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2.3. Sfericheskaya sistema koordinat

Dlya resheniya mnogih zadach okazyvaetsya udobnee vmesto dekartovoi sistemy ispol'zovat' krivolineinye sistemy koordinat. V obshem sluchae ispol'zuyutsya tri funkcii $ f_1(x,y,z)$, $ f_2(x,y,z)$, $ f_3(x,y,z)$ dlya opredeleniya polozheniya tela v prostranstve. Koordinatnaya setka sostoit iz peresekayushihsya krivyh $ f_i(x,y,z)=\textrm{const}, i=1,2,3$ vmesto setki pryamyh linii $ x=\textrm{const}, y=\textrm{const}, z=\textrm{const}$ v dekartovoi sisteme. Esli funkcii vybrany podhodyashim obrazom, to polozhenie ob'ekta mozhet byt' odnoznachno opredeleno s pomosh'yu krivolineinyh koordinat $ (f_1,f_2,f_3)$ vmesto dekartovyh koordinat $ (x,y,z)$.

K takim sistemam otnositsya i sfericheskaya sistema koordinat, shiroko ispol'zuemaya ne tol'ko v astronomii, no i drugih naukah. Sfericheskie koordinaty (sm. ris. 2.5): $ r$ -- radius-vektor ob'ekta, $ \theta$ -- polyarnoe rasstoyanie, kotoroe inogda nazyvayut koshirotoi, i $ \lambda$ -- dolgota svyazany s dekartovymi koordinatami $ x,y,z$ uravneniyami:

$\displaystyle x$ $\displaystyle =r\sin\theta \cos\lambda,$    
$\displaystyle y$ $\displaystyle =r\sin\theta \sin\lambda,$ (20)
$\displaystyle z$ $\displaystyle =r\cos\theta.$    

Polyarnoe rasstoyanie izmenyaetsya ot $ 0^\circ$ do $ 180^\circ$, dolgota -- ot $ 0^\circ$ do $ 360^\circ$.

Ris. 2.5. Opredelenie sfericheskih koordinat tochki

Sistema uravnenii (2.20) predstavlyaet preobrazovanie mezhdu sfericheskoi i dekartovoi sistemami koordinat. Sledovatel'no, funkcii $ f_1,f_2,f_3$ ravny:

$\displaystyle f_1$ $\displaystyle =r=\sqrt{x^2+y^2+z^2},$    
$\displaystyle f_2$ $\displaystyle =\theta=\textrm{arcctg}\frac{z}{\sqrt{x^2+y^2}}, \quad 0\leq\theta\leq\pi$    
$\displaystyle f_3$ $\displaystyle =\lambda=\begin{cases}\textrm{arctg}\frac{y}{x},& \quad 0\leq \lambda\lt \pi/2,\ \textrm{esli}\ x\gt 0, y\geq 0, \\ \pi+\textrm{arctg}\frac{y}{x},& \pi/2\leq \lambda\lt 3\pi/2,\ \textrm{esli}\ x\leq 0, \\ 2\pi+\textrm{arctg}\frac{y}{x},& 3\pi/2\leq \lambda\lt 2\pi,\ \textrm{esli}\ x\geq 0, y\lt 0.\end{cases}$    

Vernemsya k ris. 2.5. Cherez proizvol'no vybrannye tochki $ A$ i $ C$ provedem bol'shoi krug. Polyusy oboznachim kak $ P$ i $ \cal N$. Provedem teper' cherez polyusy i tochku $ A$ bol'shoi krug (analogichno provedem bol'shoi krug cherez tochku $ C$). Oboznachim cherez $ \theta$ central'nyi ugol mezhdu napravleniem na tochku $ P$ i napravleniem na proizvol'nuyu tochku $ A'$, lezhashuyu na sfere v ploskosti bol'shogo kruga $ PA\cal N$. Provedem cherez tochku $ A'$ ploskost', parallel'nuyu bol'shomu krugu $ OAC$. Poluchennaya ploskost' yavlyaetsya malym krugom, i radius $ O'A'=\rho$ okruzhnosti raven, esli $ OA'=R$:

