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Sliding on a Parabolic Track

This Demonstration shows an object sliding with damping on a parabolic track with equation . It explores the effect of the damping coefficient and the parameter on the swing period of the object and the constraining force that keeps it on the track.

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This Demonstration was inspired by a question in the Wolfram Community site, "Simulating Mechanics of a Cylinder Rolling on a Parabola".
Lagrangian mechanics can be used to derive the equations of motion [1].
The potential energy and kinetic energy are repectively
,
,
where is the position of the object at time , is the mass of the object, is the Lagrange multiplier, and is the damping coefficient.
This gives the equations of motion:
,
,
.
Reference
[1] S. Timoshenko and D. H. Young, Advanced Dynamics Chapter III, p. 281, Lagrangian Equations for Impulsive Forces.
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