a i b j a j b i Obgovorennya formul variacij geodezichnoyi liniyi Redaguvati V formulu 9 dlya pershoyi variaciyi vhodit skalyarnij dobutok geodezichnoyi krivini na variaciyu koordinati Yaksho poblizu geodezichnoyi liniyi provesti hvilyastu liniyu blizku do sinusoyidi z chastotoyu w displaystyle omega to dlya ciyeyi hvilyastoyi liniyi matimemo priblizno taku geodezichnu krivinu k i w 2 d u i displaystyle k i approx omega 2 delta u i U comu vipadku skalyarnij dobutok bude vid yemnim v evklidovomu prostori k i d u i w 2 d u i 2 displaystyle k i delta u i approx omega 2 delta u i 2 a persha variaciya 9 vidpovidno dodatnya d S gt 0 displaystyle delta S gt 0 Ce oznachaye sho hvilyasta liniya zavzhdi dovsha za geodezichnu Zvichajno v psevdoevklidovomu prostori ce ne tak oskilki kvadrat vektora mozhe buti yak dodatnim tak i vid yemnim U zagalnij teoriyi vidnosnosti tila ruhayutsya po geodezichnij ne tomu sho tak korotshe a z inshoyi prichini za interferencijnim principom Gyujgensa dlya hvil adzhe nulova persha variaciya oznachaye sho pri rusi dvoh hvil blizkimi trayektoriyami faza hvil zbigayetsya U formuli drugoyi variaciyi 10 dlya geodezichnoyi liniyi pershij dodanok u pidintegralnomu virazi peretvoryuyetsya na nul Drugij dodanok zavzhdi dodatnij yak kvadrat bivektora t i D t j displaystyle tau i wedge D tau j Tretij dodanok mozhe buti yak dodatnim tak i vid yemnim Zokrema u ploskomu prostori tenzor Rimana dorivnyuye nulyu R i j k l 0 displaystyle R ijkl 0 tomu druga variaciya zavzhdi dodatna a otzhe bud yakij vidrizok geodezichnoyi ye lokalno najkorotshoyu liniyeyu Yaksho zh tretij dodanok vid yemnij to mozhe trapitis sho mizh tochkami mozhna provesti inshu liniyu yaka bude korotshoyu za pershu Napriklad duga velikogo kola na dvovimirnij sferi ye geodezichnoyu liniyeyu yaksho taka duga korotsha za p R displaystyle pi R to vona bude najkorotshim shlyahom mizh dvoma tochkami yaksho duga dorivnyuye p R displaystyle pi R to mizh dvoma tochkami polyusami mozhna provesti bagato odnakovih za dovzhinoyu geodezichnih linij meridianiv yaksho zh dovzhina dugi velikogo kola bilsha p R displaystyle pi R to kincevi tochki mozhna spoluchiti inshoyu dugoyu blizkoyu do geodezichnoyi yaka matime menshu dovzhinu Vzagali mozhna dovesti sho na bud yakomu mnogovidi dosit korotkij vidrizok geodezichnoyi zavzhdi bude najkorotshim shlyahom na odinichnij sferi dosit korotkij oznachaye dovzhinu menshe p Rivnyannya geodezichnoyi dlya dovilnogo parametra Redaguvati Formula 6 spravedliva dlya naturalnogo parametra tobto parametra dovzhini liniyi abo dlya parametra sho proporcijnij dovzhini liniyi z odnim i tim zhe koeficiyentom proporcijnosti v usih tochkah liniyi Ale nam mozhe znadobitisya takozh i ne naturalnij parametr geodezichnoyi liniyi napriklad yaksho na dvomirnomu mnogovidi poverhni zadano koordinati x y displaystyle x y i mi shukayemo rivnyannya geodezichnoyi u formi y y x displaystyle y y x Pohidni po parametru t displaystyle t budemo poznachati krapkoyu vgori Mayemo takij zv yazok z pohidnimi po naturalnomu parametru d u i d s u i s displaystyle du i over ds dot u i over dot s d 2 u i d s 2 d d s u i s 1 s d d t u i s 1 s 2 u i s s u i displaystyle d 2 u i over ds 2 d over ds left dot u i over dot s right 1 over dot s d over dt left dot u i over dot s right 1 over dot s 2 ddot u i ddot s over dot s dot u i Pidstavlyayuchi ci pohidni v formulu 6 i pomnozhuyuchi na s 2 displaystyle dot s 2 oderzhimo 13 u i s s u i G j k i u j u k 0 displaystyle 13 qquad ddot u i ddot s over dot s dot u i Gamma jk i dot u j dot u k 0 Zauvazhimo sho formula 13 ne rozv yazana vidnosno drugih pohidnih oskilki drugi pohidni koordinat vhodyat v s displaystyle ddot s Geodezichna liniya dvomirnoyi poverhni Redaguvati Viberemo na poverhni zadanoyi rivnyannyam z z x y displaystyle z z