Solving Constants of Motion for Particle in 3D - No Quotes

[abeen]

[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..

 

[Chris Hillman]

Hi, abeen,

Welcome to PF.

From the way you phrased your questions, it sounds like you are seeking advice for how to attack a homework problem. If so, there's a special forum at PF for that (look up above for the "sticky"). If not, we can help you right here.

 

[siddharth]

. I don't have much idea about the rest. Do u people have any suggestions? Thanks in advance..

Hint: Poisson bracket

 

[abeen]

Hint: Poisson bracket

Thanks.Can you please give me a brief account of the methods for identifying a constant of motion.

 

[siddharth]

Thanks.Can you please give me a brief account of the methods for identifying a constant of motion.

That was the hint. If F(p,q) is a constant of motion and H is the hamiltonian, what can you conclude about {H,F} ?