Rolling Object on Curved Surface: Lagrangian Mechanics + Constraint

[physicsnewblol]

Homework Statement

I want to be able to plot a trajectory wrt time of a ball that rolls without slip on a curved surface.

Known variables:
-radius/mass/moment of inertia of the ball.
-formula for the curvature of the path (quadratic)
-formula relating path length and corresponding height above the ground. (linearly decreasing, i.e. as you travel down the curve, you decrease in height).

Homework Equations

Euler-Lagrange Equations

The Attempt at a Solution

See attached for further details.

I've come up with an answer for the acceleration of the ball along the path length. However, it isn't a function of the starting height as I expected. I suspect I may have made an assumption that I'm not supposed to but I can't see what it is.

 

[physicsnewblol]

After some though and comparison to classical mechanics, I've determined that the assumption that the height of the ball decreases linearly with arc length is not correct.

It seems my potential term is going to be some complicated function of arc length.

Anyone else have thoughts?