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Logarythmic
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Homework Statement
Using spherical coordinates [tex](r, \theta, \phi)[/tex], 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 [tex]p_{\phi}(0) = 0[/tex] and [tex]\phi (0) = 0[/tex]
The Attempt at a Solution
I have obtained a Hamiltonian
[tex]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)[/tex]
and from this also the equations of motion
[tex]\dot{r} = \frac{p_r}{m}[/tex]
[tex]\dot{\theta} = \frac{p_{\theta}}{mr^2}[/tex]
[tex]\dot{\phi} = \frac{p_{\phi}^2}{r^2 \sin^2{(\theta)}}[/tex]
[tex]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}[/tex]
[tex]m^2 \left( 2r \dot{r} \dot{\theta} + r^2 \ddot{\theta} \right) = \frac{p_{\phi}^2 \cos{(\theta)}}{r^3 \sin^3{(\theta)}}[/tex]
[tex]2 \dot{r} \sin{(\theta)} \dot{\phi} + r \left( 2 \cos{(\theta)}\dot{\theta} \dot{\phi} + \sin{(\theta)} \ddot{\phi} \right) = 0[/tex]
But how should I proceed with the last part of the problem?