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abeen
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[SOLVED] Constants Of Motion
A Particle of mass m moves in three dimensions under the action of a conservative force with potential energy V(r).Using the sperical coordinates r, \theta,\phi, obtain the hamiltonian function for the system.
Show that P_{\phi} , \frac{P^{2}_{r}}{2m} + \frac{P^{2}_{\phi}}{2mr^{2}sin^{2}\theta} + V(r) and P^{2}_{\theta} + \frac{P^{2}_{\phi}}{sin^{2}\theta} are constants of motion.
I found the hamiltonian, H = \frac{P^{2}_{r}}{2m} + \frac{P^{2}_{\theta}}{2mr^{2}} + \frac{P^{2}_{\phi}}{2mr^{2}sin^{2}\theta} + V(r).
Since \phi is cyclic we have P_{\phi}^{'}=0 or P_{\phi} is a constant of motion. I don't have much idea about the rest. Do u people have any suggestions? Thanks in advance..
A Particle of mass m moves in three dimensions under the action of a conservative force with potential energy V(r).Using the sperical coordinates r, \theta,\phi, obtain the hamiltonian function for the system.
Show that P_{\phi} , \frac{P^{2}_{r}}{2m} + \frac{P^{2}_{\phi}}{2mr^{2}sin^{2}\theta} + V(r) and P^{2}_{\theta} + \frac{P^{2}_{\phi}}{sin^{2}\theta} are constants of motion.
I found the hamiltonian, H = \frac{P^{2}_{r}}{2m} + \frac{P^{2}_{\theta}}{2mr^{2}} + \frac{P^{2}_{\phi}}{2mr^{2}sin^{2}\theta} + V(r).
Since \phi is cyclic we have P_{\phi}^{'}=0 or P_{\phi} is a constant of motion. I don't have much idea about the rest. Do u people have any suggestions? Thanks in advance..
