1. a) Obtain Hamilton's principal function $S(z,t) = W(z,t) - Et$, for a particle of mass
m and total energy $E$ which moves vertically in the uniform gravitational field $g$ near the
surface of the earth, by integrating the time-independent Hamilton-Jacobi equation.
b) Use $S(z, t)$ to generate the general solution to the dynamical problem: $(z(t), p(t))$