Si la ine rciyi sila sprotivu tila aktivnij sili yaka namagayetsya jogo priskoriti F m a displaystyle mathbf F m mathbf a de F displaystyle mathbf F sila inerciyi m masa tila a displaystyle mathbf a priskorennya tila yake zdijsnila zovnishnya sila Sili inerciyi realni bo voni v neinercijnij sistemi koordinat mozhut zdijsnyuvati robotu 1 Vsi realno isnuyuchi sistemi vidliku neinercijni i u vsih nih diyut realni pasivni sili inerciyi u povnij vidpovidnosti z tretim zakonom Nyutona Zmist 1 Sila inerciyi v sistemi sho obertayetsya 2 Vivedennya formul vihodyachi z klasichnoyi mehaniki 2 1 Koordinati i radius vektor 2 2 Shvidkist 2 3 Priskorennya 2 4 Sili 3 Vivid formul vihodyachi iz zagalnoyi teoriyi vidnosnosti 3 1 Chotirivimirni koordinati 3 2 Trivimirnij obraz sil inerciyi 3 3 Metrika v neinercijnij sistemi vidliku 3 4 Prodovzhennya obchislen sil inerciyi 4 Primitki 5 DzherelaSila inerciyi v sistemi sho obertayetsya RedaguvatiU sistemi sho obertayetsya dovkola osi sila inerciyi nabiraye viglyad 1 F 2 m w v m w w r m d w d t r displaystyle 1 qquad mathbf F 2m boldsymbol omega times mathbf v m boldsymbol omega times boldsymbol omega times mathbf r m frac d boldsymbol omega dt times mathbf r nbsp de w displaystyle boldsymbol omega nbsp kutova shvidkist a v shvidkist ob yekta v sistemi sho obertayetsya Pershij dodanok u formuli 1 nazivayetsya siloyu Koriolisa cya sila perpendikulyarna do shvidkosti Drugij dodanok ce vidcentrova sila a tretij vrahovuye kutove priskorennya neinercijnoyi sistemi koordinat Vivedennya formul vihodyachi z klasichnoyi mehaniki RedaguvatiKoordinati i radius vektor Redaguvati Nehaj mi mayemo inercijnu sistemu koordinat x y z displaystyle x y z nbsp yaku budemo vvazhati neruhomoyu i radius vektor vid pochatku ciyeyi sistemi koordinat do dovilnoyi tochki prostoru poznachimo velikoyu bukvoyu R displaystyle mathbf R nbsp Odnochasno budemo rozglyadati i ruhomu sistemu koordinat x 1 x 2 x 3 displaystyle x 1 x 2 x 3 nbsp pochatok koordinat yakoyi R 0 displaystyle mathbf R 0 nbsp ruhayetsya z chasom 2 R 0 R 0 t displaystyle 2 qquad mathbf R 0 mathbf R 0 t nbsp a koordinatni vektori a 1 a 2 a 3 displaystyle mathbf a 1 mathbf a 2 mathbf a 3 nbsp yakoyi utvoryuyut ortonormovanij bazis yakij yakos obertayetsya z chasom 3 a 1 a 1 t a 2 a 2 t a 3 a 3 t displaystyle 3 qquad mathbf a 1 mathbf a 1 t mathbf a 2 mathbf a 2 t mathbf a 3 mathbf a 3 t nbsp Radius vektor r displaystyle mathbf r nbsp vidnosno pochatku ruhomoyi sistemi koordinat mozhna rozklasti za cim bazisom koeficiyentami rozkladu budut koordinati ruhomoyi sistemi koordinat 4 r a 1 x 1 a 2 x 2 a 3 x 3 A x displaystyle 4 qquad mathbf r mathbf a 1 x 1 mathbf a 2 x 2 mathbf a 3 x 3 A mathbf x nbsp Ostannya rivnist ce zapis formuli 4 v matrichnij formi matricya A A t displaystyle A A t nbsp skladayetsya z koordinat bazisnih vektoriv nastupnim chinom 5 A a 1 a 2 a 3 a 1 x a 2 x a 3 x a 1 y a 2 y a 3 y a 1 z a 2 z a 3 z displaystyle 5 qquad A left mathbf a 1 mathbf a 2 mathbf a 3 right begin bmatrix a 1x amp a 2x amp a 3x a 1y amp a 2y amp a 3y a 1z amp a 2z amp a 3z end bmatrix nbsp Yak vidomo