A particle of mass $m$ is constrained to move without friction on the inner surface of a revolution paraboloid under the action of gravity. The equation of this revolution paraboloid is $x^2 + y^2 = az$, where $a$ is a constant (gravity acts in the direction of negative $z$).
a) Write the Lagrangian of the system using cylindrical coordinates $\rho$ and $\phi$ as generalized coordinates, which will involve a cyclic coordinate. Then obtain the equations of motion, one of which should be homogeneous (i.e., involving only one coordinate).
b) Show that there exists a solution where the particle moves at $z > 0$ constant and find the values of $\rho$ and $\phi$ for this solution. Consider a small perturbation from this stable motion ($\rho = \rho_0 + \eta(t)$ and $\phi = \phi_0 + \omega(t)$ where $\eta << 1$ and $\omega << 1$), and solve completely for the particle's motion in this regime. What does this motion consist of?
