How do I use the Lagrangian to determine an orbital path?

Question

Ideally, I'd like to find some polar function for the orbit of a point particle that takes time as its argument, and for given boundary conditions (e.g.: initial radial velocity of particle) will give future radial displacement (etc.) in polar co-ordinates. Here's what I have for the equations of my two degrees of freedom ($r(t)$ and $\theta(t)$), using Lagrange's equation $\frac{d}{dt}\left[\frac{\partial L}{\partial\dot q_i}\right]=\frac{\partial L}{\partial q_i}$:

$$\ddot\theta r+2\dot\theta\dot r=0$$ $$\ddot r=r\dot\theta^2+\frac{GM}{r^2}$$

Does anyone know how I might go about solving these differential equations? Solutions would be helpful, especially those leaving boundary conditions as clearly expressed variables.

Answer 1

There is no closed form solution to these coupled differential equations.

The best you can do is combine them to get an analytical expression for $r$ in terms of $\theta$, producing the familiar equations for parabolae, ellipses, or hyperbolae, defining the orbit's shape. The remainder is found using some form of numerical integration on one of the differential equations for $r(t)$ or $\theta(t)$, with the constraint from the orbit's shape.