Consider N classical harmonic oscillators with coordinates and momenta {qi, p;}, and subject to the Hamiltonian N - 2 Pi [ + 2m 2 H(qi,pi) = mwq (a ) Calculate the entropy S as function of the total energy E using the microcanonical ensemble approach. (Hint. By appropriate change of scale the surface of constant energy can be deformed into a sphere. You may then ignore the difference between the surface area and volume for N 1. A more elegant method is to implement the deformation by a canonical transformation.) (b) Calculate the energy E, and heat capacity C, as functions of temperature T, and N. (c) Find the joint probability density P(q, p) for a single oscillator. Hence calculate the mean kinetic energy, and mean potential energy, for each oscillator.