A spherical pendulum consists of a mass (m) attached by a rigid rod of negligible mass of length, L, to a fixed point in

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A spherical pendulum consists of a mass (m) attached by a rigid
rod of negligible mass of
length, L, to a fixed point in space in the presence of a
gravitational field, g. This constrains
the mass to move on the surface of a sphere of radius, L. Use the
spherical coordinates θ and
φ as your generalized coordinates with the gravitational field
pointing along the negative z
axis.
(a) Find the Lagrangian of the system
(b) Identify any conserved quantities which you can deduce directly
from the form of the
Lagrangian. If the conserved quantity has physical significance,
tell me what it is. For
example, if a conserved quantity were to represent the total linear
momentum in the x
direction, then you should tell me that.
(c) Find the equations of motion of the system.
(d) Find the Hamiltonian of the system.
(e) Is the Hamiltonian equal to the mechanical energy of the
system? If not, explain why.
(f) Is the mechanical energy conserved? Explain.
(g) Is the Hamiltonian conserved? Explain.