\sqrt{r^2 + (vt)^2} = r \sqrt{1 + (vt/r)^2} = r + (1/2) (vt)^2 / r + o(t^2), так что вторая производная равна
v^2/r = (vr)^2 / r^3.
Числитель — (vr)^2 — это квадрат углового момента. Так что он вдоль орбиты всегда один и тот же! А знаменатель r^3 — как раз и соответствует закону всемирного тяготения: куб, как и раньше. Так что, если взять константу l = (vr)^2/ (GM), то для разницы (r-l) будет (r-l)’’ = r’’ = hr + (vr)^2 / r^3 = - GM r / r^3 + (vr)^2 / r^3 = - GM / r^3 * (r- l) = h (r-l).
Так что трёхмерный вектор R = (x,y,r-l) подчиняется центральному закону R’’ = h R, h = -GM/r^3.
\sqrt{r^2 + (vt)^2} = r \sqrt{1 + (vt/r)^2} = r + (1/2) (vt)^2 / r + o(t^2), так что вторая производная равна
v^2/r = (vr)^2 / r^3.
Числитель — (vr)^2 — это квадрат углового момента. Так что он вдоль орбиты всегда один и тот же! А знаменатель r^3 — как раз и соответствует закону всемирного тяготения: куб, как и раньше. Так что, если взять константу l = (vr)^2/ (GM), то для разницы (r-l) будет (r-l)’’ = r’’ = hr + (vr)^2 / r^3 = - GM r / r^3 + (vr)^2 / r^3 = - GM / r^3 * (r- l) = h (r-l).
Так что трёхмерный вектор R = (x,y,r-l) подчиняется центральному закону R’’ = h R, h = -GM/r^3.
Ура — теорема доказана!
BY Математические байки
Warning: Undefined variable $i in /var/www/group-telegram/post.php on line 260
"The result is on this photo: fiery 'greetings' to the invaders," the Security Service of Ukraine wrote alongside a photo showing several military vehicles among plumes of black smoke. "There is a significant risk of insider threat or hacking of Telegram systems that could expose all of these chats to the Russian government," said Eva Galperin with the Electronic Frontier Foundation, which has called for Telegram to improve its privacy practices. On Telegram’s website, it says that Pavel Durov “supports Telegram financially and ideologically while Nikolai (Duvov)’s input is technological.” Currently, the Telegram team is based in Dubai, having moved around from Berlin, London and Singapore after departing Russia. Meanwhile, the company which owns Telegram is registered in the British Virgin Islands. Groups are also not fully encrypted, end-to-end. This includes private groups. Private groups cannot be seen by other Telegram users, but Telegram itself can see the groups and all of the communications that you have in them. All of the same risks and warnings about channels can be applied to groups. Overall, extreme levels of fear in the market seems to have morphed into something more resembling concern. For example, the Cboe Volatility Index fell from its 2022 peak of 36, which it hit Monday, to around 30 on Friday, a sign of easing tensions. Meanwhile, while the price of WTI crude oil slipped from Sunday’s multiyear high $130 of barrel to $109 a pop. Markets have been expecting heavy restrictions on Russian oil, some of which the U.S. has already imposed, and that would reduce the global supply and bring about even more burdensome inflation.
from ms
