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Logarythmic
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Homework Statement
Using spherical coordinates (r, \theta, \phi), obtain the Hamiltonian and the Hamilton equations of motion for a particle in a central potential V(r).
Study how the Hamilton equations of motion simplify when one imposes the initial conditions p_{\phi}(0) = 0 and \phi (0) = 0
The Attempt at a Solution
I have obtained a Hamiltonian
H = \frac{1}{2m} \left( p_r^2 + \frac{p_{\theta}^2}{r^2} + \frac{p_{\phi}^2}{r^2 \sin^2{(\theta)}} \right) + V(r)
and from this also the equations of motion
\dot{r} = \frac{p_r}{m}
\dot{\theta} = \frac{p_{\theta}}{mr^2}
\dot{\phi} = \frac{p_{\phi}^2}{r^2 \sin^2{(\theta)}}
m \ddot{r} = \frac{1}{m} \left( \frac{p_{\theta}^2}{r^3} + \frac{p_{\phi}^2}{r^3 \sin^2{(\theta)}} \right) - \frac{\partial V}{\partial r}
m^2 \left( 2r \dot{r} \dot{\theta} + r^2 \ddot{\theta} \right) = \frac{p_{\phi}^2 \cos{(\theta)}}{r^3 \sin^3{(\theta)}}
2 \dot{r} \sin{(\theta)} \dot{\phi} + r \left( 2 \cos{(\theta)}\dot{\theta} \dot{\phi} + \sin{(\theta)} \ddot{\phi} \right) = 0
But how should I proceed with the last part of the problem?