$\displaystyle \rho=OA'\sin \theta = R\sin\theta.$ (2.21)

Vvedem dekartovu sistemu koordinat: os' $ x$ napravim vdol' radiusa $ OC$, os' $ z$ -- vdol' radiusa $ OP$. Oboznachim edinichnye vektory osei $ x$ i $ z$ kak $ \bf i$ i $ \bf k$, sootvetstvenno. Napravlenie osi $ y$ zadadim edinichnym vektorom $ \bf j$ soglasno uravneniyu:

$\displaystyle {\bf j} ={\bf k}\times {\bf i}.$ (2.22)

Vektornoe proizvedenie (2.22) vektorov $ \bf k$ i $ \bf i$ opredelyaet pravuyu dekartovu sistemu koordinat $ Oxyz$.

Oboznachim cherez $ \lambda$ dvugrannyi ugol mezhdu ploskostyami $ PC\cal N$ i $ PA\cal N$. Chisla $ R, \theta, \lambda$ nazyvayutsya sfericheskimi koordinatami tochki $ A'$. Pri $ R=1$ dostatochno znat' dve koordinaty $ \theta, \lambda$ dlya opredeleniya polozheniya tochki na sfere. V sleduyushei glave budut opredeleny razlichnye sistemy sfericheskih koordinat. V kazhdoi iz nih koordinaty $ \theta, \lambda$ imeyut raznye nazvaniya i mogut oboznachat'sya drugimi bukvami.

Pust' tochka $ C'$ lezhit na sfere i yavlyaetsya tochkoi peresecheniya bol'shogo kruga $ PC\cal N$ i malogo kruga $ A'O'C'$ (ris. 2.5). Naidem dlinu dugi $ \widehat{A'C'}$. Tak kak central'nyi ugol $ A'O'C'$ raven $ \lambda$, to

$\displaystyle \widehat{A'C'}=O'A'\cdot \lambda= R\lambda \sin\theta.$ (2.23)

Rassmotrim bolee podrobno vopros preobrazovaniya koordinat vektora v krivolineinyh koordinatah.

V krivolineinoi sisteme koordinat v otlichie ot dekartovoi vozmozhny dva sposoba vybora bazisnoi troiki vektorov: 1) bazisnye vektory yavlyayutsya kasatel'nymi v tochke $ (x_0, y_0, z_0)$ k krivym $ f_1(x_0, y=\textrm{const}, z=\textrm{const})$, $ f_2(x=\textrm{const}, y_0, z=\textrm{const})$, $ f_3(x=\textrm{const}, y=\textrm{const}, z_0)$; oboznachim ih kak $ \mathbf{e}_1$, $ \mathbf{e}_2$, $ \mathbf{e}_3$ i 2) bazisnye vektory perpendikulyarny v tochke $ (x_0, y_0, z_0)$ k poverhnostyam, zadavaemym funkciyami $ f_1,f_2,f_3$, t.e. $ f_1(x_0, y_0, z_0)=\textrm{const}$, $ f_2(x_0, y_0, z_0)=\textrm{const}$, $ f_3(x_0, y_0, z_0)=\textrm{const}$; oboznachim ih kak $ \mathbf{e}^1$, $ \mathbf{e}^2$, $ \mathbf{e}^3$. Eshe odnim otlichiem ot dekartovoi sistemy yavlyaetsya to, chto napravlenie, a takzhe dlina bazisnyh vektorov mozhet razlichat'sya v raznyh tochkah prostranstva.

