How Can Friction Be Incorporated into a Brachistochrone Model?

[hy23]

Hello all, I'm currently an undergrad in my first year doing an experiment with the brachistochrone and I wish to model it mathematically and introduce friction into my model. I understand that the original brachistochrone was solved using the action principle and seems to require Lagrangian mechanics. I just want to know what sort of equations I would have to consider in my model and whether or not this will be beyond my level.

 

[Dickfore]

Let $x = x(p), y = y(p)$ be a parametric form of the line. Then, the arc length is:

$$ds = \sqrt{\dot{x}^{2} + \dot{y}^{2}} \, dp$$

The unit tangent vector is:

$$\hat{T} = \langle \frac{d x}{d s}, \frac{d y}{d s}\rangle$$

and the curvature $\kappa$ and the unit normal is given by:

$$\kappa \, \hat{N} = \langle \frac{d^{2} x}{d s^{2}}, \frac{d^{2} y}{d s^{2}}\rangle$$

The equations of motion become ($v = ds/dt$):

$$m \, (\mathbf{g}\cdot \hat{T}) - F_{\mathrm{fr}} = m \frac{d v}{d t}$$

$$F_{N} + m (\mathbf{g} \cdot \hat{N}) = m \, \kappa \, v^{2}$$

$$F_{\mathrm{fr}} = \mu \, |F_{N}|$$

 

[hy23]

Thanks for your reply.

I understand the equations of motion but the solution seems to be beyond my level.
Is there a way to solve it numerically? Preferably some simple method.

 

[Dickfore]

I don't know. I tried it myself several times.