I derived the equations of motion for a particle constrained on the surface of a sphere Parametrizing the trajectory as a function of time through the usual $\theta$ and $\phi$ angles, these equations read:
$$ \ddot{\theta} = \dot{\phi}^2 \sin \theta \cos \theta $$
$$ \ddot{\phi} = – 2 \dot{\phi} \dot{\theta} \frac{1}{\tan \theta} $$
I've obtained them starting from the Lagrangian of the system and using the Euler-Lagrange equations.
My question is simple: is there a way (a clever substitution, maybe), to go on and solve the differential equations? I would be interested even in a simpler, partially integrated solution. Or is a numerical solution the only way?
Best Answer
Note that you can rewrite your second equation as $$ \frac{\ddot{\phi}}{\dot{\phi}} = -2\cot{(\theta)}\dot{\theta} $$ Each side is an exact differential in one variable, so we can integrate, and Wolfram|Alpha gives $$ \ln{(\dot{\phi})}=-2\ln{(\sin{(\theta)})}+C $$ for some integration constant $C$. We can exponentiate to get $$ \dot{\phi}=\frac{B}{\sin{(\theta)}^2} $$
Substituting this into the first equation yields $$ \ddot{\theta}=B^2\frac{\cos{(\theta)}}{\sin{(\theta)}^3} $$ This, too, can be integrated via the "energy trick": multiply by $ \theta $, then integrate. The LHS integrates by parts to $\dot{\theta}^2$ but the RHS looks sufficiently complicated I don't want to type it out on my phone.