z kursu linijnoyi algebri taka matricya bude ortogonalnoyu i obernena do neyi matricya zbigayetsya z transponovanoyu Dijsno mnozhachi matricyu A displaystyle A nbsp zliva na yiyi transponovanu A T displaystyle A T nbsp oderzhimo matricyu Gramma yaka skladayetsya zi skalyarnih dobutkiv 6 A T A a 1 x a 1 y a 1 z a 2 x a 2 y a 2 z a 3 x a 3 y a 3 z a 1 x a 2 x a 3 x a 1 y a 2 y a 3 y a 1 z a 2 z a 3 z a 1 a 1 a 1 a 2 a 1 a 3 a 2 a 1 a 2 a 2 a 2 a 3 a 3 a 1 a 3 a 2 a 3 a 3 displaystyle 6 qquad A T A begin bmatrix a 1x amp a 1y amp a 1z a 2x amp a 2y amp a 2z a 3x amp a 3y amp a 3z end bmatrix begin bmatrix a 1x amp a 2x amp a 3x a 1y amp a 2y amp a 3y a 1z amp a 2z amp a 3z end bmatrix begin bmatrix mathbf a 1 cdot mathbf a 1 amp mathbf a 1 cdot mathbf a 2 amp mathbf a 1 cdot mathbf a 3 mathbf a 2 cdot mathbf a 1 amp mathbf a 2 cdot mathbf a 2 amp mathbf a 2 cdot mathbf a 3 mathbf a 3 cdot mathbf a 1 amp mathbf a 3 cdot mathbf a 2 amp mathbf a 3 cdot mathbf a 3 end bmatrix nbsp a matricya Gramma dorivnyuye odinichnij matrici oskilki nashi bazisni vektori vzayemno ortogonalni i mayut odinichni dovzhini Otzhe 7 A 1 A T A A T E 1 0 0 0 1 0 0 0 1 displaystyle 7 qquad A 1 A T qquad AA T E begin bmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end bmatrix nbsp Pidsumovuyuchi skazane zapishemo radius vektor dovilnoyi tochki prostoru cherez koordinati ruhomoyi sistemi koordinat 8 R A x R 0 displaystyle 8 qquad mathbf R A mathbf x mathbf R 0 nbsp Shvidkist Redaguvati Prodiferenciyuyemo formulu 8 po chasu 9 R A x A x R 0 displaystyle 9 qquad dot mathbf R dot A mathbf x A dot mathbf x dot mathbf R 0 nbsp Poznachimo cherez v 0 displaystyle mathbf v 0 nbsp shvidkist ruhu pochatku koordinat 10 v 0 R 0 displaystyle 10 qquad mathbf v 0 dot mathbf R 0 nbsp Dali serednij dodanok v formuli 8 ye vektorom shvidkosti tochki z koordinatami x 1 t x 2 t x 3 t displaystyle x 1 t x 2 t x 3 t nbsp vidnosno ruhomoyi sistemi koordinat poznachimo yiyi bukvoyu v displaystyle mathbf v nbsp 11 v A x a 1 d x 1 d t a 2 d x 2 d t a 3 d x 3 d t displaystyle 11 qquad mathbf v A dot mathbf x mathbf a 1 dx 1 over dt mathbf a 2 dx 2 over dt mathbf a 3 dx 3 over dt nbsp Zalishilosya rozibratisya z pershim dodankom u formuli 9 Ochevidno sho pohidna matrici A displaystyle A nbsp maye buti proporcijnoyu vektoru kutovoyi shvidkosti w displaystyle boldsymbol omega nbsp Ale yak same Sprobuyemo zapisati taku matrichnu rivnist 12 A W A displaystyle 12 qquad dot A Omega A nbsp de W displaystyle Omega nbsp deyaka matricya Yasno sho mi zavzhdi mozhemo zapisati 12 oskilki matricya A displaystyle A nbsp nevirodzhena i tomu W displaystyle Omega nbsp odnoznachno znahoditsya za vidomoyu matriceyu A displaystyle A nbsp ta yiyi pohidnoyu 13 W A A 1 A A T displaystyle 13 qquad Omega dot A A 1 dot A A T nbsp Cya matricya antisimetrichna oskilki 14 W T A A T A A T d d t A A T A A T W displaystyle 14 qquad Omega T left dot A A T right A dot A T d over dt left AA T right dot A A T Omega nbsp V antisimetrichnij matrici tretogo poryadku ye lishe N C n 2 C 3 2 3 displaystyle left N C n 2 C 3 2 3 right nbsp tri nezalezhni