A particle moves with a cylindrically symmetric potential energy \(U=U(ho, z)\) where \(ho, \varphi, z\) are cylindrical coordinates.

(a) Write the Lagrangian for an unconstrained particle of mass \(m\) in this case.

(b) Are there any cyclic coordinates? If so, what symmetries do they correspond to, and what are the resulting constants of the motion?

(c) Write the Lagrange equation for each cyclic coordinate.

(d) Find the Hamiltonian \(H\). Is it conserved?

(e) Find the total energy \(E\). Is \(E=H\) ? is \(E\) conserved?

(f) Write the simplest (i.e., lowest-order) complete set of differential equations of motion of the particle.