# A bead of mass m slides without friction on a ring

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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#### Solution Preview

See attached file for solution.

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:

â€¢ 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.