vidminni vid nulya komponenti Yaksho mi yih poznachimo nastupnim chinom 15 W 0 w z w y w z 0 w x w y w x 0 displaystyle 15 qquad Omega begin bmatrix 0 amp omega z amp omega y omega z amp 0 amp omega x omega y amp omega x amp 0 end bmatrix nbsp to diya takoyi matrici na vektor dorivnyuvatime vektornomu dobutku w displaystyle boldsymbol omega nbsp na cej vektor 16 W r 0 w z w y w z 0 w x w y w x 0 x y z w y z w z y w z x w x z w x y w y x w r displaystyle 16 qquad Omega mathbf r begin bmatrix 0 amp omega z amp omega y omega z amp 0 amp omega x omega y amp omega x amp 0 end bmatrix begin pmatrix x y z end pmatrix begin pmatrix omega y z omega z y omega z x omega x z omega x y omega y x end pmatrix boldsymbol omega times mathbf r nbsp Teper formulu 9 mi mozhemo perepisati tak 17 v a b s R A x A x R 0 W A x v v 0 w r v v 0 displaystyle 17 qquad mathbf v abs dot mathbf R dot A mathbf x A dot mathbf x dot mathbf R 0 Omega A mathbf x mathbf v mathbf v 0 boldsymbol omega times mathbf r mathbf v mathbf v 0 nbsp Pri zapisi ostannoyi rivnosti mi skoristalisya formulami 4 i 16 Yak bachimo spravzhnya absolyutna shvidkist materialnoyi tochki skladayetsya z troh dodankiv shvidkosti w r displaystyle boldsymbol omega times mathbf r nbsp pov yazanoyi z obertannyam ruhomoyi sistemi koordinat shvidkosti v displaystyle mathbf v nbsp vidnosno ciyeyi sistemi koordinat ta postupalnoyi shvidkosti v 0 displaystyle mathbf v 0 nbsp z yakoyu ruhayetsya pochatok koordinat R 0 displaystyle mathbf R 0 nbsp Priskorennya Redaguvati Prodiferenciyuyemo formulu 9 she raz oderzhimo 18 R A x 2 A x A x R 0 displaystyle 18 qquad ddot mathbf R ddot A mathbf x 2 dot A dot mathbf x A ddot mathbf x ddot mathbf R 0 nbsp Obchislimo spochatku pershij dodanok formuli 18 19 A x d d t W A x W A x W W A x W r W W r displaystyle 19 qquad ddot A mathbf x d over dt left Omega A right mathbf x dot Omega A mathbf x Omega left Omega A right mathbf x dot Omega mathbf r Omega left Omega mathbf r right nbsp Perehodyachi vid matrichnih poznachen do vektornih za formuloyu 16 znahodimo 20 A x w r w w r displaystyle 20 qquad ddot A mathbf x dot boldsymbol omega times mathbf r boldsymbol omega times left boldsymbol omega times mathbf r right nbsp Dali obchislyuyemo drugij dodanok vrahuvavshi formulu 11 21 2 A x 2 W A x 2 W v 2 w v displaystyle 21 qquad 2 dot A dot mathbf x 2 Omega A dot mathbf x 2 Omega mathbf v 2 left boldsymbol omega times mathbf v right nbsp Tretij dodanok dorivnyuye priskorennyu a displaystyle mathbf a nbsp vidnosno ruhomoyi sistemi koordinat 22 A x a 1 d 2 x 1 d t 2 a 2 d 2 x 2 d t 2 a 3 d 2 x 3 d t 2 displaystyle 22 qquad A ddot mathbf x mathbf a 1 d 2 x 1 over dt 2 mathbf a 2 d 2 x 2 over dt 2 mathbf a 3 d 2 x 3 over dt 2 nbsp Nareshti ostannij dodanok vrahovuye postupalne priskorennya a 0 displaystyle mathbf a 0 nbsp pochatku koordinat ruhomoyi sistemi Sili Redaguvati Liva chastina formuli 18 ye priskorennyam a a b s displaystyle mathbf a abs nbsp v neruhomij inercialnij sistemi koordinat a tomu dlya cogo priskorennya mi mozhemo zapisati drugij zakon Nyutona 23 F m a a b s m R displaystyle 