Assignment 7

Problem 1:

A particle of mass m is described by the Lagrangian function

L = ½m[(dx/dt)2 + dy/dt)2 + (dz/dt)2] + ½ωl3,

where l3 is the z-component of the angular momentum and ω is a constant angular frequency.
(a)  Find the equations of motion, write them in terms of the variables (x + iy) and z and solve them.
(b)  Construct the Hamiltonian function and find the kinematic and canonical momenta. 
Show that the particle has only kinetic energy and that the latter is conserved.

Problem 2:

A string of length 2l is suspended at points A and B located on a horizontal line.  The distance between A and B is 2d (d < l).  A small, heavy bead can slide on the string without friction.  Find the period of the small-amplitude oscillations of the bead in the vertical plane containing the suspension points.  The acceleration due to gravity is g.

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Problem 3:

Consider a pendulum in a plane (i.e. a "2D world"), consisting of a mass m attached at the end of a weightless rope of length l0.  When the pendulum is set into motion the length of the rope is shortened at a constant rate dl/dt = -α = constant.  Compute the Lagrangian, write down the equation of motion, and discuss the conservation of energy for this system.  Does the sign of α matter for energy conservation.

Problem 4:

A simple pendulum of mass m2 and length l is constrained to move in a single plane.  The point of support is attached to a mass m1 which can move on a horizontal line in the same plane.  
(a)  Find the Lagrangian of the system in terms of suitable generalized coordinates.
(b)  Derive the equations of motion.
(c)  Find the frequency of small oscillations of the pendulum.

Problem 5:

Consider a system consisting of a mass m, a spring, and a rigid, massless lever arm of length L with one end fixed at the origin.  The spring has unstretched length l0 with force constant k and joins mass m with the lever arm.  The entire assembly rests on a frictionless surface.

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(a)  Calculate the Lagrangian function in terms of the lengths L and l and the angles θ and φ and their derivatives.
(b)  Now impose the additional constraints that φ = π/2.  (One can do this by making m move in a track attached to L.)  Obtain the equation of motion for this constrained system.
(c)  Consider the case where d2θ/dt2 = 0.  Describe the motion of m.