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Preserve the primary constraints using the corrected Hamiltonian from solution. After a spatial integration by parts, the $B_0$ terms in its density are $B_0{\partial }_i{\pi }_i+B_0^{\ast }{\partial }_i{\pi }_i^{\ast }-M^2{B}_0^{\ast }{B}_0$. Hamilton's equations giveWith the definitions of $E^i,E^{\ast i}$ in the paper, the actual massive Gauss law is thereforeThis is also the $\nu =0$ component of the Proca equation. The zero right-hand sides printed in the source apply only to the massless theory or to special configurations with $B_0=0$.

For a concrete counterexample, take $k\ne 0$, ${\omega }^2=k^2+M^2$, and the longitudinal on-shell plane waveIt satisfies ${\partial }_{\mu }{B}^{\mu }=0$, but $E^3=-iM{e}^{-i\omega t+ikz}$ and ${\partial }_i{E}^i=kM{e}^{-i\omega t+ikz}\ne 0$. Imposing the printed zero-divergence condition would incorrectly remove the physical longitudinal particle polarization.

A phase-space action treats canonical variables as independent fields. Varying them gives Hamilton's equations; Lagrange multipliers can impose constraints.

A Lagrangian is regular when its velocity Hessian matrix is invertible. The inverse function theorem then makes the fiber derivative $p_i=L_{v^i}$ locally invertible, allowing a local Legendre transform in mechanics to a Hamiltonian. If that fiber derivative is globally invertible, the Lagrangian is called a hyperregular Lagrangian. Regularity alone does not assert global invertibility, and singular Lagrangians may give constrained rather than ordinary unconstrained Hamilton's equations.

The type-two generating function for a canonical transformation $F_2(q,P)=\cos q\sin P$ defines $p=-\sin q\sin P$ and $Q=\cos q\cos P$. On any branch with $0<q<\pi$ and $\cos P\ne 0$, its inverse in mixed coordinates isThe square root equals $\sin q$ on this branch. Differentiating $F_2$ proves the generating equations, and $F_{2,qP}=-\sin q\cos P\ne 0$ proves local invertibility by the inverse function theorem. Also $d(F_2-PQ)=pdq-PdQ$, so the symplectic form obeys $dq\wedge dp=dQ\wedge dP$. This proves the transformation is canonical locally. It is not a single global chart across the inverse-trigonometric branch boundaries.

Since $F_2$ is independent of time, the transformed Hamiltonian is obtained by substitution, with zero generating-function time correction. An added function of time alone is an irrelevant Hamiltonian freedom, since it leaves Hamilton's equations unchanged.