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Constants Of Motion

  1. Nov 27, 2007 #1
    [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, [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..
  2. jcsd
  3. Nov 27, 2007 #2

    Chris Hillman

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    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.
  4. Nov 27, 2007 #3


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    Hint: Poisson bracket
  5. Nov 27, 2007 #4
    Thanks.Can you please give me a brief account of the methods for identifying a constant of motion.
  6. Nov 27, 2007 #5


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    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} ?
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