23 qquad mathbf F m mathbf a abs m ddot mathbf R nbsp de F displaystyle mathbf F nbsp rivnodijna usih spravzhnih sil Z formul 18 23 oderzhuyemo 24 m a F m w r m w w r 2 m w v m a 0 displaystyle 24 qquad m mathbf a mathbf F m left dot boldsymbol omega times mathbf r right m left boldsymbol omega times left boldsymbol omega times mathbf r right right 2m left boldsymbol omega times mathbf v right m mathbf a 0 nbsp Vivid formul vihodyachi iz zagalnoyi teoriyi vidnosnosti RedaguvatiFormula 1 ye formuloyu klasichnoyi mehaniki i yiyi mozhna vivoditi ne zvertayuchis do teoriyi vidnosnosti Ale vivid ciyeyi ale vzhe utochnenoyi formuli ne skladno zrobiti i v teoriyi vidnosnosti Vihodyachi z principu ekvivalentnosti v dovilnij v tomu chisli krivolinijnij sistemi koordinat dobutok masi materialnoyi tochki na priskorennya dorivnyuye 25 m d 2 x i d t 2 m G j k i d x j d t d x k d t F i displaystyle 25 qquad m d 2 x i over d tau 2 m Gamma jk i dx j over d tau dx k over d tau F i nbsp de t displaystyle tau nbsp vlasnij chas materialnoyi tochki pershij dodanok z simvolami Kristofelya v pravij storoni formuli 25 vidpovidaye silam inerciyi ta gravitaciyi a drugij dodanok ce realni sili F i displaystyle F i nbsp Zoseredimosya na silah inerciyi poklavshi F i 0 displaystyle F i 0 nbsp a takozh vvazhayuchi prostir chas ploskim tobto vidsutnya gravitaciya yaka vinikaye vnaslidok vikrivlennya prostoru chasu V ploskomu prostori chasi mozhna obrati inercijnu dekartovu sistemu koordinat x 0 x 1 x 2 x 3 displaystyle x 0 x 1 x 2 x 3 nbsp de persha koordinata napryamlena vzdovzh osi chasu x 0 c t displaystyle x 0 ct nbsp a reshta ce tri prostorovi koordinati x 1 x x 2 y x 3 z displaystyle x 1 x x 2 y x 3 z nbsp V cij sistemi koordinat metrichnij tenzor ye konstantoyu tobto metrikoyu Minkovskogo 26 g i j 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 displaystyle 26 qquad g ij begin bmatrix 1 amp 0 amp 0 amp 0 0 amp 1 amp 0 amp 0 0 amp 0 amp 1 amp 0 0 amp 0 amp 0 amp 1 end bmatrix nbsp i vsi simvoli Kristofelya dorivnyuyut nulyu V cij sistemi koordinat zgidno z 25 sili inerciyi dorivnyuyut nulyu Rozglyanemo teper inshu sistemu koordinat x 0 x 1 x 2 x 3 displaystyle hat x 0 hat x 1 hat x 2 hat x 3 nbsp v nij simvoli Kristofelya dorivnyuyut 27 G j k i x p x j x q x k 2 x i x p x q displaystyle 27 qquad hat Gamma jk i partial x p over partial hat x j partial x q over partial hat x k partial 2 hat x i over partial x p partial x q nbsp Chotirivimirni koordinati Redaguvati Budemo vvazhati cyu novu sistemu koordinat ruhomoyu i dekartovoyu shodo prostorovih koordinat tobto funkciyi perehodu vid ruhomoyi do absolyutnoyi sistemi koordinat x i x i x 0 x 1 x 2 x 3 displaystyle x i x i hat x 0 hat x 1 hat x 2 hat x 3 nbsp dayutsya formulami analogichnimi 8 28 x 0 x 0 c t displaystyle 28 qquad x 0 hat x 0 ct nbsp x 1 a 1 1 x 1 a 2 1 x 2 a 3 1 x 3 b 1 displaystyle qquad x 1 a 1 1 hat x 1 a 2 1 hat x 2 a 3 1 hat x 3 b 1 nbsp x 2 a 1 2 x 1 a 2 2 x 2 a 3 2 x 3 b 2 displaystyle qquad x 2 a 1 2 hat x 1 a 2 2 hat x 2 a 3 2 hat x 3 b 2 nbsp x 3 a 1 3 x 1 a 2 3 x 2 a 3 3 x 3 b 3 displaystyle qquad x 