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    A bead of mass m slides without friction on a ring

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    A bead of mass m slides without friction on a ring. The ring rotates with constant angular velocity w about a rotational axis that is aligned with a ring diameter, as shown in Figure 1. Find the Lagrange equations of motion, and the Hamiltonian for the bead. Is the Hamiltonian a constant of motion? Does it coincide with the energy of the system? Interpret the Hamiltonian as the sum of the kinetic energy of a bead rotating on a fixed ring and an effective potential resulting from the gravitational potential and a centrifugal potential. Plot the effective potential as a function of the angle theta for different values of w and discuss the motion of the bead for all cases. Under which conditions does the bead reach an equilibrium condition where theta remains constant?

    See attached file for diagrams.

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    In spherical coordinates, the bead speed is:

    And:

    Thus, the energies of the system are:
    (1.1)
    (1.2)
    And the Lagrangian is:

    (1.3)

    There is only one generalized coordinate (q), so we get only one equation of motion:

    (1.4)
    The Associated momentum is:
    (1.5)
    consequently,
    (1.6)
    The system's Hamiltonian is:

    (1.7)

    Recall the commuting relations between the Hamiltonian and system's parameter A:

    Therefore:

    But our Hamiltonian is not explicitly dependent on time, so we get

    Which means that the Hamiltonian is a constant of motion and it represents the system energy.
    It has two components:
    • A kinetic energy of a bead rotating on a fixed ring: ...

    Solution Summary

    This solution provides step by step calculations for a system where a bead of mass m slides without friction on a ring.

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