V sluchae sfericheskih koordinat poverhnost', zadavaemaya uravneniem $ f_1=\textrm{const}$, est' sfera radiusa $ r$, uravnenie $ f_2=\theta=\textrm{const}$ opredelyaet malyi krug, a $ f_3=\lambda=\textrm{const}$ -- ploskost' meridiana. Peresecheniya etih ploskostei so sferoi yavlyayutsya okruzhnostyami. Tak kak krivye $ f_1(x_0, y=\textrm{const}, z=\textrm{const})$, $ f_2(x=\textrm{const}, y_0, z=\textrm{const})$, $ f_3(x=\textrm{const}, y=\textrm{const}, z_0)$ takzhe yavlyayutsya okruzhnostyami, to v sluchae sfericheskih koordinat obe bazisnye troiki sovpadayut. V obshem sluchae eto ne tak.

Dva vybora bazisnyh troek dayut vozmozhnost' naiti proekcii vektora $ \mathbf{a}$ kak na osi $ \mathbf{e}_1$, $ \mathbf{e}_2$, $ \mathbf{e}_3$, tak i na osi $ \mathbf{e}^1$, $ \mathbf{e}^2$, $ \mathbf{e}^3$:

$\displaystyle \mathbf{a}=a_1\mathbf{e}^1+a_2\mathbf{e}^2+a_3\mathbf{e}^3= a^1\mathbf{e}_1+a^2\mathbf{e}_2+a^3\mathbf{e}_3. $

Ispol'zuya predlozhennoe Einshteinom pravilo summirovaniya, soglasno kotoromu v vyrazhenii summirovanie vypolnyaetsya po pare povtoryayushihsya indeksov (znak summy pri etom opuskaetsya), razlozhenie vektora po bazisnym troikam zapisyvaetsya v vide:

$\displaystyle \mathbf{a}=a_i\mathbf{e}^i,\quad \mathbf{a}= a^j\mathbf{e}_j;$ (2.24)

indeksy summirovaniya mogut oboznachat'sya lyubymi bukvami.

Chisla $ a^1,a^2,a^3$ nazyvayutsya kontravariantnymi, a $ a_1,a_2,a_3$ -- kovariantnymi proekciyami vektora $ \mathbf{a}$.

Dlya bazisnyh vektorov $ {\mathbf{e}}_i, {\mathbf{e}}^k$ spravedlivy sootnosheniya:

$\displaystyle {\mathbf{e}}_i\cdot {\mathbf{e}}^k= \delta^k_i=\begin{cases}1,& \textrm{pri}\ i=k\\ 0,& \textrm{pri}\ i\neq k. \end{cases}$ (2.25)

Simvol $ \delta^k_i$ nazyvaetsya simvolom Kronekera2.2.

Skalyarnoe proizvedenie v krivolineinyh koordinatah zapisyvaetsya v vide

$\displaystyle {\mathbf{a}}\cdot{\mathbf{b}}= (a^i\mathbf{e}_i)\cdot(b_j\mathbf{e}^j)= (a^ib_j)\mathbf{e}_i\cdot\mathbf{e}^j. $

Po opredeleniyu $ \mathbf{e}_i\cdot\mathbf{e}^j=\delta_i^j$, prichem $ \delta_i^i=1$, $ \delta_i^j=0$ pri $ i\neq j$. Znachit $ \mathbf{a}\cdot\mathbf{b}= (a^ib_j)\delta_i^j=a^ib_i$. Skalyarnoe proizvedenie mozhno vychislit', esli izvestny kovariantnye proekcii vektora $ \mathbf{a}$ i kontravariantnye proekcii vektora $ \mathbf{b}$, pri etom $ \mathbf{a}\cdot\mathbf{b}=a^ib_i=a_ib^i$.