3 a 1 3 hat x 1 a 2 3 hat x 2 a 3 3 hat x 3 b 3 nbsp de koeficiyenti a j i b i displaystyle a j i b i nbsp pri i j 1 2 3 displaystyle i j left 1 2 3 right nbsp zalezhat tilki vid chasu tobto vid nulovoyi koordinati x 0 c t displaystyle hat x 0 ct nbsp 29 a j i a j i t b i b i t displaystyle 29 qquad a j i a j i t qquad b i b i t nbsp i koeficiyenti a j i displaystyle a j i nbsp razom utvoryuyut trivimirnu ortogonalnu matricyu Pidstavlyayuchi funkciyi 28 v 27 mi mozhemo obchisliti vsi koeficiyenti Kristofelya a otzhe i trayektoriyu ruhu materialnoyi tochki za formuloyu 25 ne vdayuchis do analizu sil inerciyi Tut mi obchislimo tilki matricyu perehodu x i x j displaystyle partial x i over partial hat x j nbsp mizh cimi sistemami koordinat vidokremlyuyuchi chasovu koordinatu vid prostorovih 30 x 0 x 0 1 x 0 x i 0 x i x j a j i displaystyle 30 qquad partial x 0 over partial hat x 0 1 qquad partial x 0 over partial hat x i 0 qquad partial x i over partial hat x j a j i nbsp 31 x i x 0 1 c k 1 3 a k i x k b i u i c displaystyle 31 qquad partial x i over partial hat x 0 1 over c left sum k 1 3 dot a k i hat x k dot b i right u i over c nbsp V formulah 30 31 indeksi i j k displaystyle i j k nbsp probigayut prostorovi komponenti 1 2 3 displaystyle left 1 2 3 right nbsp U formuli 31 cherez u i displaystyle u i nbsp poznacheno shvidkist tochok ruhomoyi sistemi koordinat vidnosno neruhomoyi 32 u A x b W A x v 0 W r v 0 w r v 0 displaystyle 32 qquad mathbf u dot A hat mathbf x dot mathbf b Omega A mathbf x mathbf v 0 Omega mathbf r mathbf v 0 boldsymbol omega times mathbf r mathbf v 0 nbsp Trivimirnij obraz sil inerciyi Redaguvati Velichina z odnim indeksom 33 F i m G j k i d x j d t d x k d t displaystyle 33 qquad tilde F i m hat Gamma jk i d hat x j over d tau d hat x k over d tau nbsp podibna do 4 vektora ale nepravilno zminyuyutsya pri zamini koordinat Zafiksuvavshi nashu ruhomu sistemu koordinat x i displaystyle hat x i nbsp mi mozhemo rozglyanuti dva geometrichni ob yekti 4 vektor F i displaystyle tilde F i nbsp i trivimirnu giperpoverhnyu v danomu razi ce giperploshina yaka zalezhit vid troh parametriv x 1 x 2 x 3 displaystyle hat x 1 hat x 2 hat x 3 nbsp pri fiksovanomu chasi x 0 c o n s t displaystyle left x 0 const right nbsp Mi mozhemo ortogonalno sproektuvati F i displaystyle tilde F i nbsp na cyu giperpoverhnyu i oderzhati trivimirnij vektor sili inerciyi Koordinati cogo vektora budut virazhatisya cherez kovariantni koordinati psevdovektora 34 F i g i j F j displaystyle 34 qquad tilde F i hat g ij tilde F j nbsp Dokladnishe pro ce u statti Trivimirni tenzori vseredini chotirivimirnih Otzhe mayemo viraz sili inerciyi cherez simvoli Kristofelya z nizhnimi indeksami 35 F i m G j k i d x j d t d x k d t displaystyle 35 qquad tilde F i m hat Gamma jk i d hat x j over d tau d hat x k over d tau nbsp Cyu formulu mi rozglyadayemo obmezhivshis prostorovimi znachennyami indeksu i 1 2 3 displaystyle i 1 2 3 nbsp Simvoli Kristofelya obchislyuyutsya cherez metrichnij tenzor za formuloyu 36 G j k i 1 2 g i k x j g j i x k g j k x i displaystyle 36 qquad hat Gamma jk