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!

Homework Help: Relation between force in cartesian, polar.

  1. Mar 18, 2010 #1
    Goldstein(3rd) 1.15

    Generalized potential, U as follows.

    [tex] U( \stackrel{\rightarrow}{r} ,\stackrel{\rightarrow}{v})=V(r)+\sigma\cdot L[/tex]

    L is angular momentum and [tex]\sigma[/tex] is a fixed vector.


    (b) show thate the component of the forces in the two coordinate systems(cartesin, spherical polar) are related to each other as
    [tex]Q_{j}=F_{i}\cdot \frac{\partial r_{i}} {\partial q_{j}} \cdots (a) [/tex]

    So I did,
    [tex] Q_{j}= - \frac{\partial U}{\partial q_{j}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot q_{j}})[/tex]

    [tex]
    =- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial \dot x_{k}}{\partial \dot q_{j}} \frac{\partial U}{\partial \dot x_{k}})
    [/tex]

    [tex]
    =- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial \dot x_{k}})
    [/tex]

    [tex]
    =- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}+ \frac{\partial x_{k}}{\partial q_{j}} \frac{d}{dt}(\frac{\partial U}{\partial \dot x_{k}})
    [/tex]

    [tex]=\frac{\partial x_{k}}{\partial q_{j}}(
    - \frac{\partial U}{\partial x_{k}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot x_{k}}))+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}
    [/tex]

    [tex]=\frac{\partial x_{k}}{\partial q_{j}} (
    F_{k})+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}
    [/tex]



    If the last term in the last line vanishes, [tex] Q_{j} [/tex] and [tex] F_{k} [/tex] satisfies the relation (a), but it DOESN't vanish. What's my problem??
    I've evaluated the last term in this condition, but It doesn't....
     
    Last edited: Mar 19, 2010
  2. jcsd
  3. Mar 19, 2010 #2
    Replace x with r:

    [tex]\frac{d}{dt}(\frac{\partial r_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot r_{k}}[/tex]

    Does U change with respect to r dot?
     
  4. Mar 19, 2010 #3
    Yes, There's L(angular momentum) in the U
     
  5. Mar 19, 2010 #4
    Explicitly write the angular momentum terms and see if any have r dot terms.
     
  6. Mar 20, 2010 #5
    Isn't x the component cartesian coordinate?
     
  7. Mar 20, 2010 #6
    You could have started the problem as this:

    [tex] Q_{r}= - \frac{\partial U}{\partial r} + \frac{d}{dt} (\frac{\partial U}{\partial \dot r})[/tex]

    and expressed the generalized potential as this:

    [tex] U( \stackrel{\rightarrow}{r} ,\stackrel{\rightarrow}{v})=V(r)+\sigma_{\theta}\L_{\theta}+\sigma_{\phi}\L_{\phi}[/tex]

    Express the L terms explicitly in terms of m, r, theta, phi, and sigma and the second term on the right side of the generalized force equation will equal zero.
     
    Last edited: Mar 20, 2010
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook