物理

❴栖岸计划❵Binet's formula

有助于英语学习的推导，(可能觉得洋文会更正宗吧。原稿放结尾了

Binet's Formula

$m(\ddot{r} - r\dot{\theta}^2) = F_r = -\frac{Gmm}{r^2}$

$m(r^2\ddot{\theta} + 2r\dot{r}\dot{\theta}) = 0 \quad \text{[conservation of angular momentum]}$

即 $\beta = 0$

Also, we have  

$L = m r^2 \dot{\theta} = \text{const.} \quad \text{[same thing]}$

$\frac{d\theta}{dt} = \frac{L}{mr^2}, \quad dt = d\theta \cdot \frac{mr^2}{L}$

Then, we got  

$m \frac{d}{d\theta} \left( \frac{dr}{dt} \cdot \frac{1}{mr^2} \right) - \frac{L^2}{mr^2} - mr \cdot \frac{L^2}{m^2 r^4} = -\frac{Gmm}{r^2}$

$\frac{dr}{d\theta} \cdot \frac{L}{mr^2} = -\frac{L}{m} \cdot \frac{d}{d\theta} \left( \frac{1}{r} \right)$

$\frac{d^2(1/r)}{d\theta^2} + \frac{1}{r} = -\frac{Gmm^2}{L^2}$

$-\frac{d^2(1/r)}{d\theta^2} - \frac{1}{r} = -\frac{Gmm^2}{L^2}$

Then, we let  

$\frac{1}{r} = u$

$-\frac{d^2 u}{d\theta^2} - u = -\frac{Gmm^2}{L^2} \quad \text{Common Binet's Formula}$

$\frac{d^2 u}{d\theta^2} = \frac{Gmm^2}{L^2} - u$

It's easy to found out this formula's characteristic roots  

$u = \frac{Gmm^2}{L^2} + A \cos(\theta - \theta_0)$

$r = \frac{\frac{L^2}{Gmm^2}}{1 + \frac{L^2}{Gmm^2} A \cos(\theta - \theta_0)}$

Let  $ep = \frac{L^2}{Gmm^2}, \quad A = \frac{1}{p}$

$r = \frac{ep}{1 + e \cos(\theta - \theta_0)}$

It's easy to prove Kepler's idea: the route of planets is a ellipse

Tips:  

$\frac{a^3}{T^2} = \frac{G(M + m_0)}{4\pi^2}$

For all closing route