i 1 over 2 left partial hat g ik over partial hat x j partial hat g ji over partial hat x k partial hat g jk over partial hat x i right nbsp Otzhe nam treba spochatku obchisliti metrichnij g i j displaystyle hat g ij nbsp tenzor v ruhomij sistemi koordinat Metrika v neinercijnij sistemi vidliku Redaguvati Oskilki v absolyutnij sistemi koordinat metrichnij tenzor dorivnyuye metrici Minkovskogo 26 mi mozhemo za tenzornimi pravilami pererahuvati cej tenzor v ruhomu sistemu koordinat 37 g i j x k x i x l x j x 0 x i x 0 x j k 1 3 x k x i x k x j displaystyle 37 qquad hat g ij partial x k over partial hat x i partial x l over partial hat x j partial x 0 over partial hat x i partial x 0 over partial hat x j sum k 1 3 partial x k over partial hat x i partial x k over partial hat x j nbsp Yaksho obidva indeksi i j displaystyle i j nbsp nabuvayut prostorovih znachen i 1 2 3 displaystyle i 1 2 3 nbsp to pershij dodanok dorivnyuvatime nulyu zgidno z 30 Znahodimo 38 g i j k 1 3 x k x i x k x j k 1 3 a i k a j k A T A i j d i j displaystyle 38 qquad hat g ij sum k 1 3 partial x k over partial hat x i partial x k over partial hat x j sum k 1 3 a i k a j k A T A ij delta ij nbsp oskilki matricya A displaystyle A nbsp ortogonalna Dali znahodimo mishani prostorovo chasovi komponenti metrichnogo tenzora tut takozh pershij dodanok v pravij chastini formuli 37 peretvoryuyetsya v nul 39 g 0 i k 1 3 x k x 0 x k x i k 1 3 u k c a i k 1 c A 1 u i displaystyle 39 qquad hat g 0i sum k 1 3 partial x k over partial hat x 0 partial x k over partial hat x i sum k 1 3 u k over c a i k 1 over c A 1 mathbf u i nbsp tobto dorivnyuyut komponentam shvidkosti u c displaystyle mathbf u over c nbsp v ruhomij sistemi koordinat Nareshti chasova komponenta metrichnogo tenzora dorivnyuye 40 g 00 x 0 x 0 2 k 1 3 x k x 0 2 1 u 2 c 2 displaystyle 40 qquad hat g 00 left partial x 0 over partial hat x 0 right 2 sum k 1 3 left partial x k over partial hat x 0 right 2 1 mathbf u 2 over c 2 nbsp Formuli 38 40 povnistyu opisuyut metrichnij tenzor yakij mi teper mozhemo zobraziti u viglyadi matrici 41 g i j 1 u 2 c 2 u x c u y c u z c u x c 1 0 0 u y c 0 1 0 u z c 0 0 1 displaystyle 41 qquad hat g ij begin bmatrix 1 mathbf u 2 over c 2 amp u x over c amp u y over c amp u z over c u x over c amp 1 amp 0 amp 0 u y over c amp 0 amp 1 amp 0 u z over c amp 0 amp 0 amp 1 end bmatrix nbsp Koristuyuchis metrichnim tenzorom mi mozhemo obchisliti diferencial vlasnogo chasu materialnoyi tochki 42 c 2 d t 2 g i j d x i d x j 1 u 2 c 2 d x 0 2 2 i 1 3 u i c d x 0 d x i i 1 3 d x i 2 c 2 u v 2 d t 2 displaystyle 42 qquad c 2 d tau 2 hat g ij d hat x i d hat x j left 1 mathbf u 2 over c 2 right d hat x 0 2 2 sum i 1 3 left u i over c right d hat x 0 d hat x i sum i 1 3 d hat x i 2 left c 2 mathbf u mathbf v 2 right dt 2 nbsp 43 d t 1 v a b s 2 c 2 d t displaystyle 43 qquad d tau sqrt 1 mathbf v abs 2 over c 2 dt nbsp Prodovzhennya obchislen sil inerciyi Redaguvati Rozdilimo sumu v pravij chastini formuli 35 na tri dodanki vidokremlyuyuchi dodanki z prostorovimi koordinatami vid dodankiv z chasovoyu koordinatoyu 44 F i