two nonholonomic particles in mutual potentialby coolnessitself Tags: mutual, nonholonomic, particles, potential 

#1
Nov1810, 01:37 PM

P: 32

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 radiallysymmetric 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_2x_1) = (y_2y_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_2x_1)(\ddot{x}_2  \ddot{x}_1) + (\dot{y}_2  \dot{y}_1)^2 + (y_2y_1)(\ddot{y}_2  \ddot{y}_1) + (\dot{z}_2  \dot{z}_1)^2 + (z_2z_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 headinganglepotential, I'd appreciate your input! 


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