Dlya togo, chtoby poluchit' yavnoe vyrazhenie kovariantnyh i kontravariantnyh koordinat vektora, umnozhim skalyarno pervoe iz uravnenii (2.24) na $ \mathbf{e}_j$, a vtoroe -- na $ \mathbf{e}^i$. Uchityvaya opredelenie (2.25), naidem:

$\displaystyle {\mathbf{a}}\cdot \mathbf{e}_j$ $\displaystyle =a_i(\mathbf{e}^i\cdot\mathbf{e}_j)= a_i\delta^i_j =a_j$    
$\displaystyle {\mathbf{a}}\cdot \mathbf{e}^i$ $\displaystyle =a^j(\mathbf{e}_j\cdot\mathbf{e}^i)= a^j\delta^i_j =a^i.$    

Znachit,

$\displaystyle a_i ={\mathbf{a}}\cdot \mathbf{e}_i, \quad a^i={\mathbf{a}}\cdot \mathbf{e}^i.$ (2.26)

Ispol'zuya formuly (2.26), perepishem (2.24) v vide:

$\displaystyle \mathbf{a}=({\mathbf{a}}\cdot \mathbf{e}_i)\mathbf{e}^i,\quad \mathbf{a}= ({\mathbf{a}}\cdot \mathbf{e}^i)\mathbf{e}_i.$ (2.27)

Sootnosheniya spravedlivy dlya lyubogo vektora $ \mathbf{a}$. Esli vmesto $ \mathbf{a}$ v (2.27) podstavit' bazisnye vektory, to poluchim:

$\displaystyle \mathbf{e}_k=(\mathbf{e}_k\cdot \mathbf{e}_i)\mathbf{e}^i,\quad \mathbf{e}^k= (\mathbf{e}^k\cdot \mathbf{e}^i)\mathbf{e}_i.$ (2.28)

Vvodya oboznacheniya

$\displaystyle g_{ki}=\mathbf{e}_k\cdot \mathbf{e}_i,\quad g^{ki}= \mathbf{e}^k\cdot \mathbf{e}^i,$ (2.29)

perepishem sootnosheniya (2.28) takim obrazom:

$\displaystyle \mathbf{e}_k=g_{ki}\mathbf{e}^i,\quad \mathbf{e}^k= g^{ki}\mathbf{e}_i.$ (2.30)

Dlya postroeniya bazisnoi troiki $ \mathbf{e}_k$ po vektoram $ \mathbf{e}^i$ neobhodimo znat' matricu s elementami $ [g_{ki}]$; i, naoborot, dlya postroeniya bazisa $ \mathbf{e}^k$ po bazisu $ \mathbf{e}_i$ -- matricu s elementami $ [g^{ki}]$. Eti matricy vzaimno obratny, t.e.

$\displaystyle g^{ki}g_{ij}= \delta^k_j=\begin{cases}1,& \textrm{pri}\ j=k\\ 0,& \textrm{pri}\ j\neq k. \end{cases}$

Velichiny $ g^{ki}$ i $ g_{ij}$ nazyvayutsya komponentami dvazhdy kontravariantnogo i kovariantnogo metricheskogo tenzora, sootvetstvenno.

Chto iz sebya predstavlyaet tenzor v matematike? Kak my videli, zadanie bazisnoi troiki opredelyaet sistemu koordinat, v kotoroi mozhno naiti koordinaty proizvol'nyh vektorov, t.e. ih proekcii na bazisnye vektory. No tak kak pri perehode v druguyu tochku prostranstva napravlenie i velichina bazisnyh vektorov mozhet menyat'sya, to neobhodimo reshit' zadachu o preobrazovanii proekcii proizvol'nyh vektorov iz odnoi bazisnoi troiki v druguyu. Eta zadacha reshaetsya metodami tenzornogo analiza. Tenzory predstavlyayut soboi sistemu velichin, preobrazuyushihsya po lineinomu zakonu pri perehode ot odnoi sistemy koordinat k drugoi. Sootnosheniya, zapisannye v tenzornoi forme, sohranyayut svoyu formu v lyuboi koordinatnoi sisteme.

