Details

Lagrangian mechanics can be used to derive the equations of motion [1].

Potential energy:

Kinetic energy:

Lagrangian:

Here is the interpolating polynomial representing the curve of the track, is the position of the object at time , is the mass of the object, is the Lagrangian multiplier, and is the damping coefficient.

This gives the equations of motion

,

,

subject to the constraint .

Reference

[1] S. Timoshenko and D. H. Young, Advanced Dynamics, New York: McGraw–Hill, 1948 p. 281 (eq. 150).