As described in lectures, the Poisson Bracket [F, G] between two functions F and G of the generalized positions q; and momenta p; is defined as: [F, G] = OF OG _OG OF aq; ap; aq; api where the sum is over all the generalized coordinates/momenta. Consider a system with Hamiltonian H= P2/2m - y/r = (Px2+Py2+Pz2)/(2m) - y (x^2 + y^2 + z/2)-1/2 where y is a constant. a) Evaluate [Lz, H] and interpret the result in two ways i.e. what it says about Lz, and what it says about H. b) Using the Poisson Bracket and the given Hamiltonian, find the value of a that makes the quantity Px Z - ar a constant of the motion (i.e. invariant in time). Pand Z are respectively the momentum and angular momentum. The existence of this non-obvious conserved quantity implies there is a non-obvious symmetry in this system