Naidem teper' rasstoyanie $ d{\mathbf{r}}$ mezhdu dvumya beskonechno blizkimi tochkami prostranstva. Dekartovy koordinaty vektora $ d{\mathbf{r}}$ ravny $ dx, dy,dz$. Dlya etogo, schitaya $ x,y,z$ v formulah (2.20) funkciyami peremennyh $ r,\theta,\lambda$ naidem differencialy $ dx$, $ dy$, $ dz$. Po pravilu vychisleniya differencialov funkcii mnogih peremennyh, snachala fiksiruem peremennye $ \theta, \lambda$ i nahodim izmenenie funkcii (chastnuyu proizvodnuyu $ \partial/\partial{r}$) v zavisimosti ot prirasheniya $ dr$, zatem fiksiruem peremennye $ r$ i $ \lambda$ i nahodim izmenenie funkcii v zavisimosti ot prirasheniya $ d\theta$, i nakonec pri postoyannyh $ r$ i $ \theta$ nahodim chastnuyu proizvodnuyu $ \partial/\partial{\lambda}$. V rezul'tate poluchim:

$\displaystyle dx$ $\displaystyle =\frac{\partial{x}}{\partial{r}}dr+\frac{\partial{x}}{\partial{\theta}}d\theta +\frac{\partial{x}}{\partial{\lambda}}d\lambda,$    
$\displaystyle dy$ $\displaystyle =\frac{\partial{y}}{\partial{r}}dr+\frac{\partial{y}}{\partial{\theta}}d\theta +\frac{\partial{y}}{\partial{\lambda}}d\lambda,$ (31)
$\displaystyle dz$ $\displaystyle =\frac{\partial{z}}{\partial{r}}dr+\frac{\partial{z}}{\partial{\theta}}d\theta +\frac{\partial{z}}{\partial{\lambda}}d\lambda,$    

Po opredeleniyu chastnye proizvodnye $ \partial{x}/\partial{r},\partial{x}/\partial{\theta}$ i dr. yavlyayutsya kasatel'nymi k funkciyam $ x,y,z$, t.e. predstavlyayut soboi komponenty bazisnyh vektorov $ \mathbf{e}_1$, $ \mathbf{e}_2$, $ \mathbf{e}_3$ vdol' napravlenii $ r,\theta,\lambda$:

$\displaystyle \mathbf{e}_1=\begin{pmatrix}\frac{\partial{x}}{\partial{r}}\\ \frac{\partial{y}}{\partial{r}} \\ \frac{\partial{z}}{\partial{r}} \end{pmatrix}, \quad \mathbf{e}_2=\begin{pmatrix}\frac{\partial{x}}{\partial{\theta}}\\ \frac{\partial{y}}{\partial{\theta}} \\ \frac{\partial{z}}{\partial{\theta}} \end{pmatrix}, \quad \mathbf{e}_3=\begin{pmatrix}\frac{\partial{x}}{\partial{\lambda}}\\ \frac{\partial{y}}{\partial{\lambda}} \\ \frac{\partial{z}}{\partial{\lambda}} \end{pmatrix}. $

Eto oznachaet, chto prirasheniya $ dr,d\theta,d\lambda$ yavlyayutsya kontravariantnymi proekciyami vektora $ d{\mathbf{r}}$ v sfericheskih koordinatah; uravneniya (2.31), poetomu, predstavlyayut preobrazovanie kontravariantnogo vektora.

Sledovatel'no, differencial funkcii yavlyaetsya kontravariantnym vektorom. Pereoboznachiv beskonechno malye prirasheniya kak $ dx^1=dx$, $ dx^2=dy$, $ dx^3=dz$, naidem kvadrat rasstoyaniya $ ds^2$ mezhdu dvumya beskonechno blizkimi tochkami, kotoryi ravnyaetsya v dekartovoi sisteme koordinat

$\displaystyle (ds)^2=(dx^1)^2+(dx^2)^2+(dx^3)^2.$ (2.32)

V bolee obshem vide s uchetom pravila summirovaniya vyrazhenie dlya kvadrata rasstoyaniya mezhdu dvumya tochkami prostranstva zapisyvaetsya v vide:

$\displaystyle (ds)^2=(d{\mathbf{r}})^2=dx^idx^j({\mathbf{e}}_i\cdot{\mathbf{e}}_j)= g_{ij}dx^idx^j,$ (2.33)

gde $ g$ -- metricheskii tenzor. Zakon vychisleniya rasstoyaniya (2.33) nazyvaetsya metrikoi prostranstva.

