 Solution

Apply Jacobi's reconstruction to the polynomial complete integral in solution. Its parameter derivatives are constant:ThereforeAt $t=0$ the constants $\mathbf{\beta }$ are the initial positions and $\mathbf{\alpha }$ are the initial momenta. Direct differentiation of $S$ gives $p_x={\alpha }_x$, $p_y={\alpha }_y$, $p_z={\alpha }_z-mgt$, confirming the same motion. The six constants provide arbitrary initial data; the solution covers both ascent and descent, not merely one local square-root branch of the separated action.