A particle of mass m is described by the Lagrangian function

L = ½m[(dx/dt)^{2} + dy/dt)^{2} + (dz/dt)^{2}] +
½ωl_{3},

where l_{3} 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.

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.^{ }

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 l_{0}. 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.

A simple pendulum of mass m_{2}
and length l is constrained to move in a single plane. The point of
support is attached to a mass m_{1} 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.

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 l_{0}
with force constant k and joins mass m with the lever arm.
The entire assembly rests on a frictionless surface.

(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
d^{2}θ/dt^{2} = 0. Describe the motion of m.