行星运动的Lagrange函数

动能项:

\[ T = \frac{1}{2}m(v_{x}^{2}+v_{y}^{2}) = \frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}) \]

势能项:

\[ V = -\frac{k^{2}m}{r}\hspace{0.4cm}(r = \sqrt{x^{2}+y^{2}}) \]

$k^{2}=Gm_{s}$是一个与行星无关的量,叫做高斯常数

则Lagrange函数:

\[ L=T-V=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}) + \frac{\alpha}{r}\hspace{0.4cm}(\alpha = Const) \]

Lagrange方程

将Lagrange函数带入到Lagrange方程,有:

\[ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0 \]

分别令$q=x,q=y$,即

\[ \begin{equation} \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x} = 0 \end{equation} \]
\[ \begin{equation} \frac{d}{dt}\frac{\partial L}{\partial \dot{y}} - \frac{\partial L}{\partial y} = 0 \end{equation} \]

运动方程

\[ \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x} = 0 \]

式中,$\frac{\partial L}{\partial \dot{x}} = m\dot{x}$, 则$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=m\ddot{x}$, $\frac{\partial L}{\partial x} = -\alpha x(x^{2}+y^{2})^{-\frac{3}{2}}$

则:

\[ \begin{equation} m\ddot{x} + \frac{\alpha x}{(x^{2}+y^{2})^{\frac{3}{2}}} = 0 \end{equation} \]

同理,有：

\[ \begin{equation} m\ddot{y} + \frac{\alpha y}{(x^{2}+y^{2})^{\frac{3}{2}}} = 0 \end{equation} \]

此即二维直角坐标系下的行星运动方程