 Solution

The required Hamilton-Jacobi equation isBecause $x,y$ and time are cyclic in the Hamiltonian, use separation $S=ax+by-Et+W(z)$. The remaining equation isFor $g\ne 0$, each sign branch has the separated complete integralIt is real in the classically allowed region $Q\ge 0$, and $p_z=s\sqrt{Q}$. At a turning point one matches the ascending and descending branches. If $g=0$, the integral is simply $W=p_{z}z$ with constant $p_z$.

A convenient equivalent complete integral, expressed using initial-momentum constants $\mathbf{\alpha }$, is globally polynomial in time:Substitution verifies that its time derivative cancels the kinetic term and $mgz$. It avoids a square-root branch change when extracting the complete trajectory.