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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, [tex]\theta[/tex],[tex]\phi[/tex], obtain the hamiltonian function for the system.

Show that [tex]P_{\phi}[/tex] , [tex]\frac{P^{2}_{r}}{2m}[/tex] + [tex]\frac{P^{2}_{\phi}}{2mr^{2}sin^{2}\theta}[/tex] + V(r) and [tex]P^{2}_{\theta}[/tex] + [tex]\frac{P^{2}_{\phi}}{sin^{2}\theta}[/tex] are constants of motion.

I found the hamiltonian, H = [tex]\frac{P^{2}_{r}}{2m}[/tex] + [tex]\frac{P^{2}_{\theta}}{2mr^{2}}[/tex] + [tex]\frac{P^{2}_{\phi}}{2mr^{2}sin^{2}\theta}[/tex] + V(r).

Since [tex]\phi[/tex] is cyclic we have [tex]P_{\phi}^{'}[/tex]=0 or [tex]P_{\phi}[/tex] is a constant of motion. I dont have much idea about the rest. Do u people have any suggestions? Thanks in advance..

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# Homework Help: Constants Of Motion

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