As described in class, the poisson bracket [f, g] between two functions f and g of the generalized
positions qi and momenta pi is defined as:
[f, g] = ∑i� ∂f/∂qi ∂g/∂pi- ∂g/∂qi ∂f∂pi
consider a system with hamiltonian h= p2 / 2m - γ/r = (px^2+py^2+pz^2) / (2m) - γ (x ^2 + y^2 + z^2) -1/2
where γ 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 α that makes the quantity
pxl-at a constant of the motion (i. e. invariant in time).

Aparticle's position is given by x = 3.00 - 9.00t + 3t2, in which x is in meters and t is in seconds. (a) what is its velocity at t = 1 s? (b) is it moving in the positive or negative direction of x just then? (c) what is its speed just then? (d) is the speed increasing or decreasing just then? (try answering the next two questions without further calculation.) (e) is there ever an instant when the velocity is zero? if so, give the time t; if not, answer "0". (f) is there a time after t = 3 s when the particle is moving in the negative direction of x? if so, give the time t; if not, answer "0".