Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Two nonholonomic particles in mutual potential

  1. Nov 18, 2010 #1
    Hi all,
    1. The problem statement, all variables and given/known data
    Not actually homework, but hopefully it fits well in this forum.
    I have two particles that can only move in the direction of their respective heading angles. I'm trying to relate a radially-symmetric mutual potential between the particles to the acceleration of this heading angle. For example,
    [tex]
    \begin{bmatrix}
    \dot{x}_i \\
    \dot{y}_i \\
    \dot{z}_i \\
    \dot{\theta}_i \\
    \dot{\Theta}_i \\
    \dot{\phi}_i \\
    \dot{\Phi}_i
    \end{bmatrix} = \begin{bmatrix} v\sin\theta_i\cos\phi_i \\ v\sin\theta_i\sin\phi_i \\ v\cos\theta_i \\ \Theta_i \\ f(r) \\ \Phi_i \\ g(r)\end{bmatrix}
    [/tex]
    for the distance r between 1 and 2. Normally we might say that [tex]\ddot{r} = -\nabla G(r)[/tex], but here in order for r to change, the heading angles must change. So my question is how to find f(r) and g(r) (the acceleration of the heading angles) so that the same behavior occurs as with [tex]\ddot{r} = -\nabla G(r)[/tex]. If we require that the direction of r is constant throughout this acceleration, ie [tex]\dot{r} = (\ldots) \hat{r} + 0\hat{\theta} + 0\hat{\phi}[/tex] then it seems to me like there should be a unique solution (if one exists).

    2. Relevant equations



    3. The attempt at a solution
    If the direction of r doesn't change,
    [tex] (\dot{z}_2 - \dot{z}_1)r = (z_2 - z_1)\dot{r}[/tex] and [tex] (\dot{y}_2 - \dot{y}_1)(x_2-x_1) = (y_2-y_1)(\dot{x}_2 - \dot{x}_1)[/tex]. My hope would be to then expand out [tex]\ddot{r}[/tex] until equations for [tex]\ddot{\theta}[/tex] and [tex]\ddot{\phi}[/tex] appear, then use that condition to find one solution. Something like
    [tex] (\dot{r})^2 + r\ddot{r} = (\dot{x}_2 - \dot{x}_1)^2 + (x_2-x_1)(\ddot{x}_2 - \ddot{x}_1) + (\dot{y}_2 - \dot{y}_1)^2 + (y_2-y_1)(\ddot{y}_2 - \ddot{y}_1) + (\dot{z}_2 - \dot{z}_1)^2 + (z_2-z_1)(\ddot{z}_2 - \ddot{z}_1)[/tex]
    But after plugging in [tex]\dot{x}[/tex], etc. with phis and thetas, It doesn't seem to lead anywhere.
    So I get the feeling I'm going about this incorrectly. Physics gurus, if you have any suggestions for an alternative approaches to finding this heading-angle-potential, I'd appreciate your input!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted