Two nonholonomic particles in mutual potential

[coolnessitself]

Hi all,

Homework Statement

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,
$$\begin{bmatrix} \dot{x}_i \\ \dot{y}_i \\ \dot{z}_i \\ \dot{\theta}_i \\ \dot{\Theta}_i \\ \dot{\phi}_i \\ \dot{\Phi}_i \end{bmatrix} = \begin{bmatrix} v\sin\theta_i\cos\phi_i \\ v\sin\theta_i\sin\phi_i \\ v\cos\theta_i \\ \Theta_i \\ f(r) \\ \Phi_i \\ g(r)\end{bmatrix}$$
for the distance r between 1 and 2. Normally we might say that $$\ddot{r} = -\nabla G(r)$$, 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 $$\ddot{r} = -\nabla G(r)$$. If we require that the direction of r is constant throughout this acceleration, ie $$\dot{r} = (\ldots) \hat{r} + 0\hat{\theta} + 0\hat{\phi}$$ then it seems to me like there should be a unique solution (if one exists).

Homework Equations

The Attempt at a Solution

If the direction of r doesn't change,
$$(\dot{z}_2 - \dot{z}_1)r = (z_2 - z_1)\dot{r}$$ and $$(\dot{y}_2 - \dot{y}_1)(x_2-x_1) = (y_2-y_1)(\dot{x}_2 - \dot{x}_1)$$. My hope would be to then expand out $$\ddot{r}$$ until equations for $$\ddot{\theta}$$ and $$\ddot{\phi}$$ appear, then use that condition to find one solution. Something like
$$(\dot{r})^2 + r\ddot{r} = (\dot{x}_2 - \dot{x}_1)^2 + (x_2-x_1)(\ddot{x}_2 - \ddot{x}_1) + (\dot{y}_2 - \dot{y}_1)^2 + (y_2-y_1)(\ddot{y}_2 - \ddot{y}_1) + (\dot{z}_2 - \dot{z}_1)^2 + (z_2-z_1)(\ddot{z}_2 - \ddot{z}_1)$$
But after plugging in $$\dot{x}$$, 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!

 

[mark726]

Hi there,

Thank you for posting your question on this forum. I can offer some suggestions for finding the heading-angle potential that you are looking for.

Firstly, it seems like you are on the right track with your attempt at a solution. However, instead of expanding out \ddot{r}, I would suggest using the chain rule to find expressions for \ddot{\theta} and \ddot{\phi}. You can start by writing out the expressions for \dot{\theta} and \dot{\phi} in terms of \dot{x}, \dot{y}, and \dot{z}, and then taking the derivatives with respect to time. This should give you equations for \ddot{\theta} and \ddot{\phi} in terms of \ddot{x}, \ddot{y}, and \ddot{z}. Then, you can use the equations you have already derived for \dot{x}, \dot{y}, and \dot{z} to substitute these values in and ultimately find expressions for \ddot{\theta} and \ddot{\phi} in terms of \ddot{r}.

Once you have these expressions, you can use them to find the acceleration of the heading angles as a function of the radial distance between the particles. This can be done by equating the equations you have derived for \ddot{\theta} and \ddot{\phi} to the equations you have for the acceleration of the heading angles in your original post. This should give you a system of equations that you can solve for f(r) and g(r).

I hope this helps guide you in finding the solution you are looking for. Let me know if you have any further questions or if you need any clarification. Good luck!