??????????? ????????? , ????????????? ? ???????? ??????? ???????? , ??? ??????? ? ??????????? ????? ???? ?????????? ????? ??????? ???????????? ? ??????????? ?????????? ???? ?????????? ?????????? ??????????????? ? ????????? ?????????? ? ${\displaystyle \varphi }$ ?????? ??????? ? ??????????? ????? ????????? ?????? ?????? ????????? ???? ????????? ????????? ? ??? ???? ???????? ??????????????? ??????????? ?????????? ?????????????? ?????? ????? ????????? ? ?????? ???? ???????? 18-?? ???? ???????

???????????? ??????????? ???????????????? ??????????? ?????????? ?????? ????????? ?????????? ??????? ??? ??????, ???????????? ???????? ???????? ?????????????? ????????????????? ?????? ????????????? ??????????? ?????????? ???? ?????? ? ?????? ??????? ?

??????????? ????????? ? ??????? ????????????? [ ???????? | ???????? ???? ]

??????? ???????????? ??????? ???????? ??????????? ??????? ???????? ? ???????? ??????????? , ??? ???????? ?????????? ??????????? ??????? ????? ?????

${\displaystyle L={\frac {m{\dot {q}}^{2}}{2}}-m\varphi }$, ????? ${\displaystyle m}$?? ??????? ???????? ?, ${\displaystyle q}$??? ????????????? , ${\displaystyle \varphi }$??? ??????????? ????? ???????????

L ??????????? ????? ???????????????? ?????????? ???????? ?????????? ????

${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}}}-{\frac {\partial L}{\partial q}}=0}$,

??????? ??? ??????? ?????????????? ?

${\displaystyle {\ddot {q}}=-grad(\varphi )}$:

??????????? ?????????? ? ?????????????? ????????? [ ???????? | ???????? ???? ]

??????? ???????????? ??????????? ??????? ??????? ??????? ?????????????? ??? ??????????? ???????? ??? ??????? ?????????? ??? ??????????? ?? ??????????? ????? ???????? ??????????? ?????????????? ????????? ???????????????? ??

??????? ??????? ? ????????? ?????? ??????????? ?????????? [ ???????? | ???????? ???? ]

??????? ??????? ??????????? ?????????? ??????? ?

${\displaystyle \varphi =-{\frac {Gm}{R}}}$,

????? ${\displaystyle G}$?? ??????????? ?????????? ?, ${\displaystyle m}$??? ??????? ???????? , ${\displaystyle R}$??? ??????????????? ????????? ??? ????? ??????? ???????? ? ???????? ?????? ??????????? ?????????? ?????, ??? ??????? ?????????? ??????? ???????? ? ????????

????????? ${\displaystyle \rho }$ ????????? ??????? ????????? ?????? ????? ??????????? ?????????? ?????????? ? ????????? ??????????? ?

${\displaystyle \Delta \varphi =-4\pi G\rho }$,

????? ${\displaystyle \Delta }$?? ??????? ????????? ?, ${\displaystyle \rho }$??? ???????? ??????? ????????? ?????????? ????????? ??????? ??? ?????????? ????????? ???????? ???? ??????? ??????

${\displaystyle \varphi =-G\int _{V}{\frac {\rho dV}{r}},}$,

????? r?? dV ?????? ????? ??????????????? ? ???? ????????? ?????, ??? ?????????? ????????? ? ????? ??????? ?????? ?????? ???????? ???????? ?????? ??????????? ?????????? ???????? ??

??????????? ?????????? ? ????????? ???????? [ ???????? | ???????? ???? ]

??????? ????????? ???????? ??????????? ??????? ??????? ? ??? ???????? ? ????? ?????????? ???????????? ???????? ????????? ??????? ??????? ????????? ????????? ????? ???? ?

${\displaystyle U={\frac {1}{2}}\int {\mu \varphi dV},\qquad \qquad (1)}$

????????????????, ????? ${\displaystyle \mu }$?? ?????? ???????? ?????????? ?, ${\displaystyle \varphi }$??? ??????????? ??????????, ${\displaystyle V}$??? ?????? ???????

