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Lagrange for uniform thin disk

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Derive Lagrange’s equations and find the generalized for uniform thin disk that rolls without slipping on a horizontal plane.
Classical Mechanics - Derive Lagrange’s equations and find the generalized force. Classical Mechanics - Derive Lagrange’s equations and find the generalized for uniform thin disk force.
Consider a uniform thin disk that rolls without slipping on a horizontal plane. A horizontal force is applied to the center of the disk and in a direction parallel to the plane of the disk.
  • Derive Lagrange’s equations and find the generalized force ?
  • Discuss the motion if the force is not applied parallel to the plane of the disk ?

(Source: Herbert Goldstein, Classical Mechanics - Chapter 01)
 
To find Lagrangian’s equations, we need to first find the Lagrangian.
Lagrangian



Therefore

Plug into the Lagrange equations:





If the motion is not applied parallel to the plane of the disk, then there might be some slipping, or another generalized coordinate would have to be introduced, such as theta to describe the y-axis motion. The velocity of the disk would not just be in the x-direction as it is here.
 
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