m G 00 i d x 0 d t d x 0 d t 2 m j 1 3 G 0 j i d x 0 d t d x j d t m j k 1 3 G j k i d x j d t d x k d t displaystyle 44 qquad tilde F i m hat Gamma 00 i d hat x 0 over d tau d hat x 0 over d tau 2m sum j 1 3 hat Gamma 0j i d hat x 0 over d tau d hat x j over d tau m sum j k 1 3 hat Gamma jk i d hat x j over d tau d hat x k over d tau nbsp Pochnemo analiz ciyeyi formuli z ostannogo dodanka Oskilki simvoli Kristofelya obchislyuyutsya za formuloyu 36 a prostorova chastina metrichnogo tenzora ye konstantoyu 38 to simvoli Kristofelya peretvoryuyutsya v nul i ostannij dodanok u formuli 44 znikaye Dali rozglyanemo serednij dodanok vin proporcijnij shvidkosti a tomu ye siloyu Koriolisa Znahodimo vidpovidnij simvol Kristofelya 45 G 0 j i 1 2 g i j x 0 g 0 i x j g 0 j x i displaystyle 45 qquad hat Gamma 0j i 1 over 2 left partial hat g ij over partial hat x 0 partial hat g 0i over partial hat x j partial hat g 0j over partial hat x i right nbsp Pershij dodanok u formuli 45 dorivnyuye nulyu vnaslidok 38 a reshta dva dodanki v sumi dayut deyaku trivimirnu antisimetrichnu za indeksami i j displaystyle ij nbsp matricyu Cya matricya ye po pershe komponentami rotora vektornogo polya u displaystyle mathbf u nbsp obchislenimi v ruhomij sistemi koordinat a po druge cya matricya z tochnistyu do postijnogo mnozhnika 1 c displaystyle 1 c nbsp zbigayetsya z matriceyu W displaystyle Omega nbsp formula 13 ale komponenti yakoyi obchisleni v ruhomij sistemi koordinat 46 G 0 j i 1 2 x j 1 c A T u i x i 1 c A T u j displaystyle 46 qquad hat Gamma 0j i 1 over 2 left partial over partial hat x j 1 over c A T mathbf u i partial over partial hat x i 1 over c A T mathbf u j right nbsp 1 2 c x j A T A x A T b i x i A T A x b i 1 c A T A i j 1 c A T W A i j displaystyle 1 over 2c left partial over partial hat x j A T dot A hat mathbf x A T dot mathbf b i partial over hat partial x i A T dot A hat mathbf x dot mathbf b i right 1 over c left A T dot A right ij 1 over c left A T Omega A right ij nbsp Otzhe sila Koriolisa dorivnyuye 47 F i 2 m j 1 3 G 0 j i d x 0 d t d x j d t 2 m 1 c j 1 3 A T W A i j c d t d t 2 d x j d t displaystyle 47 qquad tilde F i 2m sum j 1 3 hat Gamma 0j i d hat x 0 over d tau d hat x j over d tau 2m 1 over c sum j 1 3 left A T Omega A right ij c dt over d tau 2 d hat x j over dt nbsp Vrahovuyuchi formulu 43 mi mozhemo zapisati cyu formulu u vektornomu viglyadi 48 F C o r r 2 m w v 1 v a b s 2 c 2 displaystyle 48 qquad mathbf F Corr 2m boldsymbol omega times mathbf v over 1 mathbf v abs 2 over c 2 nbsp Obchislimo nareshti pershij dodanok u formuli 44 Dlya cogo znahodimo vidpovidnij simvol Kristofelya 49 G 00 i 1 2 2 g 0 i x 0 g 00 x i 1 c t 1 c A T u i 1 2 x i 1 u 2 c 2 displaystyle 49 qquad hat Gamma 00 i 1 over 2 left 2 partial hat g 0i over partial hat x 0 partial hat g 00 over partial hat x i right 1 over c partial over partial t left 1 over c A T mathbf u right i 1 over 2 partial over partial hat x i left 1 mathbf u 2 over c 2 right nbsp Rozpishemo dokladnishe obidva dodanki ciyeyi formuli pidstavlyayuchi viraz dlya u displaystyle mathbf u nbsp iz formuli 32 i