Some particle of mass m is constrained to move on the surface of a vertical cylinder defined by 2 2 2 x ? y ? R , where R is the radius of the cylinder. The particle is subject to a force directed towards the origin and proportional to the distance of the particle from the origin F ?kr. The particle is also subject to gravitational force.
a) Find Lagrangian for this particle.
b) Obtain Lagrange’s equations of motion for this particle.
c) Find Hamiltonian for this particle.
d) Obtain Hamilton’s canonical equations of motion for this particle
e) Show that the angular momentum of this particle about the z-axis is conserved.
f) Use Lagrange’s multiplier method to find the force of the constrain acting on this particle.
g) Find the equilibrium position for this particle.
h) Find the period of small oscillations of this particle near the equilibrium position.