????????? ??????????? ????? ??????????? ????????? [ ???????? | ???????? ???? ]

????????? ?????? ????? ??????????? ?????????? ??????? ????

${\displaystyle \varphi =-G\left({\frac {M}{R_{0}}}+{\frac {1}{6}}D_{\alpha \beta }{\frac {\partial ^{2}}{\partial {X_{\alpha }}\partial {X_{\beta }}}}{\frac {1}{R_{0}}}+...\right)\qquad \qquad (2),}$

?????, ????? ${\displaystyle M=\int \mu dV}$?? ????????? ???? ???????? ?, ???

${\displaystyle D_{\alpha \beta }=\int \mu (3x_{\alpha }x_{\beta }-r^{2}\delta _{\alpha \beta })dV}$

?????????????? ?????? ? ??????? ???????? ????????????????? ??????? ?????????

${\displaystyle J_{\alpha \beta }=\int \mu (r^{2}\delta _{\alpha \beta }-x_{\alpha }x_{\beta })dV}$

????????? ??????? ??????? ??? ??? ?????? ?

${\displaystyle D_{\alpha \beta }=J_{\gamma \gamma }\delta _{\alpha \beta }-3J_{\alpha \beta }}$

?????????????????

??????????? ??????????? ?????????? [ ???????? | ???????? ???? ]

????????? ??????? ???????? ?????????? ??????? ??????????? ?????????? ?????? ? ????????? ??? ?????? ?????????????

${\displaystyle U={\frac {GM}{r}}\left(1-\sum _{n=2}^{\infty }J_{n}\left({\frac {R}{r}}\right)^{n}P_{n}(\sin \theta )+\sum _{n=2}^{\infty }\sum _{k=2}^{n}\left({\frac {R}{r}}\right)^{n}(C_{nm}\cos(m\lambda )+S_{nm}\sin(m\lambda ))P_{n}^{k}(\sin \theta )\right)}$?

?????? ${\displaystyle r,\theta ,\lambda }$?? ????????? ???? ?????? ????????????? ??, ${\displaystyle P_{n}}$??? ???????? n-?? ????? ??????????, ${\displaystyle P_{n}^{k}}$??? ???????? ???????? ?????????????, ${\displaystyle J_{n},C_{nm},S_{nm}}$??? ??????????? ?????????? ^[1] ?

??????????? ?????????? ? ?????? ??????????? ???????? [ ???????? | ???????? ???? ]

?????? ??????????? ????????? ???????? ?? ?????????? (1) ???????????????? ?????? ???????, ?????????? ????? ???????????? (2) ????????????????? m ?????????, a ????????, ???????? ???????? ??????????? ????????? ???? ????? ???????? ? ?????? U ??????????? ????????? ?????????????????

${\displaystyle U={\frac {-3Gm^{2}}{5a}}}$?

??????????? ?????????? ? ????????????????? ????????? ??????????? [ ???????? | ???????? ???? ]

????????????????? ????????? ????????? ??? ????????? ???? ??????? ??????????? ??????????? ??????? ????

${\displaystyle {\frac {d^{2}x^{i}}{ds^{2}}}+\Gamma _{rs}^{i}{\frac {dx^{r}}{ds}}{\frac {dx^{s}}{ds}}=0}$,

?????, ????? ${\displaystyle \Gamma _{rs}^{i}={\frac {g^{ik}}{2}}({\frac {dg_{kr}}{dx^{s}}}+{\frac {dg_{ks}}{dx^{r}}}-{\frac {dg_{rs}}{dx^{k}}})}$?? ?????????? ?????????? ??, ${\displaystyle g_{ik}}$?? ?????? ??????? ?, ??? ????????????????? ????????? ????????? ??? ??????????? ? ??????????? ??????

??? ??????? ?????????????? ??????????? ${\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}=-{\frac {d\varphi }{dx^{i}}}}$ ??????? ?????????????? ??? ??????? ???, ?? ${\displaystyle \varphi }$ ?????????? ?????????? ???? ????????????????? ????????? ????????? ??? ?????? ? ?????? ???????? ?????? ????????????? ???? ??????????????? ? ????? ????????? ??????????? ??????? ??????? ??????? ?????????????? ????????? ??