x y koordinati u 1 x u 2 y displaystyle u 1 x u 2 y Kvadrat elementa dovzhini zapishetsya chastinni pohidni poznachayemo indeksom vnizu z x z x z y z y displaystyle z x partial z partial x z y partial z partial y d s 2 d x 2 d y 2 d z 2 1 z x 2 d x 2 2 z x z y d x d y 1 z y 2 d y 2 displaystyle qquad ds 2 dx 2 dy 2 dz 2 1 z x 2 dx 2 2z x z y dxdy 1 z y 2 dy 2 Zvidki metrichnij tenzor g i j d i j z i z j displaystyle qquad g ij delta ij z i z j Cej tenzor maye dva vlasni vektori z i displaystyle z i i ortogonalnij do nogo a i displaystyle a i Mayemo j g i j z j z i z i j z j 2 l z i displaystyle qquad sum j g ij z j z i z i sum j z j 2 lambda z i de vlasne chislo l 1 i z i 2 1 z x 2 z y 2 displaystyle qquad lambda 1 sum i z i 2 1 z x 2 z y 2 Dlya ortogonalnogo vektora a i displaystyle a i vlasne chislo dorivnyuye odinici j g i j a j a i z i j z j a j a i displaystyle qquad sum j g ij a j a i z i sum j z j a j a i Viznachnik metrichnogo tenzora dorivnyuye dobutku cih dvoh vlasnih chisel g det g i j 1 l 1 z x 2 z y 2 displaystyle qquad g det g ij 1 cdot lambda 1 z x 2 z y 2 Teper mi mozhemo znajti simvoli Kristofelya G j k i 1 2 j g i k k g i j i g j k 1 2 j z i z k k z i z j i z j z k z i z j k displaystyle qquad Gamma jk i 1 over 2 partial j g ik partial k g ij partial i g jk 1 over 2 partial j z i z k partial k z i z j partial i z j z k z i z jk Oskilki vektor gradiyenta z i displaystyle z i bude vlasnim vektorom dlya obernenoyi matrici g i j displaystyle g ij z vlasnim chislom 1 l 1 g displaystyle 1 over lambda 1 over g to legko znahodyatsya i simvoli Kristofelya z verhnimi indeksami 14 G j k i g i s G j k s g i s z s z j k z i z j k g displaystyle 14 qquad Gamma jk i g is Gamma jk s g is z s z jk z i z jk over g Koristuyuchis shojno napisanoyu formuloyu mi mozhemo zapisati formulu 13 geodezichnoyi liniyi z parametrom t x u 1 displaystyle t x u 1 pomitivshi sho u 1 x 1 u 2 y u 1 0 u 2 y displaystyle qquad dot u 1 x 1 qquad dot u 2 y qquad ddot u 1 0 qquad ddot u 2 y Takim chinom mayemo dva rivnyannya 15 u 1 s s u 1 G j k 1 u j u k s s z x g F 0 displaystyle 15 qquad ddot u 1 ddot s over dot s dot u 1 Gamma jk 1 dot u j dot u k ddot s over dot s z x over g Phi 0 16 u 2 s s u 2 G j k 2 u j u k y s s y z y g F 0 displaystyle 16 qquad ddot u 2 ddot s over dot s dot u 2 Gamma jk 2 dot u j dot u k y ddot s over dot s y z y over g Phi 0 De vvedeno poznachennya F z j k u i u k z x x 2 z x y y z y y y 2 displaystyle qquad Phi z jk dot u i dot u k z xx 2z xy y z yy y 2 Obchislyuyuchi s s displaystyle ddot s over dot s mozhna pokazati sho rivnyannya 15 i 16 ekvivalentni mizh soboyu i ekvivalentni prostishomu rivnyannyu yake utvoryuyetsya pri vidnimanni vid 16 rivnyannya 15 domnozhenogo na pohidnu y displaystyle y y z y z x y F g 0 displaystyle qquad y z y z x y Phi over g 0 zvidki g y z x y z y F displaystyle qquad gy z x y z y Phi 17 1 z x 2 z y 2 y z x y z y z x x 2 z x y y z y y y 2 displaystyle 17 qquad 1 z x 2 z y 2 y z x y z y z xx 2z xy y z yy y 2 Div takozh Redaguvati nbsp Portal Matematika Simvoli Kristofelya Pole Yakobi Spryazheni tochkiPrimitki Redaguvati GeographicLib Geodesics on a triaxial ellipsoid geographiclib sourceforge io Procitovano 2 serpnya 2023 Literatura RedaguvatiAdler Ronald Bazin Maurice Schiffer Menahem 1975 Introduction to General Relativity vid 2nd New York McGraw Hill ISBN 978 0 07 000423 8 Div rozdil 2 Volkov Yu A 2001 Geodesic line U Hazewinkel Michiel Matematichna enciklopediya Springer ISBN 978 1 55608 010 4 Bronshtejn I N Semendyaev K A Spravochnik po matematike dlya inzhenerov i uchashihsya vtuzov M Nauka 1980 976 s il Mala girnicha enciklopediya u 3 t za red V S Bileckogo D Donbas 2004 T 1 A K 640 s ISBN 966 7804 14 3 Otrimano z https uk wikipedia org w index php title Geodezichna liniya amp oldid 40259225
