The Hamilton-Jacobi theory to be presented in Chapter 19 allows us to find generating functions that

The Hamilton-Jacobi theory to be presented
in Chapter 19 allows us to find generating functions that simplify particular
Hamiltonians. The harmonic oscillator in one dimension has an extended
Hamiltonian in the q, p system

(a) Find the transformed momenta P0(q, p) and P1(q, p), and demonstrate
that, when expressed in the Q, P system, the extended Hamiltonian becomes
simply K(Q, P) = P0.

(b) Show that all
of the coordinates and momenta of the Q, P system are constants of the motion,
except for Q0 = q0 = t.

(c) Use the
inverse relation q1(Q, P) derived from the generating function to give a
general solution for q1 as a function of time and suitable contants to be
determined at time zero.