${\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}=-c^{2}\Gamma _{44}^{i}}$

????? ${\displaystyle i=1,2,3}$ ????????? ? ${\displaystyle x^{4}=ct}$ ??????????? ????????????? ?????? ????????? ??? ???????? ????????????, ${\displaystyle \Gamma _{44}^{i}}$?? ??????? ?????? ? ???????? ${\displaystyle -{\frac {1}{2}}{\frac {dg_{44}}{dx^{i}}}}$ ? ???????? ??????

${\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}=-{\frac {d\varphi }{dx^{i}}}}$

??????? ?????????? ??????????????? ?????? ${\displaystyle \varphi }$ ??????????? ?????????? ? ?????? ??????? ${\displaystyle g_{44}}$ ???????????? ?????? ??

${\displaystyle \varphi =-{\frac {1}{2}}c^{2}(g_{44}+1)}$, ${\displaystyle g_{44}=-\left(1+{\frac {2\varphi }{c^{2}}}\right)}$

?????????????????

???? ?? ?????? ???????? ?????? ?????????? ????????????? ???? ??????? ?

${\displaystyle ds^{2}=g_{44}(dx^{4})^{2}}$,

??? ?????????

${\displaystyle t={\frac {x^{4}}{c}}}$,

??? ?????????? ?????????? ??????????? ??????? ?????

${\displaystyle t_{g}={\frac {t}{\sqrt {-g_{44}}}}={\frac {t}{\sqrt {1+{\frac {2\varphi }{c^{2}}}}}}\approx t(1-{\frac {\varphi }{c^{2}}})}$?

??????????? ?????????? ????? ???? ??????? ?????? ???????? ??????? ??????????? ?????????? ????????? ????? ??? ??????????? ??????????? ???? ??? ??????? ? ??? ???????? ??????????? ?????????? ???????????? ? ?????? ?????????? ?????????? ?????????????? ^[2] ?

??? ??? [ ???????? | ???????? ???? ]

• ???????? ??????? ?????????? ??????????

?????????????????? [ ???????? | ???????? ???? ]

????????????? [ ???????? | ???????? ???? ]

• Ландау Л. Д., Лифшиц Е. М., ≪Теоретическая физика≫, учебное пособие для вузов, в 10 т. / т. 1, ≪Механика≫, 5-е изд., стереотип., М., ≪Физматлит≫, 2002, 224 с., ISBN 5-9221-0055-6 (т. 1), гл. 1 ≪Уравнения движения≫, п. 2 ≪Принцип наименьшего действия≫, с. 10-14;
• Ландау Л. Д., Лифшиц Е. М., ≪Теоретическая физика≫, уче. пособ. для вузов, в 10 т. / т. 2, ≪Теория поля≫, 8-е изд., стереотип., М., ≪Физматлит≫, 2001, 536 с., ISBN 5-9221-0056-4 (т. 2), гл. 10 ≪Частица в гравитационном поле≫, п. 81 ≪Гравитационное поле в нерелятивистской механике≫, с. 304?306; гл. 12 ≪Поле тяготеющих тел≫, п. 99 ≪Закон Ньютона≫, с. 397?401;
• С. Вейнберг, ≪Гравитация и космология≫, Принципы и приложения общей теории относительности, пер. с англ. В. М. Дубовика и Э. А. Тагирова, под ред. Я. А. Смородинского, ≪Платон≫, 2000, ISBN 5-80100-306-1 , ч. 2 ≪Общая теория относительности≫, гл. 3 ≪Принцип эквивалентности≫, п. 4 ≪Ньютоновское приближение≫, с. 92-93;
• К. В. Холшевников, И. И. Никифоров Свойства гравитационного потенциала в примерах и задачах: Учебное пособие. ? С-Пб., 2008. ? 72 с., ББК 22.6.
• Жарков В. Н. Внутреннее строение Земли и планет. ? М.: Наука, 1978. ? 192 с.

??????????? ???? ???????, ????? ?????????? ?? ≪??????????? ?????????≫ ?????????

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