1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lagrange - Mass under potential in spherical

  1. Aug 9, 2009 #1

    CNX

    User Avatar

    1. The problem statement, all variables and given/known data

    A particle of mass [itex]m[/itex] moves in a force field whose potential in spherical coordinates is,

    [tex]U = \frac{-K \cos \theta}{r^3}[/tex]

    where [itex]K[/itex] is constant.

    Identify the two constants of motion of the system.

    3. The attempt at a solution

    [tex]L = T - V = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta ~\dot{\phi}^2) + \frac{K \cos \theta}{r^3}[/tex]

    I don't see how there are two constants of motion if the Lagrangian is missing only [itex]\phi[/itex], i.e.,

    [tex]\frac{ \partial L}{\partial \phi} = 0 \Rightarrow \frac{\partial L}{\partial \dot{\phi}} = constant[/tex]
     
  2. jcsd
  3. Aug 10, 2009 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    I'm not 100% sure that this is what the questioner has in mind, but I can think of one quantity that is always a constant of motion whenever the Lagrangian has no explicit time dependence....:wink:
     
  4. Aug 11, 2009 #3

    CNX

    User Avatar

    Energy function/Hamiltonian?

    [tex]\frac{\partial L}{\partial t} = 0 = - \frac{dH}{dt}[/tex]

    So H = constant.
     
  5. Aug 11, 2009 #4

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Lagrange - Mass under potential in spherical
Loading...