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.
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 . 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:
 S. Timoshenko and D. H. Young, Advanced Dynamics Chapter III, p. 281, Lagrangian Equations for Impulsive Forces.