vikonuyuchi diferenciyuvannya Pershij dodanok dorivnyuye 50 1 c 2 t x c o n s t A T A x b i 1 c 2 A T A x b A T A x b i displaystyle 50 qquad 1 over c 2 partial over partial t bigg hat x const left A T dot A hat mathbf x dot mathbf b right i 1 over c 2 left dot A T dot A hat mathbf x dot mathbf b A T ddot A hat mathbf x ddot mathbf b right i nbsp a drugij 51 1 2 c 2 x i A x b 2 1 c 2 A T A x b i displaystyle 51 qquad 1 over 2c 2 partial over partial hat x i dot A hat mathbf x dot mathbf b 2 1 over c 2 left dot A T dot A hat mathbf x dot mathbf b right i nbsp Yak bachimo dodanok 51 znishuyetsya z pershim dodankom v pravij chastini formuli 50 Otzhe dlya simvolu Kristofelya mayemo 52 G 00 i 1 c 2 A T A x b i displaystyle 52 qquad hat Gamma 00 i 1 over c 2 left A T ddot A hat mathbf x ddot mathbf b right i nbsp Vrahovuyuchi formulu 20 formula 52 ye prosto koordinatoyu vidnosno ruhomoyi sistemi koordinat nastupnogo trivimirnogo vektora 54 1 c 2 w r w w r a 0 displaystyle 54 qquad 1 over c 2 left dot boldsymbol omega times mathbf r boldsymbol omega times boldsymbol omega times mathbf r mathbf a 0 right nbsp Otzhe u vektornomu vidi pershij dodanok 44 zapishetsya tak 55 m G 00 d x 0 d t d x 0 d t m c 2 w r w w r a 0 c d t d t 2 m w r m w w r m a 0 1 v a b s 2 c 2 displaystyle 55 qquad m hat boldsymbol Gamma 00 d hat x 0 over d tau d hat x 0 over d tau m over c 2 left dot boldsymbol omega times mathbf r boldsymbol omega times boldsymbol omega times mathbf r mathbf a 0 right left c dt over d tau right 2 m dot boldsymbol omega times mathbf r m boldsymbol omega times boldsymbol omega times mathbf r m mathbf a 0 over 1 mathbf v abs 2 over c 2 nbsp Pidstavlyayuchi 48 i 55 v formulu 44 i zgaduyuchi sho tretij dodanok v pravij chastini 44 dorivnyuye nulyu oderzhuyemo ostatochnij viraz dlya sil inerciyi 56 F m w r m w w r 2 m w v m a 0 1 v a b s 2 c 2 displaystyle 56 qquad tilde mathbf F m left dot boldsymbol omega times mathbf r right m left boldsymbol omega times left boldsymbol omega times mathbf r right right 2m left boldsymbol omega times mathbf v right m mathbf a 0 over 1 mathbf v abs 2 over c 2 nbsp Porivnyayemo cyu formulu z formuloyu 24 oderzhanoyu v klasichnij mehanici Yedinoyu vidminnistyu ye znamennik v 56 yakij vrahovuye upovilnennya chasu formula 43 sho pov yazane z ruhom materialnoyi tochki Cikavo sho v formuli 56 dlya sistemi koordinat sho obertayetsya znamennik mozhe peretvoritisya v nul abo stati vid yemnim Adzhe daleko vid osi obertannya shvidkist ruhomoyi sistemi koordinat vidnosno neruhomoyi mozhe perevishiti shvidkist svitla Yasno sho na takih vidstanyah ne mozhe isnuvati materialnogo tila yake b ruhalosya razom iz sistemoyu koordinat v comu razi i sistema koordinat i sila 56 stayut ne bilshe nizh matematichnoyu abstrakciyeyu sho ne maye fizichnogo traktuvannya Primitki Redaguvati S E Hajkin Sily inercii i nevesomost Izd vo Nauka Glavnaya redakciya fiz mat literatury M 1967 312s Dzherela RedaguvatiFedorchenko A M 1975 Teoretichna mehanika Kiyiv Visha shkola 516 s Otrimano z https uk wikipedia org w index php title Sila inerciyi amp oldid 31413867
