Consider a particle of mass m moving in a vertical x-y plane along a curve y=acos(2xπ/λ). free fall acceleration is g. First consider two coordinates x and y:
a) write down lagrangian of system
b) obtain lagrange's equation of motion with undetermined multipliers.
c) define the forces of the constraint. what is the minimal velocity of a particle at the top of a curve when the force of constrain vanishes? compare your expressions for the forces of constraint with the forces you would introduce to formulate the problem in terms of newtons second law.
Now choose a single generalized coordinate corresponding to a horizontal coordinate x.
d) write down the lagrangian of the particle and lagrange's equations of motion.
e) Define the generalized momentum and write down the hamiltonian of the particle.
f) derive the canonical(Hamilton's) equation of motion.
g) demonstrate that the canonical equations of motions can be reduced to lagrange's equation of motion.