How does a disk roll down an incline?

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SUMMARY

The discussion focuses on the dynamics of a uniform disk of mass m and radius R rolling down an incline at an angle θ. The Lagrangian is formulated as L = (m(a φ̇)^2)/2 + mgA sin(θ), where φ̇ represents the angular velocity and a is the radius R. Participants clarify that the Hamiltonian represents the total energy, H = T + U, and seek to understand Hamilton's equations of motion, specifically H = ṗ q - L. The conversation emphasizes the need to distinguish between translational and rotational kinetic energy in the context of rolling motion.

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Gogsey
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A uniform disk of mass m and radius R rolls without slipping down a ramp inclined at angle q to the horizontal.
Using the angle f through which it turns as a generalised coordinate, write the lagrangian, and then the Hamiltonian.
Write out and solve Hamilton’s equations of motion.

Ok, so I'm not really sure what to do here at all. Is ther 2 parts to the kinetic enerdy of the disk, one due to the disk spinning, and the other due to moving down the incline?
 
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So I have my Lagrangian as:

L = (m(a phidot)^2)/2 + mga Phi sin(theta)

I did use a an example to start me off so I'm not sure what a is. Is a just the radius, which would be R for this example?

Also, whta is the definition of the Hamiltonian? Is it just the total energy T + U?

Lastly, what is Hamilton's equation of motion? Is it just H = pdot q - L?
 

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