Vybor toi ili inoi sistemy koordinat daet vozmozhnost' opredelit' polozhenie tela v prostranstve i uprostit' uravneniya dvizheniya tela, no ne opredelyaet svoistva samogo prostranstva. Zadanie metriki sovmestno s opredeleniem sistemy koordinat polnost'yu opisyvaet prostranstvo. Eto oznachaet, chto, znaya metricheskii tenzor, mozhno vychislit' rasstoyanie mezhdu dvumya tochkami. V sluchae evklidova prostranstva, kotoroe nazyvaetsya ploskim, rasstoyanie nahoditsya po formule (2.32) i metricheskii tenzor raven

$\displaystyle g=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}.$ (2.34)

V sluchae ploskogo prostranstva metricheskii tenzor $ g$ yavlyaetsya diagonal'nym i simmetrichnym: $ g_{ij}=g_{ji}$. V obshem sluchae tenzor mozhet imet' nediagonal'nye elementy, kotorye zavisyat ot koordinat, no tenzor $ g$ vsegda yavlyaetsya simmetrichnym, tak kak velichiny $ g_{ij}$ opredelyayutsya iz simmetrichnoi formy (2.33).

Esli prostranstvo ne yavlyaetsya ploskim, to dlya vychisleniya rasstoyanii uzhe nel'zya ispol'zovat' zakon Pifagora (2.32). V chastnosti, pri vychisleniyah na sfere (v krivom prostranstve) dlina dugi mezhdu dvumya tochkami ne ravna dline hordy (rasstoyaniyu v ploskom prostranstve).

Kvadrat elementa dliny v sfericheskoi sisteme koordinat legko naiti, vychisliv chastnye proizvodnye $ \partial{x}/\partial{r},\partial{x}/\partial{\theta}$ i t.d. i podstaviv ih v (2.32). Ispol'zuya uravneniya (2.20), nahodim, chto $ \partial{x}/\partial{r}=\sin\theta\cos\lambda$, $ \partial{x}/\partial{\theta}=r\cos\theta\cos\lambda$ i t.d. V rezul'tate posle privedeniya podobnyh chlenov poluchim, chto

$\displaystyle ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\lambda^2. $

Metricheskii tenzor, sledovatel'no, raven

$\displaystyle g=\begin{pmatrix}1&0&0\\ 0&r^2&0\\ 0&0&r^2\sin^2\theta \end{pmatrix}. $

Tenzor ne soderzhit nediagonal'nyh chlenov. Eto govorit o tom, chto sfericheskie koordinaty ortogonal'ny: sfera s radiusom $ r=\textrm{const}$, konus s uglom $ \theta=\textrm{const}$ i meridional'naya ploskost' $ \lambda=\textrm{const}$ peresekayutsya pod pryamymi uglami drug k drugu.

Takim obrazom, svoistva geometrii v krivolineinoi sisteme koordinat opredelyayutsya komponentami $ g_{ij}$ metricheskogo tenzora. V dal'neishem my budem rassmatrivat' chetyrehmernoe prostranstvo-vremya dlya vychisleniya effektov teorii otnositel'nosti (izmeneniya hoda chasov, nahodyashihsya v gravitacionnom pole, otkloneniya lucha sveta). V chetyrehmernom prostranstve-vremeni imeetsya, sledovatel'no, 16 komponent tenzora, iz nih tol'ko 10 razlichny iz-za simmetrichnosti tenzora (chetyre s odinakovymi indeksami i $ 12/2=6$ s razlichnymi indeksami).


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