Lagrange multipliers

Problem:

A heavy particle with mass m is placed on top of a vertical hoop.  Calculate the reaction of the hoop on the particle by means of the Lagrange undetermined multipliers and Lagrange's equations.  Find the height at which the particle falls off.

Solution:

Problem:

A bead of mass m slides without friction on a circular loop of radius a.  The loop lies in a vertical plane.

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(a)  Use the Lagrange multiplier method and find the appropriate Lagrangian including terms expressing the constraints. 
(b)  Apply the Euler-Lagrange equations to obtain the equations of motion and solve for θ << 1.
(c)  Find the force of constraint.

Solution:

Problem:

Two masses m1 and m2 are connected by a massless string that runs over a frictionless pulley.  The length of the string, l, somehow increases at a constant rate, i.e. l(t) = l0 + l1 t.  Use the method of Lagrange multipliers to determine the tension of the string at time t.

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Solution:

Problem:

Consider a particle of mass m constrained to remain on the surface of a cylinder of radius b.   Let the axis of the cylinder be the z-axis.  The particle is subject to the force of gravity mg in the negative z-direction. 
(a)  Use the Lagrange multiplier method and find the appropriate Lagrangian including a term expressing the constraint. 
(b)  Apply the Euler-Lagrange equations to obtain the equations of motion. 
(c)  Find the force of constraint and briefly discuss why this value is reasonable and to be expected.
(d)  Now, repeat parts (a) and (b) without using the Lagrange multiplier method.  Instead, build the constraint into the general coordinates.

Solution: